Tom Robbins WW Prob Lib2 Summer 2001

Transcription

1 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment LogEp due 4//05 at :00 AM..( pt) Evaluate the following epressions. (a) log 7 7 (b) log 3 7 (c) log 4 04 (d) log ( pt) Evaluate the following epressions. (a) log 3 43 (b) log (c) log 4 64 (d) 4 log ( pt) Evaluate the following epressions. log log log 65 5 log ( pt) Evaluate the following epressions. (a) lne 5 (b) e ln4 (c) e ln 4 (d) ln e 5.( pt) (a) If log, then. (b) If log 5 3, then. 6.( pt) (a) If log 6, then. (b) If log 9, then. 7.( pt) (a) If 3 3, then. (b) If 5, then. 8.( pt) Solve the given equation for ( pt) Solve the given equation for ( pt) If ln 6 4, then..( pt) If e 7 4, then..( pt) Match the statements defined below with the letters labeling their equivalent epressions.. ln y. ln y 3. ln y 4. ln y A. ln lny B. yln C. ln lny D. ln y 3.( pt) Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. You must get all of the answers correct to receive credit.. log a b log b a. lna b blna 3. ln y ln lny 4. log ab log a log b 4.( pt) is equal to ln r 3 s 4 4 r 6 s 7 Alnr Blns where A and where B 5.( pt) Evaluate the following epressions. (a) e ln5 (b) 0 4log 0 5 (c) log 8 4 [NOTE: Your answers cannot be algebraic epressions. ] 6.( pt) If ln ln 4 ln3, then. 7.( pt) Solve the given equation for. log 0 log 0 8.( pt) Solve the given equation for ( pt) Solve the given equation for ( pt) Rewrite the epression in terms of ln log 4 5.( pt) Rewrite the epression in terms of ln log 6 9.( pt) The equation 8 0 has two roots. The smaller root is and the bigger root is 3.( pt) The equation 7 e e 9 0 has two roots. The smaller root is and the bigger root is. 4.( pt) The equation e e 30 0 has two solutions. The smaller one is: and the larger one is:. 5.( pt) If e e 6, then. 6.( pt) For each of the following, find the base b if the graph of y b contains the given point. 3 5 b b 3 8 b b 0 b

2 0 04 b 3 b 3 8 b 0 5 b 3 7 b 7.( pt) Determine the smallest integer that satisfies the given inequality ( pt) Starting with the graph of f 7, write the equation of the graph that results from (a) shifting f units downward. y (b) shifting f 6 units to the right. y (c) reflecting f about the -ais and the y-ais. y (d) reflecting f about the line y = -. y

3 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment LogEpApplications due 4//05 at :00 AM..( pt) A bacteria culture initially contains 000 bacteria and doubles every half hour. Find the size of the baterial population after 0 minutes. Find the size of the baterial population after 6 hours..( pt) A certain bacteria population is known to doubles every 90 minutes. Suppose that there are initially 00 bacteria. What is the size of the population after t hours? 3.( pt) If a bateria culture starts with 9000 bateria and doubles every 30 minutes, how many minutes will it take the population to reach 8000? 4.( pt) The count in a bateria culture was 00 after 0 minutes and 700 after 35 minutes. What was the initial size of the culture? Find the doubling period. Find the population after 85 minutes. When will the population reach ( pt) The doubling period of a baterial population is 0 minutes. At time t 0 minutes, the baterial population was What was the initial population at time t 0? Find the size of the baterial population after 5 hours. 6.( pt) A bacteria culture initially contains 000 bacteria and doubles every half hour. The formula for the population is p t 000e kt for some constant k. (You will need to find k to answer the following.) Find the size of the baterial population after 00 minutes. Find the size of the baterial population after 5 hours. 7.( pt) The doubling period of a baterial population is 5 minutes. At time t 80 minutes, the baterial population was For some constant A, the formula for the population is p t Ae kt where k ln 5. What was the initial population at time t 0? Find the size of the baterial population after 3 hours. 8.( pt) The rat population in a major metropolitan city is given by the formula n t 56e 0 035t where t is measured in years since 99 and n t is measured in millions. What was the rat population in 99? What is the rat population going to be in the year 003? 9.( pt) The half-life of Radium-6 is 590 years. If a sample contains 500 mg, how many mg will remain after 000 years? 0.( pt) The half-life of Palladium-00 is 4 days. After 6 days a sample of Palladium-00 has been reduced to a mass of 4 mg. What was the initial mass (in mg) of the sample? What is the mass 4 weeks after the start?.( pt) If 7000 dollars is invested in a bank account at an interest rate of 4 per cent per year, find the amount in the bank after 5 years if interest is compounded annually Find the amount in the bank after 5 years if interest is compounded quaterly Find the amount in the bank after 5 years if interest is compounded monthly Finally, find the amount in the bank after 5 years if interest is compounded continuously.( pt) The 906 San Francisco earthquake had a magnitude of 8.3 on the Richter scale. At the same time in South America there was an eathquake with magnitude 4. that caused only minor damage. How many times more intense was the San Francisco earthquake than the South American one? 3.( pt) The ph scale for acidity is defined by ph log 0 H where H is the concentration of hydrogen ions measured in moles per liter (M). A substance has a hydrogen ion concentration of H M. Calculate the ph of the substance.

4 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment PolarCoordPoints due 5//05 at :00 AM..( pt) Decide if the points given in polar coordinates are the same. If so, enter T. If not, enter F. π π 3 43π 43π 4 4 5π 0 7π π π 4 4 3π π 3 7 5π 7 5π.( pt) For each set of Cartesian coordinates y, match the equivalent set of Polar coordinates r θ, with π θ π A B C D ( pt) For each set of Polar coordinates, match the equivalent Cartesian coordinates A. 7 5 B. 4 4 C D E. 9 6 F

5 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment PolarCoordCurves due 5//05 at :00 AM..( pt) A curve in polar coordinates is given by: r 8 5cosθ Point P is at θ π 6 Find polar coordinate r for P, with r 3π 0 and π θ r Find cartesian coordinates for point P, y 3 How may times does the curve pass through the origin when 0 θ π?.( pt) A curve with polar equation 6 r 8sinθ 4cosθ represents a line. This line has a Cartesian equation of the form y m b,where m and b are constants. Give the formula for y in terms of. For eample, if the line had equation y 3 then the answer would be 3. 3.( pt) A circle C has center at the origin and radius 7. Another circle K has a diameter with one end at the origin and the other end at the point 0 7 The circles C and K intersect in two points. Let P be the point of intersection of C and K which lies in the first quadrant. Let r θ be the polar coordinates of P, chosen so that r is positive and 0 θ Find r and θ. r θ 4.( pt) Match each polar equation below to the best description. Possible answers are C,E,H,L,P,R,S,V,and Z. DESCRIPTIONS C. Circle centered at origin, E. Ellipse, H. Hyperbola, L. Line neither vertical nor horizontal, P. Parabola, R. Circle not centered at origin, S. Spiral, V. Vertical Line, Z. Horizontal Line POLAR EQUATIONS. r 5. r sinθ Hint: sinθ sinθcosθ 3. r 5sinθ 8cosθ 4. r 8cosθ 5. r 8sinθ 5.( pt) Match each polar equation below to the best description. Possible answers are C,E,F,H,L,P,and S. DESCRIPTIONS C. Circle, E. Ellipse, F. Figure eight, H. Hyperbola, L. Line, P. Parabola, S. Spiral Hint: it may help to change back to rectangular coordinates. POLAR EQUATIONS. r 8 3cosθ. r 8sinθ 3cosθ 3. r 3 8cosθ 4. r 8 8cosθ 5. r 8sinθ 3cosθ 6.( pt) Match each polar equation below to the best description. Each answer should be C,F,I,L,M,O,or T. DESCRIPTIONS C. Cardioid, F. Rose with four petals, I. Inwardly spiraling spiral, L. Lemacon, M. Lemniscate, O. Outwardly spiraling spiral, T. Rose with three petals POLAR EQUATIONS. r 8cosθ. r 4cos3θ 3. r 3sinθ 4. r 4θ r 0 5. r 4 θ r 0 6. r 4 8cosθ 7. r 4 4sinθ 7.( pt) Match each polar equation below to the best description. Each answer should be C,E,F,H,L,O,P,R,S,T,or W. DESCRIPTIONS C. Cardioid, E. Ellipse, F. Lemniscate, H. Hyperbola, L. Line, O. Oval, P. Parabola, R. Rose with four petals, S. Spiral, T. Three-petaled rose, POLAR EQUATIONS. r θ r 0. tanθ 3. r cscθ 4. r 9sinθ 5. r sinθ 6. r 8 8sinθ 7. r sin3θ 8.( pt) Match each polar equation below to the best description. Each answer should be C,F,I,L,M,O,or T. DESCRIPTIONS C. Cardioid, F. Rose with four petals, I. Inwardly spiraling spiral, L. Lemacon, M. Lemniscate, O. Outwardly spiraling spiral, T. Rose with three petals POLAR EQUATIONS. r 6cosθ. r 8 8sinθ 3. r cosθ 4. r sinθ 5. r 8θ r 0 6. r 8cos3θ

6 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment LimitsRates0Theory due //06 at :00 AM..( pt) Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several eamples, especially eamples which might show that the statement is false!. For reference you can find some definitions here. You must get all of the answers correct to receive credit.. The sequence, -,, -,, -,... does not have a convergent subsequence.. The sequence of rational numbers 3., 3.4, 3.4, 3.459,... which approimates the ratio of the circumference of a circle and its diameter, has a rational number as its it point. 3. The sequence of rational numbers 3., 3.4, 3.4, 3.459,... which approimates the ratio of the circumference of a circle and its diameter, has a it point but it is not a rational number. 4. The sequence,, 3, 4,... has a finite accumulation point..( pt) Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several eamples, especially eamples which might show that the statement is false! For reference you can find some definitions here. You must get all of the answers correct to receive credit.. Every differentiable function on the interval must have both a maimum and a minimum.. Every continuous function on the interval 3 6 must have a maimum. 3. Every function on the interval 3 must have both a maimum and a minimum. 4. Every continuous function on the interval 3 must have both a maimum and a minimum. 3.( pt) Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several eamples, especially eamples which might show that the statement is false! For reference you can find some definitions here. You must get all of the answers correct to receive credit.. If the linear approimation of a differentiable function is increasing at a point a then the function is also increasing near the point a.. Every differentiable function whose domain is a bounded, closed interval has a maimum value. 4.( pt) Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several eamples, especially eamples which might show that the statement is false! For reference you can find some definitions here. You must get all of the answers correct to receive credit.. Every continuous function whose domain is a bounded, closed interval has a maimum value.. Every continuous function is differentiable. 3. Every continuous function has a maimum value. 4. If the linear approimation of a differentiable function is constant at a point a then the function could be increasing near the point a. 5.( pt) Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several eamples, especially eamples which might show that the statement is false! For reference you can find some definitions here. You must get all of the answers correct to receive credit.. If a function is increasing near a point a then its linear approimation at a cannot be decreasing.. If the linear approimation of a differentiable function is constant at a point a then the function could be increasing near the point a. 3. Every continuous function whose domain is a bounded, closed interval has a maimum value. 4. If a differentiable function has a maimum value then it also has a minimum value. 5. If a continuous function has a maimum value then its domain must be a bounded, closed interval. 6. Every continuous function is differentiable. 7. Every differentiable function is continuous. 8. Every continuous function whose domain is a bounded, closed interval and which has a maimum value also has a minimum value.

7 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment LimitsRatesTangentVelocity due //06 at :00 PM..( pt) If the tangent line to y f at (-5, 4) passes through the point (-, 5), find A. f 5 B. f 5.( pt) The point P 5 36 lies on the curve y 6. If Q is the point 6, find the slope of the secant line PQ for the following values of. If 5, the slope of PQ is: and if 5 0, the slope of PQ is: and if 4 9, the slope of PQ is: and if 4 99, the slope of PQ is: Based on the above results, guess the slope of the tangent line to the curve at P ( pt) The point P 5 9 lies on the curve y 4. If Q is the point 4, find the slope of the secant line PQ for the following values of. If 5, the slope of PQ is: and if 5 0, the slope of PQ is: and if 4 9, the slope of PQ is: and if 4 99, the slope of PQ is: Based on the above results, guess the slope of the tangent line to the curve at P ( pt) The point P 0 0 lies on the curve y. If Q is the point, find the slope of the secant line PQ for the following values of. If 0 3, the slope of PQ is: and if 0, the slope of PQ is: and if 0, the slope of PQ is: and if 0 9, the slope of PQ is: Based on the above results, guess the slope of the tangent line to the curve at P ( pt) If a ball is thrown straight up into the air with an initial velocity of 65 ft/s, it height in feet after t second is given by y 65t 6t. Find the average velocity for the time period begining when t and lasting (i) 0 5 seconds (ii) 0 seconds (iii) 0 0 seconds Finally based on the above results, guess what the instantaneous velocity of the ball is when t. 6.( pt) A ball is thrown into the air by a baby alien on a planet in the system of Alpha Centauri with a velocity of 33 ft/s. Its height in feet after t seconds is given by y 33t 3t. A. Find the average velocity for the time period beginning when t=3 and lasting.0 s:.005 s:.00 s:.00 s: NOTE: For the above answers, you may have to enter 6 or 7 significant digits if you are using a calculator. B. Estimate the instanteneous velocity when t=3. 7.( pt) The eperimental data in the table below define y as a function of y A. Let P be the point (3, 0.9). Find the slopes of the secant lines PQ when Q is the point of the graph with coordinate slope B. Draw the graph of the function for yourself and estimate the slope of the tangent line at P. 8.( pt) Below is an oracle function. An oracle function is a function presented interactively. When you type in an t value, and press the f button and the value f t appears in the right hand window. There are three lines, so you can easily calculate three different values of the function at one time. The function f(t) represents the height in feet of a ball thrown into the air, t seconds after it has been thrown. Calculate the velocity 0.9 seconds after the ball has been thrown. The velocity at 0 9 = You can use a calculator t Enter t Enter t Enter t f(t) result: f t result: f t result: f t Remember this technique for finding velocities. Later we will use the same method to find the derivative of functions such as f t. 9.( pt) The position of a cat running from a dog down a dark alley is given by the values of the table. t(seconds) s(feet) A. Find the average velocity of the cat (ft/sec) for the time period beginning when t= and lasting a) 3 s b) s c) s B. Draw the graph of the function for yourself and estimate the instantaneous velocity of the cat (ft/sec) when t=

8 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment LimitsRates 5Graphs due //06 at 3:00 PM..( pt) Let F be the function below. If you are having a hard time seeing the picture clearly, click on the picture. It will epand to a larger picture on its own page so that you can inspect it more clearly. hand window. There are three lines, so you can easily calculate three different values of the function at one time. Determine the its for the function f at = 4 86 f 4 86 = = 4 86 Are all of these values the same?: (Y or N). If so then the function is continuous at 4 86 Are the left and right its the same at 4 86?: (Y or N). If so then this function is almost continuous and could be made continuous by redefining one value of the function namely f Enter Enter Enter f() result: f result: f result: f 3.( pt) Evaluate each of the following epressions. Note: Enter DNE if the it does not eist or is not defined. a) F = b) F = c) F = d) F = e) F = f) F = g) F = h) 3 F = i) F 3 =.( pt) Below is an oracle function. An oracle function is a function presented interactively. When you type in an value, and press the f button and the value f appears in the right f() g() The graphs of f and g are given above. Use them to evaluate each quantity below. Write DNE if the it or value does not eist (or if it s infinity).. f g. f g 3. f g 4. f g

9 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment LimitsRatesLimits due //06 at :00 AM..( pt) The slope of the tangent line to the graph of the function y 5 3 at the point 3 35 is By trying values of near 3, find the slope of the tangent line..( pt) Evaluate the it ( pt) Evaluate the it 4.( pt) Evaluate the it ( pt) Evaluate the it ( pt) Evaluate the it 3 y y 4y y 3 7.( pt) Evaluate the it 8.( pt) Evaluate the it ( pt) Evaluate the it 4.( pt) Evaluate the it s 6 s 6 s 6 b b b 5.( pt) Let f 6 if 3 and f 6 if 3. Sketch the graph of this function for yourself and find following its if they eist (if not, enter N). f f 3. f 3 6.( pt) Let f 4 if 7, f 8 if 7, f 0 if 7 7, f 7 if 7. Sketch the graph of this function and find following its if they eist (if not, enter DNE).. f 7. f 7 3. f 7 4. f f 6. 7 f 9.( pt) Evaluate the it ( pt) Evaluate the it y 3 y y y.( pt) Evaluate the it 3.( pt) Evaluate the it a a 8 a 7.( pt) Determine the its for the function f at 3 5.

10 f = 3 5 f 3 5 = f = 3 5 Is this function continuous at 3 5?: (Y or N) Can this function be made continuous by changing its value at 3 5?: (Y or N) 8.( pt) Let h 0, g, f 5. a a a Find following its if they eist. If not, enter DNE ( does not eist ) as your answer.. h g a. h g a 3. h f a 4. a 5. a 6. a 7. a a 9.( pt) h g h f f h g g a g f a a f DNE f DNE a f a a g DNE g DNE a g a Using the table above calcuate the its below. Enter DNE if the it doesn t eist OR if it can t be determined from the information given.. f g 3. f 3 g 3 3. f 3 g 3 4. f 3 g 3 0.( pt) Evaluate 5 5 Enter the letters corresponding to the Limit Laws that you used to find this it: Limit Laws A. Quotient Law B. Root Law C. Power Law D. Product Law E. Difference Law F. Sum Law G. Constant Multiple Law.( pt) If 4 7 f determine 3 f = What theorem did you use to arrive at your answer?.( pt) Use factoring to calculate this it 5 t 5 t 3 t 3 If you want a hint, try doing this numerically for a couple of values of and t. 3.( pt) Enter the integer which is the apparent it of the following sequences or enter N if the sequence does not appear to have a it sin(.8),.99505sin(.99505sin(.8)),.99505sin(.99505sin(.99505sin(.8))),.... the sequence generated by f h where h is a sequence of negative numbers approaching zero and f 9 if is greater than or equal to 0 and f 9 if is less than zero , 3.98,4.3, 4.034, 3.99, 4.00, the sequence generated by f h where h is a sequence of positive numbers approaching zero and f if is greater than or equal to 0 and f if is less than zero. 4.( pt) What is the it of the sequence f k generated by the sequence k when f 43 8? ( pt) Find an integer which is the it of sin as goes to 0. (Enter I for infinity or DNE for does not eist.) You should also try using identities to transform the epressions algebraically so that you can identify the its without using a calculator.

11 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment LimitsRates3Infinite due /3/06 at :00 AM..( pt) Evaluate the it.( pt) Evaluate the it ( pt) Evaluate the it ( pt) Find the horizontal it(s) of the following function: 6 f and 5.( pt) Evaluate the it 6.( pt) Evaluate the it 7.( pt) Evaluate 8.( pt) Evaluate t 8t 9 t 0t 5 9.( pt) The horizontal asymptotes of the curve 3 y 4 4 are given by y = y = where y y. The vertical asymptote of the curve is given by = 0.( pt) Evaluate y and.( pt) Determine the infinite it of the following functions. Enter INF for and MINF for ( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) (b) ( pt) Evaluate the following its. (a) (b) 8 e 8 e [NOTE: If needed, enter INF for and MINF for.] [HINT: Look at where the eponential fuction is going in the fraction. If you need a reminder, look up infinite its in Section.5 (in particular, see pg 38-39).] 4.( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) (b) ( pt) Evaluate the following its. If needed, enter INF for and MINF for.

12 (a) (b) ( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) (b) ( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) (b) ( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) (b) ( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) (b) ( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) 0 (b) 0.( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) (b)

13 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment LimitsRates5Continuity due /5/06 at :00 AM..( pt) For what value of the constant c is the function f continuous on where f s cs 5 if s 9 cs 5 if s 9.( pt) For what value of the constant c is the function f continuous on where f c if 9 c 5 if 9 3.( pt) Enter a letter and a number for each formula below so as to define a continuous function. The letter refers to the list of equations and the number is the value of the function f at. Letter Number A. when B. cos 4π π when C. 3 7 when D. sin π when 6 5 when cos π when when when 4.( pt) Enter a letter and a number for each formula below so as to define a continuous function. The letter refers to the list of equations and the number is the value of the function f at. Letter Number 3 when sin when 3 A. 4cos π when B. 4 3 when C. D. sin π cos when when 5.( pt) The function f is given by the formula when 0 f and by the formula f 5 a when when 6 5 when 5. What value must be chosen for a in order to make this function continuous at 5? a =

14 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment LimitsRates6Rates due /6/06 at :00 AM..( pt) The slope of the tangent line to the parabola y at the point 3 is: The equation of this tangent line can be written in the form y m b where m is: and where b is:.( pt) The slope of the tangent line to the curve y 3 3 at the point 3 8 is: The equation of this tangent line can be written in the form y m b where m is: and where b is: 3.( pt) The slope of the tangent line to the curve y 4 at the point is: The equation of this tangent line can be written in the form y m b where m is: and where b is: 4.( pt) The slope of the tangent line to the curve y at the point is: The equation of this tangent line can be written in the form y m b where m is: and where b is: 5.( pt) The slope of the tangent line to the parabola y at the point where 3 is: The equation of this tangent line can be written in the form y m b where m is: and where b is: 6.( pt) If a rock is thrown into the air on small planet with a velocity of m/s, its height (in meters) after t seconds is given by y t 4 9t. Find the velocity of the rock when t 3. 7.( pt) If an arrow is shot straight upward on the moon with a velocity of 5 m/s, its height (in meters) after t seconds is given by s t 5t 0 83t. What is the velocity of the arrow (in m/s) after 0 seconds? After how many seconds will the arrow hit the moon? With what velocity (in m/s) will the arrow hit the moon? 8.( pt) The displacement (in meters) of a particle moving in a straight line is given by s 4t 3 where t is measured in seconds. Find the average velocity of the particle over the time interval 9. Find the (instantaneous) velocity of the particle when t 9. 9.( pt) Let p Use a calculator or a graphing program to find the slope of the tangent line to the point p when. Give the answer to 3 places.

15 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives0Theory due /3/06 at :00 AM..( pt) Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A statement is true only if it is true for all possibilities. You must get all of the answers correct to receive credit If f 0 and g 7, then f g does not eist 3. If p is a polynomial, then then the it p is 6 p 6 4. If f g f 0 and g 0, then 6 6 does not eist 5. If 6 f and g, then f g 0

16 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives due //06 at :00 AM..( pt) If f 6, find f 7..( pt) If f 0 6, find f 5. 3.( pt) If f 7 5, find f. 4.( pt) If f, find f. 5.( pt) Let f 6.( pt) Let f 4 f 3 Use the it definition of the derivative on page 56 to find (i) f 7 (ii) f 5 (iii) f (iv) f To avoid calculating four separate its, I suggest that you evaluate the derivative at the point when a. Once you have the derivative, you can just plug in those four values for a to get the answers. 7.( pt) For each of the given functions f, find the derivative f c at the given point c, using Theorem, first finding a f c. f ;c a = f c = f 8 on the interval 4 ;c 7 a = f c = 8.( pt) The position of a cat running from a dog down a dark alley is given by the values of the table. t(seconds) s(feet) A. Find the average velocity tor the time period beginning when t= and lasting. 3 s. s 3. s B. Draw the graph of the function for yourself and estimate the instantaneous velocity when t= 9.( pt) This problem tests calculating new functions from old ones: From the table below calculate the quantities asked for: f g f g f g 4 f g 4 f g 4 f f 0.( pt) Constructing new functions from old ones and calculating the derivative of the new function from the derivatives of the old functions: From the table below calculate the quantities asked for: 6 6 f 3 47 g f g

17 f g f g f g f g.( pt) This problem tests calculating new functions from old ones: From the table below calculate the quantities asked for: f g f g If h g f, calculate h 3. f g f f 3 f g If h f f, calculate h.( pt) This problem tests calculating new functions from old ones: From the table below calculate the quantities asked for: f g f g f g 4 f g 4 f 4 g 4 5 If h g f, calculate h. If h f f, calculate h 4 Given the following table: 3.( pt) f() Calculate the value of f 0 00 = to two places of accuracy. To obtain more precise information about the value of f near 0 00 enter a new increment value for here rulein.0in and then press the Submit Answer button. How will you tell when your increment is small enough to give you a good answer for the problem?

18 4.( pt) Identify the graphs A (blue), B( red) and C (green) as the graphs of a function and its derivatives: is the graph of the function is the graph of the function s first derivative is the graph of the function s second derivative 5.( pt) Identify the graphs A (blue), B( red) and C (green) as the graphs of a function and its derivatives: is the graph of the function is the graph of the function s first derivative is the graph of the function s second derivative 3 6.( pt) Identify the graphs A (blue), B( red) and C (green) as the graphs of a function and its derivatives: is the graph of the function is the graph of the function s first derivative is the graph of the function s second derivative 7.( pt) Let f Use the it definition of the derivative on page 63 to calculate the derivative of f : f. Use the same formula from above to calculate the derivative of this new function (i.e. the second derivative of f ): f. 8.( pt) The oracle function f is presented below. For each value you enter the oracle will tell you the value f. Calculate the derivative of the function at 0 9 using the Newton quotient definition. f at 0 9 = You can use a calculator Enter Enter Enter f() result: f result: f result: f Remember the technique for finding instantaneous velocities from average velocities? This is the same thing. 9.( pt) Below is an oracle function. An oracle function is a function presented interactively. When you type in a t value, and press the f button the value f t appears in the right hand window. There are three lines, so you can calculate three different values of the function at one time. The function f(t) represents the height in feet of a ball thrown into the air, t seconds after it has been thrown. Calculate the average velocity 0.86 seconds after the ball has been thrown. Average velocity at 0 86 = You can use a calculator The java Script calculator was displayed here Remember this technique for finding velocities. Later we will use the same method to find the derivative of functions such as f t.

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20 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives 5Tangents due //06 at 5:00 AM..( pt) If f, find f 5. Use this to find the equation of the tangent line to the parabola y at the point 5 4. The equation of this tangent line can be written in the form y m b where m is: and where b is:.( pt) If h 3 3, find h. Use this to find the equation of the tangent line to the curve y 3 3 at the point 3. The equation of this tangent line can be written in the form y m b where m is: and where b is: 3.( pt) If f 3, find f. Use this to find the equation of the tangent line to the hyperbola y 3 at the point The equation of this tangent line can be written in the form y m b where m is: and where b is: 4.( pt) If f, find f 4. Use this to find the equation of the tangent line to the curve y at the point The equation of this tangent line can be written in the form y m b where m is: and where b is: 5.( pt) If f 5 3, find f 5. Use this to find the equation of the tangent line to the curve y 5 3 at the point The equation of this tangent line can be written in the form y m b where m is: and where b is: 6.( pt) If f 3 3, find f 5. Use this to find the equation of the tangent line to the curve y 3 3 at the point The equation of this tangent line can be written in the form y m b where m is: and where b is: 7.( pt) If find f 4. f Use this to find the equation of the tangent line to the curve y at the point The equation of this tangent line can be written in the form y m b where m is: and where b is: 8.( pt) The parabola y 6 has two tangents which pass through the point 0 4. One is tangent to the to the parabola at A A 6 and the other at A A 6. Find (the positive number) A. 9.( pt) On a separate piece of paper, sketch the graph of the parabola y 3. On the same graph, plot the point 0 4. Note that there are two tangent lines of y 3 that pass through the point 0 4. Specifically, the tangent line of the parabola y 3 at the point a a 3 passes through the point 0 4 where a 0. The other tangent line that passes through the point 0 4 occurs at the point a a 3. Find the number a. 0.( pt) The graph of f has two horizontal tangents. One occurs at a negative value of and the other at a positive value of. What is the negative value of where a horizontal tangent occurs? What is the positive value of where a horizontal tangent occurs?.( pt) For what values of does the graph of f have a horizontal tangent? Enter the values in order, smallest first, to 4 places of accuracy:.( pt) For what values of does the graph of f have a horizontal tangent? Enter the values in order, smallest first, to 4 places of accuracy: 3.( pt) For what values of is the tangent line of the graph of f parallel to the line y 4? Enter the values in order, smallest first, to 4 places of accuracy: 4.( pt) For what values of is the tangent line of the graph of f parallel to the line y 3? Enter the values in order, smallest first, to 4 places of accuracy: 5.( pt) Given f Calculate the tangent line at the point 64 7 y 64 7 For similar problems see p0: ( pt) At what point does the normal to y at 4 intersect the parabola a second time? (, ) The normal line is perpendicular to the tangent line. If two lines are perpendicular their slopes are negative reciprocals i.e.

21 if the slope of the first line is m then the slope of the second line is m 7.( pt) For what values of a and b is the line 4 y b tangent to the curve y a when 4? a b

22 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment DerivativesFormulas due //06 at :00 AM..( pt) If f 7 9 3, find f. Find f 4..( pt) If f 5 3, find f. 3.( pt) If f , find f. Find f. 4.( pt) If f , find f. Find f 4. 5.( pt) If f 5 7 5, find f..( pt) Let f f 3.( pt) If find f. Find f 3. 4.( pt) If find f. Find f. f f 4 6.( pt) If f t t 6t 4 4t 5, find f t. Find f. 7.( pt) Let f t t 7t t 4. (a) f t (b) f [NOTE: Your answer to part (a) should be a function in terms of the variable t and not a number! Your answer to part (b) should be a number.] 8.( pt) If f t t 7, find f t. Find f. 9.( pt) If find f t. Find f. f t t 3 0.( pt) If f t 6 t 4, find f t. [NOTE: Your answer should be a function in terms of the variable t and not a number! ].( pt) If f 5 4 find f. Find f. 5.( pt) If f 6 3, find f 6. 6.( pt) If find f. Find f 5. 7.( pt) Let f 3 3. f 4 f ( pt) If f, find f. Find f. 9.( pt) If f 3, find f. 0.( pt) If f 4 3 5, find f. Find f 5..( pt) If f 3 5 7, find f..( pt) Let f f [NOTE: Your answer should be a function in terms of the variable and not a number!] 3.( pt) If f 6 5, find f.

23 Find f 3. 4.( pt) If f 5, find f 9. 5.( pt) If find f. Find f 5. f ( pt) If f , find f. 7.( pt) If f , find f. 8.( pt) If f 5 3 4, find f. Find f. 9.( pt) If f , find f ( pt) Calculate G to 3 significant figures where G ( pt) Calculate f 4 to 3 significant figures where f t 4t 3t 3 8 Tip: You can enter an answer such as 3.4e- for ( pt) Find the y-intercept of the tangent line to y at ( pt) Let f 3e e. f 0 [NOTE: A small algebraic manipulation is needed first to get f() into a form so that the derivative can be taken.] 34.( pt) Given that f h h h 5 Calculate f. [HINT: Use the product rule and the power rule.] 35.( pt) Find the derivative of the function g g 5 5 e 36.( pt) Find the derivative of the function g 37.( pt) Given g e 5 f The derivative function is given by f

24 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives 5Implicit due //06 at :00 PM..( pt) If 4 5 y 3 and y 3 6, find y 3 by implicit differentiation..( pt) If y 9 and y 5 6, find y 5 by implicit differentiation. 3.( pt) If 6 and y , find y by implicit differentiation. 6 y 4.( pt) Find the slope of the tangent line to the curve at the point 7. 3 y 4y 3 39e 03 5.( pt) Use implicit differentiation to find the slope of the tangent line to the curve 3 y 4y 3 67 at the point 4. m 6.( pt) Find the slope of the tangent line to the curve y 3 y 0 0 at the point 0. 7.( pt) Use implicit differentiation to find the slope of the tangent line to the curve y 3 y 8 at the point 9. m 8.( pt) Use implicit differentiation to find the equation of the tangent line to the curve y 3 y at the point 6. The equation of this tangent line can be written in the form y m b where m is: and where b is: 9.( pt) Find the slope of the tangent line to the curve at the point 7. 4 y y ( pt) Find the slope of the tangent line to the curve (a lemniscate) y 5 y at the point 3. m.( pt) Find the equation of the tangent line to the curve (a lemniscate) y 5 y at the point 3. The equation of this tangent line can be written in the form y m b where m is: and where b is:.( pt) Use implicit differentiation to find the slope of the tangent line to the curve y 4y at the point 4. m 3.( pt) Find y by implicit differentiation. Match the epressions defining y implicitly with the letters labeling the epressions for y.. sin y 5ycos. cos y 5ycos 3. sin y 5ysin 4. cos y 5ysin A. cos y 5ysin cos y 5cos B. cos y 5ycos cos y 5sin C. sin y 5ysin 5cos sin y D. sin y 5ycos 5sin sin y 4.( pt) Find y by implicit differentiation. Match the epressions defining y implicitly with the letters labeling the epressions for y.. 3cosy 7siny 7siny. 3cosy 7cosy 7siny 3. 3siny 7cosy 7cosy 4. 3siny 7siny 7cosy 3cosy A. 3siny 4siny 7cosy 3cosy B. 3siny 4cosy 7cosy 3siny C. 4siny 3cosy 7siny 3siny D. - 3cosy 4cosy 7siny

25 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives3WordProblems due /3/06 at :00 AM..( pt) A particle moves along a straight line and its position at time t is given by s t t 3 t 60t where s is measured in feet and t in seconds. Find the velocity (in ft/sec) of the particle at time t 0: The particle stops moving (i.e. is in a rest) twice, once when t A and again when t B where A B. A is and B is What is the position of the particle at time 4? Finally, what is the TOTAL distance the particle travels between time 0 and time 4?.( pt) If a ball is thrown vertically upward from the roof of 64 foot building with a velocity of 80 ft/sec, its height after t seconds is s t 64 80t 6t. What is the maimum height the ball reaches? What is the velocity of the ball when it hits the ground (height 0)? 3.( pt) The area of a square with side s is A s s. What is the rate of change of the area of a square with respect to its side length when s 4? 4.( pt) The population of a slowly growing bacterial colony after t hours is given by p t 4t 33t 00. Find the growth rate after 3 hours. 5.( pt) The cost of producing units of stuffed alligator toys is c Find the marginal cost at the production level of 000 units. 6.( pt) A mass attached to a vertical spring has position function given by s t sin 4t where t is measured in seconds and s in inches. Find the velocity at time t. Find the acceleration at time t. 7.( pt) The mass of the part of a rod that lies between its left end and a point meters to the right is kg. The linear density of the rod at 5 meters is kg/meter and at 3 meters the density is kg/meter 8.( pt) If f is the focal length of a conve lens and an object is placed at a distance p from the lens, then its image will be at a distance q from the lens, where f, p, and q are related by the lens equation f p q What is the rate of change of p with respect to q if q 4 and f 8? (Make sure you have the correct sign for the rate.) 9.( pt) A particle moves along a straight line with equation of motion s t 4 t 3 Find the value of t (other than 0 ) at which the acceleration is equal to zero.

26 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives4Trig due /4/06 at :00 AM..( pt) Evaluate the it.( pt) Evaluate the it 3.( pt) Evaluate the it sin4 0 6 sin9 0 sin3 0 tan 3 4.( pt) If f cos 5tan, then f and f 5 5.( pt) Let f f π 3 f 8cos 8tan 6.( pt) If f 6sin cos, then f and f 3. 7.( pt) Let f sin cos f π f 3 [Note: When entering trigonometric functions into Webwork, you must include parentheses around the arguement. I.e. sin would not be accepted but sin() would.] 8.( pt) If 3sin f cos find f. Find f 4. 9.( pt) If f 5tan, find f. Find f 3. 0.( pt) If find f. f tan 3 sec Find f..( pt) Let f f π 6 f 5tan 4 sec.( pt) If f 3 sin cos, find f. Find f 5. 3.( pt) Let f f π 6 4.( pt) Let f π f f sin cos sin cos 5.( pt) If f 4sincos, find f. Find f 3. 6.( pt) Let f π f 4sincos 7.( pt) If f 3 tan sec, find f. Find f. 8.( pt) Find the equation of the tangent line to the curve y 6sin at the point π 6 3. The equation of this tangent line can be written in the form y m b where m and b 9.( pt) Find the equation of the tangent line to the curve y 3tan at the point π 4 3. The equation of this tangent line can be written in the form y m b where m is: and where b is: 0.( pt) Find the equation of the tangent line to the curve y 5sec 0cos at the point π 3 5. The equation of this tangent line can be written in the form y m b where m is: and where b is:

27 .( pt) Find the equation of the tangent line to the curve y cos at the point π π. The equation of this tangent line can be written in the form y m b where m and b.( pt) Match the functions and their derivatives:. y cos tan. y tan 3. y cos 3 4. y sin tan A. y sin tan sec B. y 3cos 3 tan C. y sin tan cos D. y tan 3.( pt) Find the 83rd derivative of sin at 5 by finding the first few derivatives and observing the pattern that occurs. 4.( pt) Let h t tan 4t 8. Then h is and h is

28 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives5ChainRule due /5/06 at :00 AM..( pt) If f 5 8, find f. Find f 3..( pt) Let f f 3 f ( pt) If f 4 4, find f. Find f 5. 4.( pt) If f 5 3, find f. Find f 3. 5.( pt) Let f f 3 f ( pt) If f sin 4, find f. Find f 3. 7.( pt) If f sin 3, find f. Find f 5. 8.( pt) Let f f 9sin 3 9.( pt) If f tan4, find f. Find f. 0.( pt) Let f f cos.( pt) If f cos 4 3, find f. Find f 3..( pt) Let f f 3sec 5 3.( pt) If f 3sec 5, find f. Find f. 4.( pt) Let f f 5cos cos 5.( pt) If f sin sin, find f. Find f 3. 6.( pt) Let F f 9 and G f 9. You also know that a 8 4 f a 3 f a f a 9 5. Find F a and G a. 7.( pt) Let F f f and G F. You also know that f 7 5 f 5 3 f 5 5 f 7 3. Find F 7 and G 7. 8.( pt) If d d f Calculate f 9.( pt) Let f 0.( pt) Let f.( pt) Let f f 6e cos f f 8cos cos 6.( pt) If f cos sin, find f. Find f 4. 3.( pt) Let f f cos e 4 cos

29 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives6InverseTrig due /6/06 at :00 AM..( pt) If f 6arcsin 3, find f..( pt) Let f f 7sin 4 3.( pt) If f 3arcsin, find f. Find f ( pt) If f 4 3 arctan 6 4, find f. 5.( pt) Let f 3 tan f NOTE: The WeBWorK system will accept arctan but not tan as the inverse of tan. 6.( pt) If f 7arctan 3, find f. Find f. 7.( pt) If f 8arctan 3e, find f. 8.( pt) Let f f tan 5 9.( pt) If f sin 4 arcsin, find f. 0.( pt) Let f 6cos sin f NOTE: The webwork system will accept arcsin and not sin as the inverse of sin..( pt) Let f tan cos 3 f.( pt) If f 5arctan 7sin 4, find f. 3.( pt) Let Then dy d = y tan 4

30 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives7Log due /7/06 at :00 AM..( pt) If f 4ln 3, find f. Find f..( pt) Let f f 5 3.( pt) Let f f e 4 4.( pt) Let f f e 4 f f 4ln 3 ln 3 3 f ln 5.( pt) If f ln, find f. Find f 5. 6.( pt) Let f f e f 3 3 ln 7.( pt) If f 4cos 5ln, find f. Find f 4. 8.( pt) If f 5ln 8 ln, find f. Find f. 9.( pt) If f 3log 9, find f. 0.( pt) Let f f 8.( pt) Let f f f 5log 4 7 log.( pt) Find the indicated derivatives. (a) d d e 4 log 4 π = (b) d d 4 ln = 3.( pt) Let f 4.( pt) Let f ln f Use logarithmic differentiation to determine the derivative. f f 4 5.( pt) If f , find f. 6.( pt) Let f 7.( pt) Let f ln f 6 Use logarithmic differentiation to determine the derivative. f f 8.( pt) If f 3 4, find f 3. 9.( pt) If f 3sin, find f 3. 0.( pt) If f sin, find f..( pt) If f 4 ln, find f 9..( pt) Let y log 4 Then dy d = Note. You must epress your answer in terms of natural logs, as Webwork doesn t understand how to evaluate logarithms to other bases. 3.( pt) Find dy d for each of the following functions 7 4 y ln 7 dy d = dy d = y cos

31 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives8RelatedRates due /8/06 at :00 AM..( pt) Let and let Find d dt when 4. y 5 dy dt.( pt) Let A be the area of a circle with radius r. If dr dt 4, find da dt when r. 3.( pt) A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0. cm/min. At what rate is the volume of the snowball decreasing when the diameter is 3 cm. (Note the answer is a positive number). 4.( pt) The altitude of a triangle is increasing at a rate of 000 centimeters/minute while the area of the triangle is increasing at a rate of square centimeters/minute. At what rate is the base of the triangle changing when the altitude is centimeters and the area is square centimeters? 5.( pt) The altitude of a triangle is increasing at a rate of 500 centimeters/minute while the area of the triangle is increasing at a rate of square centimeters/minute. At what rate is the base of the triangle changing when the altitude is centimeters and the area is square centimeters? 4 changing at 6 PM? (Note: knot is a speed of nautical mile per hour.) Note: Draw yourself a diagram which shows where the ships are at noon and where they are some time later on. You will need to use geometry to work out a formula which tells you how far apart the ships are at time t, and you will need to use distance = velocity * time to work out how far the ships have travelled after time t. 9.( pt) Gravel is being dumped from a conveyor belt at a rate of 0 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 4 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V 3 πr h. 0.( pt) Gravel is being dumped from a conveyor belt at a rate of 0 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V 3 πr h Note: The altitude is the height of the triangle in the formula Area=(/)*base*height. Draw yourself a general representative triangle and label the base one variable and the altitude (height) another variable. Note that to solve this problem you don t need to know how big nor what shape the triangle really is. 6.( pt) When air epands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV 4 C where C is a constant. Suppose that at a certain instant the volume is 450 cubic centimeters and the pressure is 95 kpa and is decreasing at a rate of 9 kpa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant? (Pa stands for Pascal it is equivalent to one Newton/(meter squared); kpa is a kilopascal or 000 Pascals. ) 7.( pt) At noon, ship A is 50 nautical miles due west of ship B. Ship A is sailing west at 5 knots and ship B is sailing north at 7 knots. How fast (in knots) is the distance between the ships changing at 6 PM? (Note: knot is a speed of nautical mile per hour.) 8.( pt) At noon, ship A is 0 nautical miles due west of ship B. Ship A is sailing west at 7 knots and ship B is sailing north at 9 knots. How fast (in knots) is the distance between the ships Note: See number on pg 58 for a picture of this..( pt) A street light is at the top of a 5 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 5 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 30 ft from the base of the pole?.( pt) A street light is at the top of a 6 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 30 ft from the base of the pole? Note: You should draw a picture of a right triangle with the vertical side representing the pole, and the other end of the hypotenuse representing the tip of the woman s shadow. Where does the woman fit into this picture? Label her position as a variable, and label the distance between her and the tip of her shadow as another variable. You might like to use similar triangles to find a relationship between these two variables. 3.( pt) A plane flying with a constant speed of 4 km/min passes over a ground radar station at an altitude of 3 km and cbs at an angle of 30 degrees. At what rate, in km/min is the distance from the plane to the radar station increasing minutes later?

32 4.( pt) Water is leaking out of an inverted conical tank at a rate of cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height meters and the diameter at the top is meters. If the water level is rising at a rate of centimeters per minute when the height of the water is 000 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute. 5.( pt) Water is leaking out of an inverted conical tank at a rate of cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 0 meters and the diameter at the top is 7 0 meters. If the water level is rising at a rate of 3 0 centimeters per minute when the height of the water is 0 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute. Note: Let R be the unknown rate at which water is being pumped in. Then you know that if V is volume of water, dv dt R Use geometry (similar triangles?) to find the relationship between the height of the water and the volume of the water at any given time. Recall that the volume of a cone with base radius r and height h is given by 3 πr h.

33 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives9Approimations due /9/06 at :00 AM..( pt) Let y 3. Find the change in y, y when 3 and 0 Find the differential dy when 3 and d 0.( pt) Let y. Find the change in y, y when 5 and 0 4 Find the differential dy when 5 and d ( pt) Let y 3 7. Find the differential dy when 4 and d 0 4 Find the differential dy when 4 and d ( pt) Let y tan 5 6. Find the differential dy when and d 0 4 Find the differential dy when and d ( pt) The linear approimation at 0 to 4 9 is A B where A is: and where B is: 6.( pt) The linear approimation at 0 to sin 9 is A B where A is: and where B is: 7.( pt) The linear approimation at 0 to 3 is A B where A is: and where B is: 8.( pt) Use linear approimation, i.e. the tangent line, to approimate 5 3 as follows: Let f. The equation of the tangent line to f at 5 can be written in the form y m b where m is: and where b is: Using this, we find our approimation for 5 3 is NOTE: For this part, give your answer to at least 9 significant figures or use fractions to give the eact answer. 9.( pt) Use linear approimation, i.e. the tangent line, to approimate 3 5 as follows: Let f 3. The equation of the tangent line to f at 5 can be written in the form y m b where m is: and where b is: Using this, we find our approimation for 3 5 is 0.( pt) Use linear approimation, i.e. the tangent line, to approimate 6 as follows: Let f and find the equation of the tangent line to f at 6. Using this, find your approimation for 6 Let f 7. The equation of the tangent line to f at 3 can be written in the form y m b where m is: and where b is: Using this, we find our approimation for 7 7 is 3.( pt) Use linear approimation, i.e. the tangent line, to approimate 0 0 as follows: Let f and find the equation of the tangent line to f at a nice point near 0 0. Then use this to approimate ( pt) Suppose that you can calculate the derivative of a function using the formula f f 5. If the output value of the function at is 3 estimate the value of the function at 0. 5.( pt) The circumference of a sphere was measured to be cm with a possible error of cm. Use linear approimation to estimate the maimum error in the calculated surface area. Estimate the relative error in the calculated surface area. 6.( pt) Use linear approimation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint cm thick to a hemispherical dome with a diameter of meters. 7.( pt) Let f t be the weight (in grams) of a solid sitting in a beaker of water. Suppose that the solid dissolves in such a way that the rate of change (in grams/minute) of the weight of the solid at any time t can be determined from the weight using the forumula: f t 5 f t 5 f t If there is 5 grams of solid at time t estimate the amount of solid second later. 8.( pt) Suppose you have a function f and all you know is that f 3 0 and the graph of its derivative is:.( pt) Use linear approimation, i.e. the tangent line, to approimate as follows: Let f 3. The equation of the tangent line to f at 4 can be written in the form y m b where m is: and where b is: Using this, we find our approimation for is.( pt) Use linear approimation, i.e. the tangent line, to approimate 7 7 as follows:

34 Click on the graph to see a bigger and better picture. Use linear approimation to estimate f 3 3 : Is your answer a little to big or a little too small?(enter TB or TS):

35 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives0MaMin due /0/06 at :00 AM..( pt) The function f has two critical numbers. The smaller one equals and the larger one equals.( pt) Consider the function f 7 5. The absolute maimum value is and this occurs at equals The absolute minimum value is and this occurs at equals 3.( pt) The function f 5 4 has an absolute maimum value of and this occurs at equals 4.( pt) Consider the function f 6 4. The absolute maimum of f is 5.( pt) The function f has one local minimum and one local maimum. This function has a local minimum at equals with value and a local maimum at equals with value 6.( pt) The function f has one local minimum and one local maimum. This function has a local minimum at equals with value and a local maimum at equals with value 7.( pt) The function f 3 9 has one local minimum and one local maimum. This function has a local minimum at equals with value and a local maimum at equals with value 8.( pt) Consider the function f f is increasing on the interval A and decreasing on the interval A where A is the critical number. Find A At A, does f have a local min, a local ma, or neither? Type in your answer as LMIN, LMAX, or NEITHER. 9.( pt) Consider the function f For this function there are four important intervals: A, A B, B C, and C where A, B, and C are the critical numbers. Find A and B and C At each critical number A, B, and C does f have a local min, a local ma, or neither? Type in your answer as LMIN, LMAX, or NEITHER. At A At B At C 0.( pt) A University of Rochester student decided to depart from Earth after his graduation to find work on Mars. Before building a shuttle, he conducted careful calculations. A model for the velocity of the shuttle, from liftoff at t = 0 s until the solid rocket boosters were jettisoned at t = 38.8 s, is given by v t t t 8 7t 8 (in feet per second). Using this model, estimate the absolute maimum value and absolute minimum value of the ACCELERATION of the shuttle between liftoff and the jettisoning of the boosters..( pt) Consider the function f The absolute maimum of f (on the given interval) is and the absolute minimum of f (on the given interval) is.( pt) Consider the function f This function has an absolute minimum value equal to and an absolute maimum value equal to 3.( pt) Consider the function f This function has an absolute minimum value equal to and an absolute maimum value equal to 4.( pt) The function f is decreasing on the interval (, ). It is increasing on the interval (, ) and the interval (, ). The function has a local maimum at. 5.( pt) The function f is increasing on the interval (, ). It is decreasing on the interval (, ) and the interval (, ). The function has a local maimum at. 6.( pt) For 4 3 the function f is defined by f 7 6 On which two intervals is the function increasing (enter intervals in ascending order)? to and to Find the region in which the function is positive: to Where does the function achieve its minimum? 7.( pt) For 4 4 the function f is defined by f On which two intervals is the function increasing?

36 to and to Find the region in which the function is positive: to Where does the function achieve its minimum? 8.( pt) Consider the function f For this function there are three important intervals: A, A B, and B where A and B are the critical numbers. Find A and B For each of the following intervals, tell whether f is increasing (type in INC) or decreasing (type in DEC). A : A B : B : 9.( pt) Consider the function f 9. For this function there are four important intervals: A, A B, B C, and C where A, and C are the critical numbers and the function is not defined at B. Find A and B and C For each of the following intervals, tell whether f is increasing (type in INC) or decreasing (type in DEC). A : A B : B C : C : 0.( pt) Consider the function f e 4. For this function there are three important intervals: A, A B, and B where A and B are the critical numbers. Find A and B For each of the following intervals, tell whether f is increasing (type in INC) or decreasing (type in DEC). A : A B : B.( pt) Answer the following questions for the function f defined on the interval 4 6. A. f is concave down on the region to B. f is concave up on the region to C. The inflection point for this function is at D. The minimum for this function occurs at E. The maimum for this function occurs at.( pt) Answer the following questions for the function f 5 5 defined on the interval 7 4. A. f is concave down on the region to B. f is concave up on the region to C. The inflection point for this function is at D. The minimum for this function occurs at E. The maimum for this function occurs at 3.( pt) Answer the following questions for the function f defined on the interval. A. f is concave down on the region to B. f is concave up on the region to C. The inflection point for this function is at D. The minimum for this function occurs at E. The maimum for this function occurs at 4.( pt) Answer the following questions for the function f 3 6 defined on the interval 8 0. Enter points, such as inflection points in ascending order, i.e. smallest values first. Enter intervals in ascending order also. A. The function f has vertical asympototes at and B. f is concave up on the region to and to C. The inflection points for this function are, and 5.( pt) Answer the following questions for the function f defined on the interval 6 6. Enter points, such as inflection points in ascending order, i.e. smallest values first. A. The function f has vertical asympototes at and B. f is concave down on the region to and to 6.( pt) Answer the following questions for the function f sin defined on the interval Enter points, such as inflection points in ascending order, i.e. smallest values first. Rememer that you can enter pi for π as part of your answer. A. f is concave down on the region to B. A global minimum for this function occurs at C. A local maimum for this function which is not a global maimum occurs at D. The function is increasing on to and on to. 4

37 7.( pt) Consider the function f f has inflection points at (reading from left to right) D, E, and F where D is and E is and F is For each of the following intervals, tell whether f is concave up (type in CU) or concave down (type in CD). D : D E : E F : F : 8.( pt) Consider the function f For this function there are two important intervals: A and A where the function is not defined at A. Find A For each of the following intervals, tell whether f is increasing (type in INC) or decreasing (type in DEC). A : A Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f is concave up (type in CU) or concave down (type in CD). A : A 9.( pt) Consider the function f For this function there are two important intervals: A and A where A is a critical number. Find A For each of the following intervals, tell whether f is increasing (type in INC) or decreasing (type in DEC). A : A : For each of the following intervals, tell whether f is concave up (type in CU) or concave down (type in CD). A : A : 30.( pt) Consider the function f For this function there are three important intervals: A, A B, and B where A and B are the critical numbers. Find A and B For each of the following intervals, tell whether f is increasing (type in INC) or decreasing (type in DEC). A : A B : B f has an inflection point at where C is C 3 Finally for each of the following intervals, tell whether f is concave up (type in CU) or concave down (type in CD). C : C 3.( pt) Consider the function f 3 4. For this function there are four important intervals: A, A B, B C, and C where A, and C are the critical numbers and the function is not defined at B. Find A and B and C For each of the following intervals, tell whether f is increasing (type in INC) or decreasing (type in DEC). A : A B : B C : C Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f is concave up (type in CU) or concave down (type in CD). B : B : 3.( pt) Consider the function f e. f has two inflection points at = C and = D with C D where C is and D is Finally for each of the following intervals, tell whether f is concave up (type in CU) or concave down (type in CD). C : C D : D 33.( pt) Consider the function e f 3 e Then f = The following questions ask for endpoints of intervals of increase or decrease for the function f. Write INF for, MINF for, and NA (ie. not applicable) if there are no intervals of that type. The interval of increase for f is from to The interval of decrease for f is from to f has a local minimum at. (Put NA if none.) f has a local maimum at. (Put NA if none.) Then f = The following questions ask for endpoints of intervals of upward and downward concavity for the function f. Write INF for, MINF for, and put NA if there are no intervals of that type. The interval of upward concavity for f is from to

38 The interval of downward concavity for f is from to f has a point of inflection at. (Put NA if none.) 4

39 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives0 5Optim due /0/06 at :00 PM..( pt) Find the point on the line 7y 7 0 which is closest to the point 5. (, ).( pt) A rectangle is inscribed with its base on the -ais and its upper corners on the parabola y. What are the dimensions of such a rectangle with the greatest possible area? Width = Height = 3.( pt) A cylinder is inscribed in a right circular cone of height 3.5 and radius (at the base) equal to. What are the dimensions of such a cylinder which has maimum volume? Radius = Height = 4.( pt) If 00 square centimeters of material is available to make a bo with a square base and an open top, find the largest possible volume of the bo. Volume = cubic centimeters. 5.( pt) A fence 6 feet tall runs parallel to a tall building at a distance of 3 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? 6.( pt) A fence 5 feet tall runs parallel to a tall building at a distance of 3 feet from the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to the wall of the building. Here are some hints for finding a solution: Use the angle that the ladder makes with the ground to define the position of the ladder and draw a picture of the ladder leaning against the wall of the building and just touching the top of the fence. If the ladder makes an angle 0.84 radians with the ground, touches the top of the fence and just reaches the wall, calculate the distance along the ladder from the ground to the top of the fence. The distance along the ladder from the top of the fence to the wall is Using these hints write a function L which gives the total length of a ladder which touches the ground at an angle, touches the top of the fence and just reaches the wall. L =. Use this function to find the length of the shortest ladder which will clear the fence. The length of the shortest ladder is feet. 7.( pt) A rancher wants to fence in an area of square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use? 8.( pt) For the given cost function C find a) The cost at the production level 00 b) The average cost at the production level 00 c) The marginal cost at the production level 00 d) The production level that will minimize the average cost. e) The minimal average cost. 9.( pt) For the given cost function C find: a) The cost at the production level 00 b) The average cost at the production level 00 c) The marginal cost at the production level 00 d) The production level that will minimize the average cost e) The minimal average cost 0.( pt) For the given cost function C and the demand fuction p 00. Find the production level that will maimaze profit..( pt) A manufacture has been selling 600 television sets a week at 480 each. A market survey indicates that for each 7 rebate offered to a buyer, the number of sets sold will increase by 70 per week. a) Find the demand function p, where is the number of the television sets sold per week. p b) How large rebate should the company offer to a buyer, in order to maimize its revenue? c) If the weekly cost function is , how should it set the size of the rebate to maimize its profit?.( pt) A baseball team plays in he stadium that holds spectators. With the ticket price at 9 the average attendence has been When the price dropped to 7, the averege attendence rose to a) Find the demand function p, where is the number of the spectators. (assume p is linear) p b) How should be set a ticket price to maimize revenue? 3.( pt) The manager of a large apartment comple knows from eperience that 80 units will be occupied if the rent is 500 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each dollar increase in rent. Similarly, one additional unit will be occupied for each dollar decrease in rent. What rent should the manager charge to maimize revenue? 4.( pt) A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 49 feet? 5.( pt) Let Q 0 6 and R be given points in the plane. We want to find the point P 0 on the -ais

40 such that the sum of distances PQ PR is as small as possible. (Before proceeding with this problem, draw a picture!) To solve this problem, we need to minimize the following function of : f over the closed interval a b where a and b. We find that f has only one critical number in the interval at where f has value Since this is smaller than the values of f at the two endpoints, we conclude that this is the minimal sum of distances. 6.( pt) Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to epand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuation. Centerville is located at 0 in the y-plane, Springfield is at 0 8, and Shelbyville is at 0 8. The cable runs from Centerville to some point 0 on the -ais where it splits into two branches going to Springfield and Shelbyville. Find the location 0 that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer. To solve this problem we need to minimize the following function of : f We find that f has a critical number at To verify that f has a minimum at this critical number we compute the second derivative f and find that its value at the critical number is, a positive number. Thus the minimum length of cable needed is

41 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment DerivativesNewton due //06 at :00 AM..( pt) Use Newton s method to approimate a root of the equation as follows. Let be the initial approimation. The second approimation is and the third approimation 3 is.( pt) Use Newton s method to approimate a root of the equation as follows. Let be the initial approimation. The second approimation is and the third approimation 3 is 3.( pt) Use Newton s method to approimate a root of the equation as follows. Let 3 be the initial approimation. The second approimation is and the third approimation 3 is 4.( pt) Use Newton s method to approimate a root of the equation sin as follows. Let be the initial approimation. The second approimation is and the third approimation 3 is 5.( pt) Use Newton s method to approimate a root of the equation cos 3 as follows. Let be the initial approimation. The second approimation is 6.( pt) Find the positive value of which satisfies 4 400sin. Give the answer to four places of accuracy. Remember to calculate the trig functions in radian mode. 7.( pt) Find the positive value of which satisfies cos. Give the answer to si places of accuracy. Remember to calculate the trig functions in radian mode. 8.( pt) Find the smallest positive value of which satisfies 000cos 000. Give the answer to four places of accuracy. Remember to calculate the trig functions in radian mode.

42 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment DerivativesMVT due //06 at :00 AM..( pt) Consider the function f 5 3 on the interval 4. Find the average or mean slope of the function on this interval, i.e. f 4 f 4 By the Mean Value Theorem, we know there eists a c in the open interval 4 such that f c is equal to this mean slope. For this problem, there is only one c that works. Find it..( pt) Consider the function f Find the average slope of this function on the interval 4. By the Mean Value Theorem, we know there eists a c in the open interval 4 such that f c is equal to this mean slope. Find the value of c in the interval which works 3.( pt) Consider the function f on the interval 5 5. Find the average or mean slope of the function on this interval. By the Mean Value Theorem, we know there eists at least one c in the open interval 5 5 such that f c is equal to this mean slope. For this problem, there are two values of c that work. The smaller one is and the larger one is 4.( pt) Consider the function f on the interval 4 0. Find the average or mean slope of the function on this interval. By the Mean Value Theorem, we know there eists a c in the open interval 4 0 such that f c is equal to this mean slope. For this problem, there are two values of c that work. The smaller one is and the larger one is 5.( pt) Consider the function f on the interval 5 7. Find the average or mean slope of the function on this interval. By the Mean Value Theorem, we know there eists a c in the open interval 5 7 such that f c is equal to this mean slope. For this problem, there is only one c that works. Find it. 6.( pt) Consider the function f 6 on the interval 3 6. Find the average or mean slope of the function on this interval. By the Mean Value Theorem, we know there eists a c in the open interval 3 6 such that f c is equal to this mean slope. For this problem, there is only one c that works. Find it.

43 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives3Higher due /3/06 at :00 AM..( pt) If f 7 5 e, find f. Find f. Find f. Find f..( pt) Let f 8 5e. (a) f (b) f 3.( pt) Let f Then f is and f is f is and f is 4.( pt) Let f Then f is f is f is and f is 5.( pt) If f , find f. Find f 3. Find f. Find f 3. 6.( pt) If f 3 4 7, find f. Find f. Find f. Find f. 7.( pt) Let h t t 3 3t 3. Then h t is h is, h t is and h is 8.( pt) Let f. Then f is and f is and f is 9.( pt) Let f (a) f 6 (b) f 6 [NOTE: There are two ways to do this problem. The first is the quotient rule. The second is much easier and does not use the quotient rule.] 0.( pt) If g t 3t 4 8t 4 find g 0 g 0 g 0 g 0 g 4 0 g 5 0.( pt) Let f sin. Find f 0 3. (Remember radian mode!).( pt) Let h t tan 4. Then h is and h is 3.( pt) Let g s 5s 3 6. Then g s is g 5 is, g s is and g 5 is 4.( pt) If g t 0 t 3 find g 0 g 0 g 0 5.( pt) Let f 8. Then f is f is, f is and f is 6.( pt) Let f 7 7.( pt) d4 d = f 8e 4 Note: There is a way of doing this problem without using the quotient rule 4 times. 8.( pt) Let f 5 4 f 4 Note: There is a way of doing this problem without using the quotient rule 4 times. 9.( pt) Let f 4 f

44 0.( pt) Let f.( pt) Let f 3ln sin f 6ln sec tan f HINT: Simplify the first derivative before you find the second derivative..( pt) Find the 74 th derivative of the function f cos. The answer is function

45 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Derivatives0Antideriv due /0/06 at :00 AM..( pt) Consider the function f Enter an antiderivative of f.( pt) Consider the function f An antiderivative of f is F A 4 B 3 C D where A is and B is and C is and D is 3.( pt) Consider the function f Enter an antiderivative of f 4.( pt) Consider the function f An antiderivative of f is F A n B m C p D q where A is and n is and B is and m is and C is and p is and D is and q is 5.( pt) Consider the function f 7 6. Let F be the antiderivative of f with F 0. Then F 6.( pt) Consider the function f 9 7. Let F be the antiderivative of f with F 0. Then F 5 equals 7.( pt) Consider the function f t 3sec t 5t. Let F t be the antiderivative of f t with F 0 0. Then F t equals 8.( pt) Consider the function f t sec t 6t 3. Let F t be the antiderivative of f t with F 0 0. Then F 5 9.( pt) Consider the function f whose second derivative is f 7 7sin. If f 0 3 and f 0 4, what is f? 0.( pt) Consider the function f whose second derivative is f 0 9sin. If f 0 4 and f 0, what is f 5?.( pt) Given f 6sin 4 and f 0 5 and f 0 3. Find f π 4 Remember: The angles for sin and cosine are always (well... almost always) in radians!.( pt) Given f 6 and f 3 and f 4. Find f and find f 3 3.( pt) Given that the graph of f passes through the point 3 7 and that the slope of its tangent line at f is 6 7, what is f? 4.( pt) A particle is moving with acceleration a t 6t. its position at time t 0 is s 0 and its velocity at time t 0 is v 0 4. What is its position at time t 4? 5.( pt) A car traveling at 40 ft/sec decelerates at a constant 3 feet per second squared. How many feet does the car travel before coming to a complete stop? 6.( pt) A ball is shot straight up into the air with initial velocity of 49 ft/sec. Assuming that the air resistance can be ignored, how high does it go? Hint: The acceleration due to gravity is 3 ft per second squared. 7.( pt) A ball is shot at an angle of 45 degrees into the air with initial velocity of 4 ft/sec. Assuming no air resistance, how high does it go? How far away does it land? Hint: The acceleration due to gravity is 3 ft per second squared. 8.( pt) A stone is thrown straight up from the edge of a roof, 800 feet above the ground, at a speed of 6 feet per second. A. Remembering that the acceleration due to gravity is -3 feet per second squared, how high is the stone 5 seconds later? B. At what time does the stone hit the ground? C. What is the velocity of the stone when it hits the ground? 9.( pt) A stone is thrown straight down from the edge of a roof, 050 feet above the ground, at a speed of 7 feet per second. A. Remembering that the acceleration due to gravity is -3 feet per second squared, how high is the stone 4 seconds later? B. At what time does the stone hit the ground? C. What is the velocity of the stone when it hits the ground? 0.( pt) A stone is dropped from the edge of a roof, and hits the ground with a velocity of -0 feet per second. How high (in feet) is the roof?

46 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment DerivativesLHospital due //06 at :00 AM..( pt) Evaluate the it 8 7.( pt) Compute the following it using l Hôpital s rule if appropriate. Use INF to denote and MINF to denote = 3.( pt) Evaluate the it using L Hospital s rule if necessary e 0 sin 5 4.( pt) Evaluate the it using L Hospital s rule if necessary 0 sin 0 sin 5.( pt) Evaluate the it using L Hospital s rule sin 7 0 tan 0 6.( pt) Evaluate the it using L Hospital s rule if necessary ( pt) Compute the following its using l Hôpital s rule if appropriate. Use INF to denote and MINF to denote. = tan = 8.( pt) Compute the following its using l Hôpital s rule if appropriate. Use INF to denote and MINF to denote. cos 7 = 0 cos = 9.( pt) Compute the following its using l Hôpital s rule if appropriate. Use INF to denote and MINF to denote. ln 7 = ln cos e 7 e 8 e 8 = 0.( pt) Compute the following it using l Hôpital s rule if appropriate. Use INF to denote and MINF to denote. sin ln = 0.( pt) Find the following its, using l Hôpital s rule if appropriate arctan 7 3 = 3 ln = 0.( pt) Evaluate the it using L Hospital s rule if necessary 6e 6 3.( pt) Evaluate the it using L Hospital s rule if necessary 3 4.( pt) Compute the following it using l Hôpital s rule if appropriate. Use INF to denote and MINF to denote. 8 = 5.( pt) Evaluate the it using L Hospital s rule if necessary 6.( pt) Evaluate the it using L Hospital s rule if necessary 4 4 ln0 ln 8 7.( pt) For each of the following forms determine whether the following it type is indeterminate, always has a fied finite value, or never has a fied finite value. In the first case answer IND, in the second case enter the numerical value, and in the third case answer DNE. For eample IND DNE 0 To discourage blind guessing, this problem is graded on the following scale 0-9 correct = correct = correct = correct =.7 Note that l Hôpital s rule (in some form) may ONLY be applied to indeterminate forms... 0

47 π π e ( pt) Find the following its, using L Hôpital s rule, if appropriate. Use INF to denote and MINF to denote (a) (b) 0 tan 4 sin = cos 5 πe 8 = ln

48 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals0Theory due 3//06 at :00 AM..( pt) You are given the four points in the plane A 4 4, B 6 6, C 0 4, and D 5. The graph of the function f consists of the three line segments AB, BC and CD. Find the integral 4 5 f d by interpreting the integral in terms of sums and/or differences of areas of elementary figures. 4 5 f d.( pt) Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry d 3.( pt) Use the Midpoint Rule to approimate with n d 4.( pt) Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry d 5.( pt) Use the Midpoint Rule to approimate the integral with n= ( pt) Consider the integral d d (a) Find the Riemann sum for this integral using right endpoints and n 3. (b) Find the Riemann sum for this same integral, using left endpoints and n 3 7.( pt) Consider the integral 7 3 d (a) Find the Riemann sum for this integral using right endpoints and n 4. (b) Find the Riemann sum for this same integral, using left endpoints and n 4 4 Find and 8.( pt) Let f d. f d 8 f 4 d 9.( pt) 3 f where a f d 8 3 f and b a b f f d 0.( pt) Consider the function f. In this problem you will calculate 0 d by using the definition b n f d a n f i The summation inside the brackets is R n which is the Riemann sum where the sample points are chosen to be the righthand endpoints of each sub-interval. Calculate R n for f on the interval 0 and write your answer as a function of n without any summation signs. You will need the summation formulas on page 38 of your tetbook (page 364 in older tets). R n R n n.( pt) The following sum 6 6 n n 6 n n 6 8 n n 6 6n n n is a right Riemann sum for a certain definite integral b i f d using a partition of the interval b into n subintervals of equal length. Then the upper it of integration must be: b = and the integrand must be the function f =.( pt) The following sum 4 n n 4 4 n n 4 is a right Riemann sum for the definite integral where b = and f = b f d n n n

49 It is also a Riemann sum for the definite integral c g d where c = and g = The it of these Riemann sums as n is 3.( pt) The following sum n 4 7 n 49 n n 49 is a right Riemann sum for the definite integral 4 0 b f d where b = and f = The it of these Riemann sums as n is 7n n 7 n 4.( pt) Suppose f is continuous and decreasing on the closed interval 4 9, that f 4 9, f 9 4 and that 4 9 Then f d f d = 5.( pt) Consider the function f By drawing a suitable picture, find a relation between the definite integrals f d and to find the second of these two integrals f d = f d. Use this relation

50 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals3Definite due 3/3/06 at :00 AM..( pt) b b 8 d =.( pt) The value of 3.( pt) The value of d is d is 4.( pt) Evaluate the definite integral d 5.( pt) Evaluate the definite integral d 6.( pt) Evaluate the definite integral d 7.( pt) Evaluate the definite integral d 8.( pt) Evaluate the definite integral d ( pt) d = 0.( pt) Evaluate the definite integral.( pt) Evaluate the integral.( pt) Evaluate the integral 0 π 6 4sin d ( pt) Evaluate the integral 0 0 sin t dt d d 4.( pt) The velocity function is v t t 5t 6 for a particle moving along a line. Find the displacement and the distance traveled by the particle during the time interval [-3,6]. displacement = distance traveled = If needed, see page 405 of your tetbook (378 in older books) for the definitions of these terms. 5.( pt) The velocity function is v t t 3t for a particle moving along a line. Find the displacement (net distance covered) of the particle during the time interval [0,6]. displacement = 6.( pt) Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. 4 The integral d MUST be evaluated by breaking it up into a sum of three integrals: where a = c = a d = c d = a d = c 4 a d c a d 4 c d Thus d = 7.( pt) Consider the function Evaluate the definite integral. f if if 3 f d

51 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals4FTC due 3/4/06 at :00 AM..( pt) If f t 4 dt then f f 3 0.( pt) If f t 3 dt then f f ( pt) If f t 8 dt then f 4.( pt) If f t 5 dt then f f 5 5.( pt) If f t dt then f f 6.( pt) If f 4 t 5 dt then f 7.( pt) If f t 3 3t 6 dt then f 8.( pt) Given 0 f 0 t 9 cos t dt At what value of does the local ma of f occur? 9.( pt) Given f t 0 t 4 45 cos d At what value of t does the local ma of f t occur? t 0.( pt) NOTE: It will be easier to see the function f() if you use the display mode typeset. Keep in mind, though, that loading the problem into your computer using this display mode will take longer. and Let f 0 if 4 4 if if if 3 g 4 f t dt Determine the value of each of the following: (a) g 6 (b) g 3 (c) g (d) g 4 (e) The absolute maimum of g occurs when and is the value It may be helpful to make a graph of f() when answering these questions..( pt) Use part I of the Fundamental Theorem of Calculus to find the derivative of f t 7 dt f = [NOTE: Enter a function as your answer. Make sure that your synta is correct, i.e. remember to put all the necessary *, (, ), etc. ].( pt) Use part I of the Fundamental Theorem of Calculus to find the derivative of f 3 t 3 7dt f = [NOTE: Enter a function as your answer. Make sure that your synta is correct, i.e. remember to put all the necessary (, ), etc. ] 3.( pt) Use part I of the Fundamental Theorem of Calculus to find the derivative of f t dt 4 f = [NOTE: Enter a function as your answer. Make sure that your synta is correct, i.e. remember to put all the necessary *, (, ), etc. ] 4.( pt) Use part I of the Fundamental Theorem of Calculus to find the derivative of F sin t dt F = [NOTE: Enter a function as your answer.]

52 5.( pt) Use part I of the Fundamental Theorem of Calculus to find the derivative of h sin 3 cos t 3 t dt h [NOTE: Enter a function as your answer. Make sure that your synta is correct, i.e. remember to put all the necessary *, (, ), etc. ] 6.( pt) Use part I of the Fundamental Theorem of Calculus to find the derivative of h sin cos t t dt h [NOTE: Enter a function as your answer. Make sure that your synta is correct, i.e. remember to put all the necessary *, (, ), etc. ] For more help see: WeBWorK functions 7.( pt) Use part I of the Fundamental Theorem of Calculus to find the derivative of g 4 u u 3 du [NOTE: Enter a function as your answer. Make sure that your synta is correct, i.e. remember to put all the necessary *, (, ), etc. ] 8.( pt) Use part I of the Fundamental Theorem of Calculus to find the derivative of 3 u 5 g u du 5 [NOTE: Enter a function as your answer. Make sure that your synta is correct, i.e. remember to put all the necessary *, (, ), etc. ] For more help see: WeBWorK functions 9.( pt) Find the derivative of the following function F 3 t 3 dt using the Fundamental Theorem of Calculus. F = 0.( pt) Find the derivative of the following function F s 4 5s 4 ds using the appropriate form of the Fundamental Theorem of Calculus. F =.( pt) Find a function f and a number a such that f t a t 4 dt 5 f a.( pt) Evaluate the definite integral 4 d 3 dt 5 3t 4 dt using the Fundamental Theorem of Calculus. You will need accuracy to at least 4 decimal places for your numerical answer to be accepted. You can also leave your answer as an algebraic epression involving square roots. 4 d 5 3t 3 dt 4 dt = 3.( pt) Evaluate the following definite integrals using the Fundamental Theorem of Calculus. s 5 s ds = 0 5π sin d = 0 0 t 5 5 t 0t 6 dt = 4.( pt) Compute the following it. Use INF to denote and MINF to denote t 3 dt =

53 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals5Trig due 3/5/06 at :00 AM..( pt) Find the value of Note: The notation 0 a 0 π 4 cos d. f d is read the integral from 0 to a of f() dee. Remember: The angles for sin and cosine are always (well... almost always) in radians! π 4.( pt) Find the value of cos 6 d. Note: The notation 0 a 0 f d is read the integral from 0 to a of f() dee. Remember: The angles for sin and cosine are always (well... almost always) in radians! π 6 3.( pt) Find the value of sin sin d. 4.( pt) Find the value of 5.( pt) Evaluate the definite integral π 3 cos sin sin d. sin 9 cos 9 d 6.( pt) Evaluate the definite integral. π sin 5 cos 0 d 7.( pt) Evaluate the definite integral. π tan 5 d 0 π 5 8.( pt) Evaluate the indefinite integral. 4cos 3 76 d 9.( pt) Evaluate the indefinite integral. 94cos 39 d 0.( pt) Evaluate the indefinite integral..( pt) 0 sin 3 6 cos 6 6 d π 4 sin 4 4 d =.( pt) Evaluate the indefinite integral. 4cos 4 6 d 3.( pt) Evaluate the indefinite integral. sin 7 sin d [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin... ] 4.( pt) Evaluate the definite integral. 0 π sec 4 4 cot 4 d [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin. ] 5.( pt) Evaluate the indefinite integral. sin 8 cos d [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin. ] 6.( pt) Evaluate the indefinite integral. 9 6 sec 4 7 d C C C

54 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals0InvTrig due 3/0/06 at :00 AM..( pt) Evaluate the definite integral. 4sin π d [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin... ].( pt) Evaluate the definite integral d [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin... ] 3.( pt) Evaluate the indefinite integral. 6 d [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin... ] 4.( pt) Evaluate the indefinite integral d [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin... ] 5.( pt) Evaluate the indefinite integral. 30 d [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin... ] 6.( pt) Evaluate the indefinite integral d [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin... ] 7.( pt) Evaluate the indefinite integral d [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin... ] 8.( pt) = +C 64 49d WeBWorK notation for sin is arcsin() or asin(), for tan it s arctan() or atan(). 9.( pt) d = +C 6 34 WeBWorK notation for sin is arcsin() or asin(), for tan it s arctan() or atan(). 0.( pt) For each of the indefinite integrals below, choose which of the following substitutions would be most helpful in evaluating the integral. Enter the appropriate letter (A,B, or C) in each blank. DO NOT EVALUATE THE INTEGRALS. A. 5tanθ B. 5sinθ C. 5secθ. 5 5 d. d d 4. d d.( pt) Match each of the trigonometric epressions below with the equivalent non-trigonometric function from the following list. Enter the appropriate letter (A,B,C,D, or E) in each blank. A. tan arcsin 5 B. cos arcsin 5 C. sin arcsin 5 D. sin arctan 5 E. cos arctan ( pt) Evaluate the indefinite integral d ( pt) Evaluate the definite integral d 0 4.( pt) Evaluate the indefinite integral d 4

55 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment IntegralsMethods due 3//06 at :00 AM..( pt) Evaluate the indefinite integral. arctan 7 d [NOTE: Remember to enter all necessary ( and )!! Enter arctan() for tan, arcsin() for sin. ].( pt) Evaluate the indefinite integral. ln 6 55 d 3.( pt) Evaluate the indefinite integral. cos 6 d 4.( pt) Evaluate the indefinite integral. arctan 7 d 5.( pt) Evaluate the indefinite integral. e 6 e 4 d 6.( pt) Find the indicated integrals. ln 5 (a) d = C e t cos e t (b) 5 6sin e t dt = C 5 6 sin 6 (c) d = 7.( pt) Find the indicated integrals (if they eist) 4 7d = e 7 e 4 d = d = ln 6 d =

56 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals4Substitution due 3/4/06 at :00 AM..( pt) Evaluate the integral d by making the substitution u 3 8. NOTE: Your answer should be in terms of and not u..( pt) Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral d Then the most appropriate substitution to simplify this integral is u = Then d f du where f = After making the substitution we obtain the integral g u du where g u = This last integral is: = C (Leave out constant of integration from your answer.) After substituting back for u we obtain the following final form of the answer: = C (Leave out constant of integration from your answer.) 3.( pt) Evaluate the integral by making the given substitution. d u ( pt) Evaluate the integral by making the given substitution. sec 6 tan 6 d u 6 5.( pt) Find F 3 3 d Give a specific function for F. F() = 6.( pt) Evaluate the indefinite integral. ln 7 d 7.( pt) C 5 e d = + C 8.( pt) Evaluate the indefinite integral. e d 9.( pt) Evaluate the indefinite integral d 0.( pt) Evaluate the indefinite integral. 6 t 9 7 dt.( pt) Evaluate the indefinite integral. cos 6sin 4 d.( pt) Evaluate the indefinite integral. 5 6 d [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin. ] 3.( pt) Evaluate the indefinite integral. 4 d ln 4 [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin. ] 4.( pt) Evaluate the indefinite integral. 5e 5 sin e 5 d 5.( pt) Evaluate the indefinite integral. 4 8 d 6.( pt) Evaluate the indefinite integral. d 7.( pt) Evaluate the indefinite integral. 4 7 d [NOTE: Remember to enter all necessary ( and )!! Enter arctan() for tan, arcsin() for sin. ] 8.( pt) Evaluate the indefinite integral. 4 d [NOTE: Remeber to enter all necessary ( and )!! Enter arctan() for tan, arcsin() for sin. ] 9.( pt) Evaluate the indefinite integral d C

57 0.( pt) Evaluate the indefinite integral. 4 4 d [NOTE: Remember to enter all necessary *, (, and )!!.( pt) Evaluate the indefinite integral. sin 6 cosd.( pt) Note: You can get full credit for this problem by just entering the answer to the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral cos 5 4t sin 4t dt Then the most appropriate substitution to simplify this integral is u = Then dt f t du where f t = After making the substitution we obtain the integral g u du where g u = This last integral is: = C (Leave out constant of integration from your answer.) After substituting back for u we obtain the following final form of the answer: = C (Leave out constant of integration from your answer.) 3.( pt) Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the definite integral π cos z sin 6 z dz π 6 Then the most appropriate substitution to simplify this integral is u = Then dz f z du where f z = After making the substitution and simplifying we obtain the b integral g u du where a g u = a = b = This definite integral has value = 4.( pt) Evaluate the indefinite integral. sin 3 cos 5 d 5.( pt) Evaluate the definite integral. π sin e cos d 6.( pt) Evaluate the definite integral. 0 C 0 3 d 7.( pt) Evaluate the definite integral. π 4 sin 4t dt 8.( pt) Evaluate the definite integral. 4 d ( pt) Evaluate the definite integral. e 3 d ln [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan() for tan, sin() for sin. ] 30.( pt) Evaluate the definite integral. 0 e 4 0 d ln 3.( pt) Verify that and use this equation to evaluate 5 4 d 3.( pt) Evaluate the indefinite integral. e 3 e 6 49 d 33.( pt) 0 7 d = 34.( pt) Use the substitution 7tan θ to evaluate the indefinite integral 58d ( pt) Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral 3 6 d Then the most appropriate substitution to simplify this integral is g t where g t = Note: We are using t as variable for angles instead of θ, since there is no standard way to type θ on a computer keyboard. After making this substitution and simplifying (using trig identities), we obtain the integral f t dt where f t = This integrates to the following function of t

58 f t dt = C After substituting back for t in terms of we obtain the following final form of the answer: C 36.( pt) Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. 3 3 Consider the definite integral 3 3 d Then the most appropriate substitution to simplify this integral is g t where g t = Note: We are using t as variable for angles instead of θ, since there is no standard way to type θ on a computer keyboard. After making this substitution and simplifying (using trig identities), we obtain the integral a b f t dt where f t = a = b = After evaluating this integral we obtain: d = 37.( pt) Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the definite integral d Then the most appropriate substitution to simplify this integral is g t where g t = Note: We are using t as variable for angles instead of θ, since there is no standard way to type θ on a computer keyboard. After making this substitution and simplifying (using trig identities), we obtain the integral f t = a = b = After evaluating this integral we obtain: d = 38.( pt) Evaluate the indefinite integral. arcsin d a b f t dt where 39.( pt) Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral 4 5 d 3 Then the most appropriate substitution to simplify this integral is u = Then d f du where f = After making the substitution and simplifying we obtain the integral g u du where g u = This last integral is: = C (Leave out constant of integration from your answer.) After substituting back for u we obtain the following final form of the answer: = C (Leave out constant of integration from your answer.) 40.( pt) Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Also the appropriate way to enter roots (ecept sqrt) into WeBWorK is to use fractional eponents. d Consider the definite integral Then the most appropriate substitution to simplify this integral is u = Then d f du where f = After making the substitution and simplifying we obtain the b integral g u du where a g u = a = b = This definite integral has value = 4.( pt) Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the definite integral 0 7 8d Then the most appropriate substitution to simplify this integral is u = Then d f du where f = After making the substitution and simplifying we obtain the b integral g u du where a g u = a = b = This definite integral has value = 4.( pt) Find the following indefinite integrals.

59 6 d = C Hint: This is similar to Problem 6 of WeBWorK Hwwk #. cos t 6sin t dt = C 43.( pt) Note: You can get full credit for this problem by just entering the answer to the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral 4 4 e d The most appropriate substitution to simplify this integral is u f where f = We then have d g u du where g u = Hint: you need to back substitute for in terms of u for this part. After substituting into the original integral we obtain h u du where h u = To evaluate this integral rewrite the numerator as 4 u u 4 simplify, then integrate, thus obtaining h u du H u where H u = + C After substituting back for u we obtain our final answer 4 4 e d = + C 44.( pt) Consider the integral 6 d Then an appropriate trigonometric substitution to simplify this integral is f t where f t = After making this substitution and simplifying, we obtain the integral g t dt where g t = Note that this problem doesn t ask you to evaluate this integral. 45.( pt) For each of the following integrals find an appropriate trigonometric substitution of the form f t to simplify the integral. 7 3 d 7 d d d 4

60 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals5ByParts due 3/5/06 at :00 AM..( pt) Use integration by parts to evaluate the integral..( pt) 0 5 e d = e 3 d 3.( pt) Use integration by parts to evaluate the integral. 5sin d 4.( pt) Use integration by parts to evaluate the integral. 3cos 4 d 5.( pt) Use integration by parts to evaluate the integral. 4ln 3 d 6.( pt) Use integration by parts to evaluate the integral. 7 cos 3 d 7.( pt) Use integration by parts to evaluate the integral. ln 5 d 8.( pt) Evaluate the indefinite integral. e 6 sin 3 d C C C C C 9.( pt) Use integration by parts to evaluate the definite integral. e t lntdt 0.( pt) Evaluate the definite integral. 8 t 4 ln t dt.( pt) Use integration by parts to evaluate the definite integral. 5 te t dt 0.( pt) Use integration by parts to evaluate the integral. 6 t lntdt 3.( pt) ytan 8y dy = Use arctan to denote tan in your answer. 4.( pt) Evaluate the indefinite integral. C arctan 5 d [NOTE: Remember to enter all necessary ( and )!! Enter arctan() for tan, arcsin() for sin. ] 5.( pt) Evaluate the indefinite integral. sin 5 d 6.( pt) First make a substitution and then use integration by parts to evaluate the integral. 7 cos 4 d 7.( pt) Evaluate the indefinite integral. ln 4 33 d C 8.( pt) A particle that moves along a straight line has velocity v t t e 3t meters per second after t seconds. How many meters will it travel during the first t seconds? 9.( pt) Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. 3 Consider the definite integral sin 3 d The first step in evaluating this integral is to apply integration by parts: udv uv vdu where u = and dv h d where h = Note: Use arcsin for sin. 0 After integrating by parts, we obtain the integral 3 f d on the right hand side where vdu f = The most appropriate substitution to simplify this integral is g t where g t = Note: We are using t as variable for angles instead of θ, since there is no standard way to type θ on a computer keyboard. After making this substitution and simplifying (using trig identities), we obtain the integral k t = a = a b k t dt where

61 b = After evaluating this integral and plugging back into the integration by parts formula we obtain: 0 3 sin 3 d =

62 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals6Tables due 3/6/06 at :00 AM..( pt) Use the Table of Integrals in the back of your tetbook to evaluate the integral. e 5 sin6d.( pt) Use the Table of Integrals in the back of your tetbook to evaluate the integral. d ln 3.( pt) Use the Table of Integrals in the back of your tetbook to evaluate the integral. d 4 6

63 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals7Approimations due 3/7/06 at :00 AM..( pt) Approimate 0 π sin d by computing L f P and U f P, using the partition 0 π 6 π 4 π 3 π. Your answers should be accurate to at least 4 decimal places. L f P = U f P = You may include a formula as an answer. π.( pt) Approimate sin d by computing L f P 0 and U f P, using the partition 0 π 6 π 4 π 3 π. Your answers should be accurate to at least 4 decimal places. L f P = U f P = You may include a formula as an answer. 3.( pt) Approimate the definite integral 0 8 t dt 7 using midpoint Riemann sums with the following partitions: (a) P Then midpoint Riemann sum = (b) Using 3 subintervals of equal length. Then midpoint Riemann sum = 4.( pt) Use the Midpoint Rule to approimate the integral with n= d 5.( pt) Given the following integral and value of n, approimate the following integral using the methods indicated (round your answers to si decimal places): e d n 4 (a) Trapezoidal Rule (b) Midpoint Rule (c) Simpson s Rule 0 6.( pt) Use Simpson s Rule and all the data in the following table to estimate the value of the integral 9 5 yd y

64 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals8Improper due 3/8/06 at :00 AM..( pt) Find the area under the curve y 3 from to t and evaluate it for t 0 t 00. Then find the total area under this curve for. (a) t = 0 (b) t = 00 (c) Total area.( pt) Find the area under the curve y from 8 to t and evaluate it for t 0, t 00. Then find the total area under this curve for 8. (a) t = 0 (b) t = 00 (c) Total area 3.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. 8e d 0 4.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. e d 5.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent d 6.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. 9 7 d 6 7.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. 4 3 d 8.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. 4 d 9.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. 3 e 4 d 0.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer -. e 3 d 5.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. 6 d.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. ln d 6 3.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer -. ln d 7 4.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. 5 d 5.( pt) Define the functions F and G by F t 5 dt G 6 t 5 dt 6 0

65 Determine whether each of the following improper integrals and its is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. (a) 5 d (b) (c) F G 6.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV d 7.( pt) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV d 8.( pt) Consider the following integrals. Label each as P, C, D, according as the integral is proper, improper but convergent, or improper and divergent.. 4 t 6 dt d 43π 3. sin tan d π 0 5 d ln 0 d se 4s ds π sin 4 d 8. tan 4 d π 0 9.( pt) Let f be a continuous function defined on the interval such that f 3 f 4 8 and f e 7 d 4 Determine the value of 3 3 f e 7 d

66 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals9Area due 3/9/06 at :00 AM..( pt) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to or y. Then find the area of the region. y y 7.( pt) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to or y. Then find the area of the region. y e 3 y e 7 3.( pt) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to or y. Then find the area of the region. y y 4.( pt) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to or y. Then find the area of the region. y y 0 5.( pt) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to or y. Then find the area of the region. y 3cos y 5sec π 4 π 4 6.( pt) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to or y. Then find the area of the region. y 3 y 4 and y ( pt) Find the area between the curves: y and y ( pt) The total area enclosed by the graphs of y 3 9.( pt) Find the area enclosed between f 0 7 and g From to 4 0.( pt) Find the area of the region enclosed between y 4sin and y 3cos from 0 to π. Hint: Notice that this region consists of two parts..( pt) Use the parametric equations of an ellipse, 5cos θ y sin θ 0 θ π to find the area that it encloses..( pt) Use the parametric equations of an ellipse 6cosθ y 3sinθ 0 θ π to find the area that it encloses. 3.( pt) Find the area of the region enclosed by the parametric equation t 3 8t y 8t 4.( pt) There is a line through the origin that divides the region bounded by the parabola y 6 8 and the -ais into two regions with equal area. What is the slope of that line? is y 9 5.( pt) Farmer Jones, and his wife, Dr. Jones, decide to build a fence in their field, to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by y 9 and y. Farmer Jones thinks this would be much harder than just building an enclosure with

67 straight sides, but he wants to please his wife. What is the area of the enclosed region? 6.( pt) Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Find the area bounded by the two curves: y y 7 y The appropriate definite integral for computing this area has integrand ; lower it of integration = ; and upper it of integration = This definite integral has value = This is the area of the region enclosed by the two curves. 7.( pt) Consider the area between the graphs 6y and 5 y. This area can be computed in two different ways using integrals First of all it can be computed as a sum of two integrals a b f d b c g d where a, b, c and f g Alternatively this area can be computed as a single integral α β h y dy where α, β and h y Either way we find that the area is.

68 Tom Robbins WW Prob Lib Summer 00 WeBWorK assignment Integrals0Volume due 3/0/06 at :00 AM..( pt) Consider the blue vertical line shown above (click on graph for better view) connecting the graphs y g sin and y f cos. Referring to this blue line, match the statements below about rotating this line with the corresponding statements about the result obtained.. The result of rotating the line about the -ais is. The result of rotating the line about the y-ais is 3. The result of rotating the line about the line y is 4. The result of rotating the line about the line is 5. The result of rotating the line about the line π is 6. The result of rotating the line about the line y is 7. The result of rotating the line about the line y π 8. The result of rotating the line about the line y π A. an annulus with inner radius sin and outer radius cos B. a cylinder of radius and height cos sin C. an annulus with inner radius π sin and outer radius π cos D. an annulus with inner radius cos and outer radius sin is E. an annulus with inner radius sin and outer radius cos F. an annulus with inner radius π cos and outer radius π sin G. a cylinder of radius π and height cos sin H. a cylinder of radius and height cos sin.( pt) Consider the blue horizontal line shown above (click on graph for better view) connecting the graphs f y sin y and g y cos 3y. Referring to this blue line, match the statements below about rotating this line with the corresponding statements about the result obtained.. The result of rotating the line about the -ais is. The result of rotating the line about the y-ais is 3. The result of rotating the line about the line y is 4. The result of rotating the line about the line is 5. The result of rotating the line about the line π is 6. The result of rotating the line about the line y is 7. The result of rotating the line about the line y π 8. The result of rotating the line about the line y π A. an annulus with inner radius sin y and outer radius cos 3y B. a cylinder of radius y and height cos 3y sin y C. a cylinder of radius π y and height cos 3y sin y D. a cylinder of radius π y and height cos 3y sin y E. an annulus with inner radius π cos 3y and outer radius π sin y is F. an annulus with inner radius sin y and outer radius cos 3y G. a cylinder of radius y and height cos 3y sin y H. a cylinder of radius y and height cos 3y sin y 3.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified ais. y 7 y 0 about the -ais 4.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified ais. y y 0 0 about the y-ais 5.( pt) Find the volume formed by rotating the region enclosed by:

69 9y and y 3 with y 0 about the y-ais 6.( pt) Find the volume of the solid formed by rotating the region enclosed by y e 4 3 y about the -ais. 7.( pt) Find the volume of the solid formed by rotating the region enclosed by y e 4, y 0, 0, about the y-ais. 8.( pt) Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y y about the -ais. 6.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified ais. y 4 y 0 6; about y 5 7.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified ais. y 0 y cos 3 π 6 0 about the ais y 6 8.( pt) The region between the graphs of y is rotated around the line y 36. and y 6 The volume of the resulting solid is 9.( pt) The region between the graphs of y is rotated around the line 6. and y 6 The volume of the resulting solid is 0.( pt) 9.( pt) Find the volume of the solid formed by rotating the region enclosed by about the -ais. 0 y 0 y 8 0.( pt) Find the volume of the solid formed by rotating the region enclosed by about the y-ais. 0 y 0 y 4 5.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified ais. y 5 y 0 3 7; about the y-ais.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified ais. y 54 6 y 0; about the y-ais 3.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified ais. y y ; about y 5 4.( pt) Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves 0, y, y 3, about the line y. The base of a certain solid is the area bounded above by the graph of y f 4 and below by the graph of y g 36. Cross-sections perpendicular to the -ais are squares. (See picture above, click for a better view.) Use the formula V a b A d to find the volume of the solid. Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. The lower it of integration is a = The upper it of integration is b = The side s of the square cross-section is the following function of : A = Thus the volume of the solid is V =.( pt) 5.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified ais. y 4 y ; about y 6

70 The base of a certain solid is the area bounded above by the graph of y f 4 and below by the graph of y g 6. Cross-sections perpendicular to the y-ais are squares. (See picture above, click for a better view.) Use the formula b V A y dy a to find the volume of the formula. Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. The lower it of integration is a = The upper it of integration is b = The side s of the square cross-section is the following function of y: A y = Thus the volume of the solid is V =.( pt) As you grow more and more tired, the world begins to spin around you. However, according to Newton, there is no difference between the world spinning around you, and you spinning around the world. Unfortunately, you are so tired that you think the world is the -ais. What is the volume of the solid you (the region) create by spinning about the -ais? 4.( pt) You wake up one morning, and find yourself wearing a toga and scarab ring. Always a logical person, you conclude that you must have become an Egyptian pharoah. You decide to honor yourself with a pyramid of your own design. You decide it should have height h 4390 and a square base with side s 450 To impress your Egyptian subjects, find the volume of the pyramid. 5.( pt) A ball of radius 0 has a round hole of radius 4 drilled through its center. Find the volume of the resulting solid. 6.( pt) Find the volume of a pyramid with height 5 and rectangular base with dimensions 4 and 9. The base of a certain solid is an equilateral triangle with altitude. Cross-sections perpendicular to the altitude are semicircles. Find the volume of the solid, using the formula V a b A d applied to the picture shown above (click for a better view), with the left verte of the triangle at the origin and the given altitude along the -ais. Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. The lower it of integration is a = The upper it of integration is b = The diameter r of the semicircular cross-section is the following function of : A = Thus the volume of the solid is V = 3.( pt) As a hardworking student, plagued by too much homework, you spend all night doing math homework. By 6am, you imagine yourself to be a region bounded by y 0 4 y ( pt) Find the volume of a pyramid with height 7 and rectangular base with dimensions 3 and 0. 9.( pt) The base of a certain solid is the triangle with vertices at 6, 6 6, and the origin. Cross-sections perpendicular to the y-ais are squares. Then the volume of the solid is. 30.( pt) A soda glass has the shape of the surface generated by revolving the graph of y 6 for 0 about the y-ais. Soda is etracted from the glass through a straw at the rate of cubic inch per second. How fast is the soda level in the glass dropping when the level is 3 inches? (Answer should be implicitly in units of inches per second. Do not put units in your answer. Also your answer should be positive, since we are asking for the rate at which the level DROPS rather than rises.) answer: 3.( pt) Coffee is poured into one of mugs above at a constant rate (constant volume per unit time). The graph below shows the depth of coffee in the mug as a function of time. (Click on images for better view.)

71 Which mug was filled with coffee? Be prepared to eplain your choice (offline). 3.( pt) As viewed from above, a swimming pool has the shape of the ellipse 3600 y 600 The cross sections perpendicular to the ground and parallel to the y-ais are squares. Find the total volume of the pool. (Assume the units of length and area are feet and square feet respectively. Do not put units in your answer.) V = 4