WeBWorK assignment 1. b. Find the slope of the line passing through the points (10,1) and (0,2). 4.(1 pt) Find the equation of the line passing
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1 WeBWorK assignment Thought of the day: It s not that I m so smart; it s just that I stay with problems longer. Albert Einstein.( pt) a. Find the slope of the line passing through the points (8,4) and (,8). b. Find the slope of the line passing through the points (0,) and (0,)..( pt) Find the equation of the line passing through the point (-8,4) with slope 5. y= 3.( pt) The equation of the line passing through the point (6, ) which is perpendicular to the line given by the equation 4x+y = is y = Ax+B where A= B= 4.( pt) Find the equation of the line passing through the point (-5,-6) and parallel to the line passing through (5,5) and (-4,-). y= 5.( pt) For each of the following functions, decide whether it is even, odd, or neither. Enter E for an EVEN function, O for an ODD function and N for a function which is NEITHER even nor odd. NOTE: You will only have four attempts to get this problem right!. f (x) = 5x 4 3x 6. f (x) = x 3. f (x) = x 4 + 3x 6 + x 7 4. f (x) = x 4 6x 6 + 3x 6 6.( pt) This problem gives you some practice identifying how more complicated functions can be built from simpler functions. Let f (x) = x 3 + and let g(x) = x +. Match the functions defined below with the letters labeling their equivalent expressions.. ( f (x)). (g(x)) 3. f (x ) 4. g(x) f (x) A. + x + x B. + x 3 + x 6 C. + x 6 D. + x + x 3 + x 4 7.( pt) Relative to the graph of y = x the graphs of the following equations have been changed in what way?. y = x + 5. y = x 5 3. y = (x )/7 4. y = (x/7) A. shifted 5 units down B. shifted 5 units up C. compressed vertically by the factor 7 D. stretched horizontally by the factor 7 8.(3 pts) Let f be the linear function (in blue) and let g be the parabolic function (in red) below. If you are having a hard time seeing the picture clearly, click on the picture. It will expand to a larger picture on its own page so that you can inspect it more closely. Note: If the answer does not exist, enter DNE :. (f o g)( ) =. (g o f)( ) = 3. (f o f)( ) = 4. (g o g)( ) = 5. (f + g)( 4 ) = 6. (f / g)( ) = 9.(3 pts) For each of the following angles, find the degree measure of the angle with the given radian measure: 5 6
2 (3 pts) For each of the followings angles (in radian measure), find the sin of the angle (your answer cannot contain trig functions, it must be an arithmetic expression or number): 6 4.(4.5 pts) Let F be the function below. If you are having a hard time seeing the picture clearly, click on the picture. It will expand to a larger picture on its own page so that you can inspect it more clearly. 4 3.(3 pts) For each of the followings angles (in radian measure), find the cos of the angle (your answer cannot contain trig functions, it must be an arithmetic expression or number): ( pts) If θ = 9 4, then sin(θ) equals cos(θ) equals tan(θ) equals sec(θ) equals 3.( pt) The angle of elevation to the top of a building is found to be 9 from the ground at a distance of 5000 feet from the base of the building. Find the height of the building. Evaluate each of the following expressions. Note: Enter DNE if the it does not exist or is not defined. a) x F(x) = b) x +F(x) = c) x F(x) = d) F( ) = e) x F(x) = f) x +F(x) = g) x F(x) = h) x 3 F(x) = i) F(3) =
3 WeBWorK problems. WW Prob Lib Math course-section, semester year WeBWorK assignment due 9/8/06 at :00 AM..( pt) Evaluate the it sin5x x 0 sin3x.( pt) Evaluate the it tanx x 0 3x 3.( pt) Evaluate the it t 3 t t t 4.( pt) Evaluate the it x 3 x x 5.( pt) Evaluate the it y 8 6.( pt) Evaluate the it 8 y 9 y s 7 s 7 s 7 7.( pt) Evaluate the it t + 4 t 4 t ( pt) Let f (x) = x + 6 if x and f (x) = 6 if x >. Sketch the graph of this function for yourself and find following its if they exist (if not, enter N).. f (x) x. f (x) x + 3. f (x) x 9.( pt) Let f (x) = 6 if x >, f (x) = 8 if x =, f (x) = x + 6 if 0 x <, f (x) = 6 if x < 0. Sketch the graph of this function and find following its if they exist (if not, enter DNE).. x f (x). x + f (x) 3. x f (x) 4. x 0 f (x) 5. x 0 + f (x) 6. x 0 f (x) 0.( pt) Evaluate the it 3(5 + h) + 4(5 + h) ( ) h 0 h.( pt) For what value of the constant c is the function f continuous on (, ) where { s c if s (,5) f (s) = cs + if s [5, ).( pt) The function f is given by the formula f (x) = x3 x 4x 3 x + when x < and by the formula f (x) = x 5x + a when x. What value must be chosen for a in order to make this function continuous at -? a = 3.( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) x x + 5x 0x +
4 (b) x x + 5x 0x + 4.( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) 8 + x (b) x x + 9x = 8 + x = + 9x 5.( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) (b) ( ) x + 0x + x = x ( ) x + 0x + x = x 6.( pt) Evaluate the following its. If needed, enter INF for and MINF for. (a) (b) x 7 0 x x 7 0x = 30x 7 0x =
5 Math 0-5, Fall 006 Assignment 3 due October Remember: Don t drink and derive!.( pt) Find the derivative of f(x)=x 6 6x 4 + 3x. f (x) =.( pt) For what values of x does the curve y = x x + 3 have: Positive slope? Negative slope? Zero slope? x= Your answer to parts and should be an interval (a,b). Use INF for +, -INF for. 3.( pt) Find the slope of the curve y = 6x 4 3x 3 at the point (,3). m = 4.( pt) If f (x) = (5x 3)(5x + 5), find f (x). [NOTE: Your answer should be a function in terms of the variable x and not a number! ] 5.( pt) If f (x) = 3x+6 3x+3, find f (x). Find f (). [NOTE: When entering functions, make sure that you put all the necessary *, (, ), etc. in your answer. ] 6.( pt) Find the equation of the line tangent to the curve y = 7x 5x + at the point y = (7,30). 7.( pt) Find all points on the graph of y = 3 x3 + x x where the tangent line has slope. (, ) (, ) Instruction: Enter the points in order of increasing x-coordinate. 8.(.5 pts) Given this graph, which describes the position of a particle wandering about on a vertical line, answer the questions using the values on the graph. (Note that, the BLUE curve depicts the position of the particle and the RED line is the tangent line to the graph.) (a) What is the average velocity from t = to t = 5? (b) What is the instantaneous velocity at t = 3? (c) When is it going the fastest? (d) At what time is it stopped (in order)? t = and t = 9.( pts) Assume that the function f (t) = t 3 4t + 3t + 3 describes the position of a particle wandering about on a vertical line. Answer the questions using calculus. (a) What is the average velocity from t = to t = 4? (b) What is the instantaneous velocity at t = 3? (c) At what time is it stopped (in order)? t = and t = 0.( pt) Let f (x) = x 6 Algebraically simplify the secant line slope f (x + h) f (x), and enter the numerator below: h f (x + h) f (x) = /((x + h 6)(x 6)) h Let h 0 to deduce the derivative, f (x) =
6 The equation of the tangent line passing through the point on the graph of f with x-coordinate 8 can be written in the form y = mx + b, where m = b =.( pt) If f (x) = x + x, find f (x). Find f (3).
7 Math 0-5, Fall 006 Assignment 4 due October 6.( pt) If f (x) = 5sinx + 0cosx, then f (x) =.( pts) At time t seconds, the center of a bobbing cork is sint centimeters above (or below) water level. What is the velocity of the cork at t = 0,/,? Velocity at t = 0: cm/s. Velocity at t = /: cm/s. Velocity at t = : cm/s. 3.( pt) If find f (x). f (x) = 4sinx 3 + cosx 4.( pt) If f (x) = (x + 5x + 4) 3, find f (x). Find f (3). 5.( pts) If find f (x). Find f (3). 6.( pts) If find f (x). Find f (). 7.( pts) Let f ( 3 ) = f (x) = x 7 x + 7 f (x) = tanx secx f (x) = 3xsinxcosx 8.( pt) If f (x) = sin(x 4 ), find f (x). Find f (). 9.( pt) If f (x) = sin 4 x, find f (x). Find f (3). 0.( pt) Let f (x) = f (x) = sin(cos(x 4 )).( pt) Let f (x) = x + x x +. Answer the following. f (x) = f (x) = f () = f () = f () =.( pt) Let f (x) = x + 3x + 8 f (x) = f (5) = 3.( pt) For xy = 3, (a) find dy dx =, as a function of x only. (b) find the slope of the tangent at (, 3) 4.( pt) For y + xy x =, (a) find dy dx =, as a function of x and y. (b) find the slope of the tangent at (, 3) 5.( pt) Let f (x) = ( x 7 ). For what values of x is f (x) = 0? Write the answers in increasing order.,. 6.( pt) A street light is at the top of a ft. tall pole. A man ft tall walks away from the pole with a speed of feet/sec along a straight path. How fast is the tip of his shadow moving when he is feet from the pole? 7.( pt) Sand falls out of the end of a slurry at the rate of 0 cc/sec. The pile forms a circular cone, the ratio of whose base diameter to height is 3. When the pile is of height 0 cm., at what rate is the height of the pile increasing? dh dt = 8.( pt) Find the slope of the tangent line to the curve
8 x xy 4y 3 = 844 at the point (, 6). 9.( pt) Find the slope of the tangent line to the curve given by the equation y + 7xy = 0 at the point ( ,). y = 0.( pt) Use linear approximation, i.e. the tangent line, to approximate as follows: Let f (x) = 3 x. The equation of the tangent line to f (x) at x = 64 can be written in the form y = mx + b where m is: and where b is: Using this, we find our approximation for is
9 Math 0-5, Fall 006 Assignment 5 due October 3.( pts) A city is hit by a flu epidemic. Officials estimate that t days after the beginning of the epidemic the number of persons sick with the flu is given by p(t) = 50t t 3, when 0 t 75. At what rate is the flu spreading at time t = 5 days? people/day At what rate is the flu spreading at time t = 60 days? people/day How many days after the outbreak are the most people sick? t = days..(.5 pts) The function f (x) = 6x 3 36x 378x is decreasing on the interval (, ). It is increasing on the interval (, ) and the interval (, ). The function has a local maximum at. 3.( pts) For x [,4] the function f is defined by f (x) = x (x 8) 3 On which two intervals is the function increasing? to and to Find the region in which the function is positive: to Where does the function achieve its minimum? 4.(.5 pts) Identify the critical points and find the maximum value and minimum value of the following function on the given interval. Recall, critical points are either endpoints, stationary points, or singular points. f (x) = 3 x, over [,7]. Critical Points:,,. Maximum:. Minimum:. Instructions: When entering the critical points, please enter them in the order that they appear on the real line. 5.( pt) Consider the function f (x) = 4 7x, 4 x. The absolute maximum value is and this occurs at x equals The absolute minimum value is and this occurs at x equals
10 Math 0-5, Fall 006 Assignment 6 due November 7 Thought: Those who won t have no advantage over those who can t..( pt) If 000 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Volume = cubic centimeters..( 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? 3.( pt) A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 5 x. What are the dimensions of such a rectangle with the greatest possible area? Width = Height = 4.( pt) A fence 3 feet tall runs parallel to a tall building at a distance of 4 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? 5.(.5 pts) The function f (x) = x 3 8x + 0x + 5 is decreasing on the interval (, ). It is increasing on the interval (, ) and the interval (, ). The function has a local maximum at. 6.( pts) For x [ 3,0] the function f is defined by f (x) = x 6 (x 5) 5 On which two intervals is the function increasing? to and to Find the region in which the function is positive: to Where does the function achieve its minimum? 7.(4 pts) Answer the following questions for the function f (x) = x3 x 6 defined on the interval [ 7,8]. Enter points, such as inflection points in ascending order, i.e. smallest x values first. Enter intervals in ascending order also. A. The function f (x) has vertical asympototes at and B. f (x) is concave up on the region to and to C. The inflection points for this function are, and 8.(4.5 pts) Consider the function f (x) = 9x+6x. 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 (x) 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 (x) is concave up (type in CU) or concave down (type in CD). (,B): (B, ): 9.( pt) Consider the function f (x) = 4x 3 x + 0x 9. Enter an antiderivative of f (x) 0.( pt) Consider the function f (x) whose second derivative is f (x) = 3x + 7sin(x). If f (0) = 3 and f (0) = 3, what is f (4)?.( pt) Given f (x) = x 4 and f ( 3) = and f ( 3) =.
11 Find f (x) = and find f (4) =.( pt) Consider the function f (t) = 7sec (t) 8t 3. Let F(t) be the antiderivative of f (t) with F(0) = 0. Then F(4) =
12 Math 0-5, Fall 006 Assignment 7 due November 3.( pt) Consider the differential equation: dy x dx = y. a) Find the general solution to the above differential equation. (Instruction: Call your integration constant C.) Answer: y =. b) Find the particular solution of the above differential equation that satisfies the condition y = 4 at x =. Answer: y =..( pt) Consider the differential equation: du dt = u3 (t 3 t). a) Find the general solution to the above differential equation. (Instruction: Write the answer in a form such that its numerator is and its integration constant is C rename your constant if necessary.) Answer: u =. b) Find the particular solution of the above differential equation that satisfies the condition u = 4 at t = 0. Answer: u =. 3.( pt) Consider the integral Z 8 ( ) 3 x + dx 4 (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.( 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. Z 7 49 x dx 7 5.( pt) Evaluate the sum: 6 i= ( i) 6.( pts) If R R 0 f (x)dx = 4, 0 f (x)dx =, and R 0 g(x)dx = 3, evaluate each integral. (a) R f (x)dx = (b) R 0 f (x)dx = (c) R 0 3 f (x)dx = (d) R 0 [g(x) 3 f (x)]dx = 7.( pt) Z 7 Z 9 Z b f (x) f (x) = f (x) 4 4 a where a = and b = 4, 8.( pt) Let Z 3 Find and Z Z f (x)dx = 9. f (x)dx = Z 3 0 (9 f (x) 4)dx = f (x)dx = 9, Z 0 f (x)dx =
13 Math 0-5, Fall 006 Assignment 8 due November 0 Thought: THE END IS NEAR! Procrastination may have its own natural consequences..( pt) Evaluate the definite integral Z 5 (4x + 0)dx.( pt) Evaluate the definite integral Z 9 3 0x + dx x 3.( pt) Use part I of the Fundamental Theorem of Calculus to find the derivative of Z x ( ) 0 f (x) = 3 t dt 3 f (x) = [NOTE: Enter a function as your answer. Make sure that your syntax is correct, i.e. remember to put all the necessary *, (, ), etc. ] 4.( pt) If f (x) = Z x 0 (t 3 + 4t + 5)dt then f (x) = 5.( pt) NOTE: It will be easier to see the function f(x) 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. Let 0 if x < 3 5 if 3 x < 0 f (x) = if 0 x < 5 0 if x 5 and g(x) = Z x 3 f (t)dt Determine the value of each of the following: (a) g( 5) = (b) g( ) = (c) g() = (d) g(6) = (e) The absolute maximum of g(x) occurs when x = and is the value It may be helpful to make a graph of f(x) when answering these questions.
14 Math 0-5, Fall 006 Assignment 9 due December 6 Thought: THE END IS NEAR! Procrastination may have its own natural consequences..( pt) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. y = 4x,y = 3x.( pt) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. y = x,y = x ( pt) Find the volume of the solid obtained by rotating the triangular region bounded by the x-axis, the line y = 4x and the line x = 6 about the x-axis. Volume = 4.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = x,x = 0,x =, about the y-axis 5.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = x 6,y = ; about y = 7 6.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = x,y = 0,x = 0,x =, about the y-axis
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