Arnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment LimitsRates0Theory due 01/01/2006 at 02:00am EST.

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1 Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment LimitsRates0Theory due 0/0/006 at 0:00am EST.. ( pt) rochesterlibrary/setlimitsrates0theory/c3sp.pg 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 examples, especially examples 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 sequence which converges is either an increasing sequence or a decreasing sequence.. Every bounded sequence has an accumulation point. 3. Every bounded sequence converges to a it point. 4. The sequence of rational numbers 3., 3.4, 3.4, 3.459,... which approximates the ratio of the circumference of a circle and its diameter, has a rational number as its it point.. ( pt) rochesterlibrary/setlimitsrates0theory/c3sp.pg 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 examples, especially examples 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 [3,5] must have a minimum.. Every differentiable function on the interval ( 4, 0] must have both a maximum and a minimum. 3. Every continuous function on the interval (0, ] must have both a maximum and a minimum. 4. Every function on the interval (0,) must have both a maximum and a minimum. 3. ( pt) rochesterlibrary/setlimitsrates0theory/c3sp3.pg 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 examples, especially examples 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 and which has a maximum value also has a minimum value.. If f (x) is a continuous function and the sequence f (a ), f (a ), f (a 3 ),... converges to a finite it, then the sequence a,a,a 3,... also converges to a it. 4. ( pt) rochesterlibrary/setlimitsrates0theory/c3sp4.pg 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 examples, especially examples 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 whose domain is a bounded, closed interval and which has a maximum value also has a minimum value.. Every continuous function whose domain is a bounded, closed interval and which has a maximum value also has a minimum value. 3. Every continuous function is differentiable. 4. If a differentiable function has a maximum value then its domain must be a bounded, closed interval. 5. ( pt) rochesterlibrary/setlimitsrates0theory/c3sp5.pg 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 examples, especially examples 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 f (x) is a continuous function and the sequence f (a ), f (a ), f (a 3 ),... converges to a finite it, then the sequence a,a,a 3,... also converges to a it.. Every differentiable function has a maximum value. 3. If a continuous function has a maximum value then it also has a minimum value. 4. If a continuous function f (x) has a maximum value on an interval then the function f (x) has a minimum on that same interval. 5. Every differentiable function is continuous. 6. If the linear approximation of a differentiable function is constant at a point a then the function could be increasing near the point a. 7. Every continuous function whose domain is a bounded, closed interval has a maximum value. 8. If the linear approximation of a differentiable function is decreasing at a point a then the function could be constant near the point a. Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

2 Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment LimitsRatesTangentVelocity due 0/0/006 at 0:00pm EST.. ( pt) rochesterlibrary/setlimitsratestangentvelocity/ns 7 4.pg If the tangent line to y = f (x) at (-8, -7) passes through the point (-5, ), find A. f ( 8) = B. f ( 8) =. ( pt) rochesterlibrary/setlimitsratestangentvelocity/s.pg The point P(,3) lies on the curve y = x + x + 7. If Q is the point (x,x + x + 7), find the slope of the secant line PQ for the following values of x. If x =., the slope of PQ is: and if x =.0, the slope of PQ is: and if x =.9, the slope of PQ is: and if x =.99, the slope of PQ is: Based on the above results, guess the slope of the tangent line to the curve at P(,3). 3. ( pt) rochesterlibrary/setlimitsratestangentvelocity/s 3.pg The point P(5,7) lies on the curve y = x+. If Q is the point (x, x+), find the slope of the secant line PQ for the following values of x. If x = 5., the slope of PQ is: and if x = 5.0, the slope of PQ is: and if x = 4.9, the slope of PQ is: and if x = 4.99, the slope of PQ is: Based on the above results, guess the slope of the tangent line to the curve at P(5,7). 4. ( pt) rochesterlibrary/setlimitsratestangentvelocity/s 4.pg The point P(0.5,0) lies on the curve y = 5/x. If Q is the point (x,5/x), find the slope of the secant line PQ for the following values of x. If x = 0.6, the slope of PQ is: and if x = 0.5, the slope of PQ is: and if x = 0.4, the slope of PQ is: and if x = 0.49, the slope of PQ is: Based on the above results, guess the slope of the tangent line to the curve at P(0.5,0). 5. ( pt) rochesterlibrary/setlimitsratestangentvelocity/s 5.pg If a ball is thrown straight up into the air with an initial velocity of 70 ft/s, it height in feet after t second is given by y = 70t 6t. Find the average velocity for the time period begining when t = and lasting (i) 0. seconds (ii) 0.0 seconds (iii) 0.00 seconds Finally based on the above results, guess what the instantaneous velocity of the ball is when t =. 6. ( pt) rochesterlibrary/setlimitsratestangentvelocity/ns 5.pg A ball is thrown into the air by a baby alien on a planet in the system of Alpha Centauri with a velocity of 50 ft/s. Its height in feet after t seconds is given by y = 50t 5t. A. Find the average velocity for the time period beginning when t= 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=. 7. ( pt) rochesterlibrary/setlimitsratestangentvelocity/s a.pg The experimental data in the table below define y as a function of x. x y A. Let P be the point (3,.3). Find the slopes of the secant lines PQ when Q is the point of the graph with x coordinate x. x slope B. Draw the graph of the function for yourself and estimate the slope of the tangent line at P. 8. ( pt) rochesterlibrary/setlimitsratestangentvelocity- /ns 5a.pg 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 0.8 seconds after the ball has been thrown. The velocity at 0.8 = You can use a calculator t f(t) Enter t result: f (t) Enter t result: f (t) Enter 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).

3 9. ( pt) rochesterlibrary/setlimitsratestangentvelocity/s 8.pg 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= Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

4 Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment LimitsRates 5Graphs due 0/0/006 at 03:00pm EST.. ( pt) rochesterlibrary/setlimitsrates 5Graphs/ur lr -5.pg Let F be the function below. 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 = x 0.63 f (0.63) = = x Are all of these values the same?: (Y or N). If so then the function is continuous at 0.63 Are the left and right its the same at 0.63?: (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 (0.63). x f(x) Enter x result: f (x) Enter x result: f (x) Enter x result: f (x) 3. ( pt) rochesterlibrary/setlimitsrates 5Graphs/ur lr -5 3.pg 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) =. ( pt) rochesterlibrary/setlimitsrates 5Graphs/ur lr -5.pg Below is an oracle function. An oracle function is a function presented interactively. When you type in an x value, and press the f > button and the value f (x) appears in the right hand f(x) g(x) 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 exist (or if it s infinity).. f (g(x))] x [. f (x)g(x)] x +[ 3. f (x)g(x)] x [ 4. f (g(x))] x +[ Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

5 Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment LimitsRatesLimits due 0/0/006 at 0:00am EST.. ( pt) rochesterlibrary/setlimitsrateslimits/s 3.pg The slope of the tangent line to the graph of the function y = 5x 3 5x at the point (3,35) is 3 35 x 3 x 3. By trying values of x near 3, find the slope of the tangent line.. ( pt) rochesterlibrary/setlimitsrateslimits/s 3 3.pg Evaluate the it x 7 (4x + 8)(7x + 8) 3. ( pt) rochesterlibrary/setlimitsrateslimits/s 3 4.pg Evaluate the it 5(4x + 3)3 x 3 4. ( pt) rochesterlibrary/setlimitsrateslimits/s 3 5.pg Evaluate the it x 4 x 4x 5x + 6. ( pt) rochesterlibrary/setlimitsrateslimits/s 3.pg Evaluate the it a 3 a a. ( pt) rochesterlibrary/setlimitsrateslimits/s 3 7.pg Evaluate the it 5 t 5 t t 5 3. ( pt) rochesterlibrary/setlimitsrateslimits/s 3 36.pg Evaluate the it b 8 b 8 b 8 5. ( pt) rochesterlibrary/setlimitsrateslimits/s 3 6.pg Evaluate the it 8x 7x + 5 x 3 x 6 6. ( pt) rochesterlibrary/setlimitsrateslimits/s 3.pg Evaluate the it 5(y ) y 7y (y ) 3 7. ( pt) rochesterlibrary/setlimitsrateslimits/s 3 6.pg Evaluate the it x + 7x + 7 x 9 x ( pt) rochesterlibrary/setlimitsrateslimits/s 3 7.pg Evaluate the it x + 0x + 00 x 0 x ( pt) rochesterlibrary/setlimitsrateslimits/s 3 8.pg Evaluate the it x 4 x 4 x + x 4 0. ( pt) rochesterlibrary/setlimitsrateslimits/s 3 9.pg Evaluate the it x 3 x x x 4. ( pt) rochesterlibrary/setlimitsrateslimits/s 3 48.pg Evaluate the it s + 9 s 9 s ( pt) rochesterlibrary/setlimitsrateslimits/s 3 48a.pg Evaluate the its. If a it does not exist, enter DNE. x + 3 x 3 + x + 3 = x + 3 x 3 x + 3 = x 3 x + 3 x + 3 = 6. ( pt) rochesterlibrary/setlimitsrateslimits/ns x.pg Let x + 4 if x 4 f (x) = 4 if x > 4 Sketch the graph of this function for yourself and find following its if they exist (if not, enter DNE).. f (x) x 4. f (x) x f (x) x 4

6 7. ( pt) rochesterlibrary/setlimitsrateslimits/ns xx.pg Let 9 if x > 6 6 if x = 6 f (x) = x + 9 if 8 x < 6 7 if x < 8 Sketch the graph of this function and find following its if they exist (if not, enter DNE).. f (x) x 6. f (x) x f (x) x 6 4. f (x) x 8 5. f (x) x f (x) x 8 g(x) 4. x a f (x) g(x) 5. x a h(x) h(x) 6. x a g(x) 7. f (x) x a 8. f (x) x a 9. x a f (x) h(x) 0. ( pt) rochesterlibrary/setlimitsrateslimits/ur lr 0.pg 8. ( pt) rochesterlibrary/setlimitsrateslimits/ns 6.pg f(x) g(x) 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 exist (or if it s infinity).. x 3 [ f (x) + g(x)]. x 3 +[ f (g(x))] 3. x [ f (x)g(x)] 4. f (3) + g(3). ( pt) rochesterlibrary/setlimitsrateslimits/ur lr 0b.pg Determine the its for the function f at.69. x.69 f (x) = f (.69) = x.69 + f (x) = Is this function continuous at.69?: (Y or N) Can this function be made continuous by changing its value at.69?: (Y or N) 9. ( pt) rochesterlibrary/setlimitsrateslimits/ns 3.pg Let g(x) = 5, f (x) = 0, h(x) = 5. x a x a x a Find following its if they exist. If not, enter DNE ( does not exist ) as your answer.. g(x) + f (x) x a. g(x) f (x) x a 3. g(x) h(x) x a a f (x) x a DNE 0 f (x) DNE x a + f 3 0 g(x) DNE 0 0 x a g(x) 3 0 DNE x a + g Using the table above calcuate the its below. Enter DNE if the it doesn t exist OR if it can t be determined from the information given.. f (g(x))] x 3 +[. f (g(3)) 3. f (x)g(x)] x 3 [ 4. f (x)g(x)] x 3 +[

7 . ( pt) rochesterlibrary/setlimitsrateslimits/ns pg Evaluate x (x )4 ( 5x ). Enter the letters corresponding to the Limit Laws that you used to find this it: Limit Laws A. Constant Multiple Law B. Difference Law C. Power Law D. Product Law E. Quotient Law F. Root Law G. Sum Law If 3. ( pt) rochesterlibrary/setlimitsrateslimits/ur lr.pg 9x f (x) x + 5x 8 determine x f (x) = What theorem did you use to arrive at your answer? 4. ( pt) rochesterlibrary/setlimitsrateslimits/ns 3 8.pg Use factoring to calculate this it a 4 b 4 a b a 5 b 5 If you want a hint, try doing this numerically for a couple of values of a and b. 5. ( pt) rochesterlibrary/setlimitsrateslimits/c0s5p.pg 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.. 3, 3, 3,.... the sequence generated by f (h) where h is a sequence of positive numbers approaching zero and f (x) = tan(x)/x. 3. the sequence generated by f (h) where h is a sequence of negative numbers approaching zero and f (x) = x + 3 if x is greater than or equal to 0 and f (x) = x 3 if x is less than zero. 4. the sequence generated by f (h) where h is a sequence of positive numbers approaching zero and f (x) = x + if x is greater than or equal to 0 and f (x) = x + if x is less than zero. 6. ( pt) rochesterlibrary/setlimitsrateslimits/c0s5p.pg What is the it of the sequence f (k) generated by the sequence k =,,3,4,5... when f (x) = (4.5x 35.5)(9.3x + 8.7) 4.3x 8.9? 7. ( pt) rochesterlibrary/setlimitsrateslimits/c0s5p6.pg Find an integer which is the it of cos(x) x 4 as x goes to 0. (Enter I for infinity or DNE for does not exist.) You should also try using identities to transform the expressions algebraically so that you can identify the its without using a calculator. 8. ( pt) rochesterlibrary/setlimitsrateslimits/ur lr 4.pg 6 x x, if x Let f (x) = x 3, if x > Calculate the following its. Enter 000 if the it does not exist. f (x) = f (x) = f (x) = x x + x 9. ( pt) rochesterlibrary/setlimitsrateslimits/ur lr 5.pg 4 x +, if x 5 Let f (x) =, if x = 5 3x + 7, if x > 5 Calculate the following its. Enter 000 if the it does not exist. f (x) = f (x) = f (x) = x 5 x 5 + x ( pt) rochesterlibrary/setlimitsrateslimits/ur lr 6.pg 3 Let f (x) = x+, if x < x + 7, if x > Calculate the following its. Enter 000 if the it does not exist. f (x) = f (x) = f (x) = x x + x 3. ( pt) rochesterlibrary/setlimitsrateslimits/ur lr 7.pg Let f (x) = x 7x+6 x +5x 6. Calculate x f (x) by first finding a continuous function which is equal to f everywhere except x =. x f (x) = 3. ( pt) rochesterlibrary/setlimitsrateslimits/ur lr 8.pg Let f (x) = x+6 x x 5. Calculate f (x) by first finding a continuous function which x 3 is equal to f everywhere except x = 3. x 3 f (x) = 3

8 33. ( pt) rochesterlibrary/setlimitsrateslimits/ur lr 9.pg Let f = 64 b 8. b Calculate f by first finding a continuous function which b 64 is equal to f everywhere except b = 64. b 64 f = 34. ( pt) rochesterlibrary/setlimitsrateslimits/ur lr.pg Let f (s) = s 6 s 6 Calculate s 6 f (s) by first finding a continuous function which is equal to f everywhere except s = 6. s 6 f (s) = 35. ( pt) rochesterlibrary/setlimitsrateslimits/ur lr 3.pg The main theorem of Ste.3 tells us that many functions are continuous so that their its can be evaluated by direct substitution. Calculate the following its by direct substitution, making use of this big theorem from Ste.3. x 3 x3 4x 0 = ( t)(t + 5) t 5 3t 7 a 0 (a + 7) 4 a + = b 7 = y y3 (5 3y ) = 4 b + (b 4) = a 3a + 4 = a a ( pt) rochesterlibrary/setlimitsrateslimits/ur lr 4.pg Let f (t) = 6 t t Calculate t f (t) by first finding a continuous function which is equal to f everywhere except t =. t f (t) = Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 4

9 Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment LimitsRates3Infinite due 0/03/006 at 0:00am EST.. ( pt) rochesterlibrary/setlimitsrates3infinite/s3 5.pg Evaluate the it 4 + 7x x 3 4x. ( pt) rochesterlibrary/setlimitsrates3infinite/s3 5 3.pg Evaluate the it x + 7 x 8x 9x ( pt) rochesterlibrary/setlimitsrates3infinite/s3 5 4.pg Evaluate the it 8x 3 6x x x 0 9x 8x 3 4. ( pt) rochesterlibrary/setlimitsrates3infinite/s3 5 4a.pg Evaluate the it 3x 3 + 5x 4x x 6x 3 5x + 5. ( pt) rochesterlibrary/setlimitsrates3infinite/s3 5 5.pg Evaluate the it (5 x)( + 6x) x (3 6x)(0 + 4x) 6. ( pt) rochesterlibrary/setlimitsrates3infinite/s3 5.pg Evaluate the it 3 + 7x x (0 + 0x) 7. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3.pg Evaluate x 4 9x 3 x 5x + 8. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3.pg Evaluate t 9 t t 8t ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 3.pg The vertical asymptote of the curve y = 3x3 x 6 is given by x =. The horizontal asymptotes of the curve 7x y = (x 4 + ) 4 is given by y = and y = (enter these two values in ascending order). 0. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 3vert.pg The vertical asymptote of the curve is given by x = y = 0x3 x 0. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 3horiz.pg The horizontal asymptotes of the curve are given by y = y = where y > y. and 7x y = (x 4 + ) 4. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 4.pg Evaluate x 3x + x x 3. ( pt) rochesterlibrary/setlimitsrates3infinite/ns xxx.pg Determine the infinite it of the following functions. Enter INF for and. x 3 + x 3. x 3 x 3 3. x 7 x (x + 7) 4. x 5 (x 5) 6

10 4. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 5.pg Evaluate the following its. If needed, enter INF for and x + 3x x = 8. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 9.pg Evaluate the following its. If needed, enter INF for and 6x 3 6x 6x x 7 3x 4x 3 = x 3x x = 6x 3 6x 6x 7 3x 4x 3 = 5. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 6.pg Evaluate the following its. 5 x e x + 6 = 5 e x + 6 = [NOTE: If needed, enter INF for and ] [HINT: Look at where the exponential fuction is going in the fraction. If you need a reminder, look up infinite its in Section.5 (in particular, see pg 38-39).] 6. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 7.pg Evaluate the following its. If needed, enter INF for and 3 + 3x x 3 x = 3 + 3x 3 x = 7. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 8.pg Evaluate the following its. If needed, enter INF for and 4x + 6 x x 7x + 7 4x + 6 x 7x ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 0.pg Evaluate the following its. If needed, enter INF for and (0 x)(5 + 9x) x (3 x)( + 6x) = (0 x)(5 + 9x) (3 x)( + 6x) = 0. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3.pg Evaluate the following its. If needed, enter INF for and 0 + 6x x 8 + x = 0 + 6x = 8 + x. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3.pg Evaluate the following its. If needed, enter INF for and x x 4 + 7x 3 0 3x = x 4 + 7x 3 0 3x =

11 . ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 3.pg Evaluate the following its. If needed, enter INF for and ( ) x 9x + x = x ( ) x 9x + x = 3. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 4.pg Evaluate the following its. If needed, enter INF for and ( 0x + 3x 3) = x ( 0x + 3x 3) = 4. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 5.pg A function is said to have a vertical asymptote wherever the it on the left or right (or both) is either positive or negative infinity. For example, the function f (x) = x+ has a vertical asymptote (x 7) at x = 7. For each of the following its, enter either P for positive infinity, N for negative infinity, or D when the it simply does not exist. x + x 7 (x 7) = x + x 7 + (x 7) = x + x 7 (x 7) = 5. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 6.pg A function is said to have a vertical asymptote wherever the it on the left or right (or both) is either positive or negative infinity. For example, the function f (x) = 3(x+4) has a vertical asymptote at x = x +x 8. For each of the following its, enter either P for positive infinity, N for negative infinity, or D when the it simply does not exist. 3(x + 4) x x + x 8 = 3(x + 4) x + x + x 8 = 3(x + 4) x x + x 8 = 6. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 7.pg A function is said to have a vertical asymptote wherever the it on the left or right (or both) is either positive or negative infinity. For example, the function f (x) = 5 has a vertical asymptote at x = x 3 (x+8) 4 8. For each of the following its, enter either P for positive infinity, N for negative infinity, or D when the it simply does not exist. 5 x 3 (x + 8) 4 = 5 x 3 (x + 8) 4 = 5 x 3 (x + 8) 4 = x 8 x 8 + x 8 7. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 8.pg A function is said to have a horizontal asymptote if either the it at infinity exists or the it at negative infinity exists. Show that each of the following functions has a horizontal asymptote by calculating the given it. 4x x 7 + x = 5x 4 x 3 + 4x 4 = x 7x 9 x 3 x = x + 3x x 9x = x + 3x 9x = 8. ( pt) rochesterlibrary/setlimitsrates3infinite/ur lr 3 9.pg A function is said to have a horizontal asymptote if either the it at infinity exists or the it at negative infinity exists. Show that each of the following functions has a horizontal asymptote by calculating the given it. 4 + x x x 7x + 4 = 5 9x + x + 9x + 5 (5x 6) = x + 6 x 5x 4 x 4 = x + x 9 x = x x + x 9 + x = Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 3

12 Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment LimitsRates5Continuity due 0/05/006 at 0:00am EST.. ( pt) rochesterlibrary/setlimitsrates5continuity/s 5 37.pg For what value of the constant c is the function f continuous on (, ) where cx + if x (,9] f (x) = cx if x (9, ). ( pt) rochesterlibrary/setlimitsrates5continuity/s 5 38.pg For what value of the constant c is the function f continuous on (, ) where t c if t (,8) f (t) = ct + 7 if t [8, ) c = 3. ( pt) rochesterlibrary/setlimitsrates5continuity/csp3.pg 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 (x )sin( x ) when x < sin(x ) x + when x < cos(xπ) x+ when x < x 3 x when x < A. x + 4 when x > sin(πx) B. x + when x > C. when x > cos(x ) D. when x > x 4. ( pt) rochesterlibrary/setlimitsrates5continuity/csp3a.pg 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 sin(x ) x + when x < x 3 x when x < x x when x < cos(xπ) x+ when x < A. when x > x B. 4x+3 x when x > C. x + 4 when x > x D. x when x > 5. ( pt) rochesterlibrary/setlimitsrates5continuity/cs5p5.pg The function f is given by the formula when x < 3 and by the formula f (x) = 5x3 5x + 9x 7 x 3 f (x) = x 3x + a when 3 x. What value must be chosen for a in order to make this function continuous at 3? a = 6. ( pt) rochesterlibrary/setlimitsrates5continuity/ur lr 5.pg A function f (x) is said to have a removable discontinuity at x = a if:. f is either not defined or not continuous at x = a.. f could either be defined or redefined so that the new function IS continuous at x = a. Let f (x) = x +3x 44 x 4 Show that f (x) has a removable discontinuity at x = 4 and determine what value for f (4) would make f (x) continuous at x = 4. Must define f (4) =. 7. ( pt) rochesterlibrary/setlimitsrates5continuity/ur lr 5.pg A function f (x) is said to have a removable discontinuity at x = a if:. f is either not defined or not continuous at x = a.. f could either be defined or redefined so that the new function IS continuous at x = a. 3x + x+ x(x 4), if x 0,4 Let f (x) =, if x = 0 Show that f (x) has a removable discontinuity at x = 0 and determine what value for f (0) would make f (x) continuous at x = 0. Must redefine f (0) =. Hint: Try combining the fractions and simplifying. The discontinuity at x = 4 is actually NOT a removable discontinuity, just in case you were wondering. 8. ( pt) rochesterlibrary/setlimitsrates5continuity/ur lr 5 3.pg A function f (x) is said to have a removable discontinuity at x = a if:. f is either not defined or not continuous at x = a.. f could either be defined or redefined so that the new function IS continuous at x = a.

13 x + x + 39, if x < 6 Let f (x) = 0, if x = 6 x x 33, if x > 6 Show that f (x) has a removable discontinuity at x = 6 and determine what value for f ( 6) would make f (x) continuous at x = 6. Must redefine f ( 6) =. Now for fun, try to graph f (x). It s just a couple of parabolas! 9. ( pt) rochesterlibrary/setlimitsrates5continuity/ur lr 5 4.pg A function f (x) is said to have a jump discontinuity at x = a if:. f (x) exists. x a. f (x) exists. x a + 3. The left and right its are not equal. 5x 8, if x < 0 Let f (x) = 5 x+7, if x 0 Show that f (x) has a jump discontinuity at x = 0 by calculating the its from the left and right at x = 0. f (x) = x 0 f (x) = x 0 + Now for fun, try to graph f (x). 0. ( pt) rochesterlibrary/setlimitsrates5continuity/ur lr 5 5.pg A function f (x) is said to have a jump discontinuity at x = a if:. f (x) exists. x a. f (x) exists. x a + 3. The left and right its are not equal. x + 4x + 3, if x < Let f (x) = 5, if x = x + 3, if x > Show that f (x) has a jump discontinuity at x = by calculating the its from the left and right at x =. f (x) = x f (x) = x + Now for fun, try to graph f (x).. ( pt) rochesterlibrary/setlimitsrates5continuity/ur lr 5 6.pg 5x 6, if x 4 Let f (x) = 7x + b, if x > 4 If f (x) is a function which is continuous everywhere, then we must have b = Now for fun, try to graph f (x).. ( pt) rochesterlibrary/setlimitsrates5continuity/ur lr 5 6b.pg mx 8, if x < 3 Let f (x) = x + x 5, if x 3 If f (x) is a function which is continuous everywhere, then we must have m = Now for fun, try to graph f (x). 3. ( pt) rochesterlibrary/setlimitsrates5continuity/ur lr 5 7.pg x + b, if x < 5 Let f (x) = 50 x b, if x 5 There are exactly two values for b which make f (x) a continuous function at x = 5. The one with the greater absolute value is b = Now for fun, try to graph f (x). 4. ( pt) rochesterlibrary/setlimitsrates5continuity/ur lr 5 8.pg Find c such that the function x f (x) = 3, x c 6x, x > c is continuous everywhere. c = Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

14 Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment LimitsRates6Rates due 0/06/006 at 0:00am EST.. ( pt) rochesterlibrary/setlimitsrates6rates/s 6.pg The slope of the tangent line to the parabola y = x + 6x + 4 at the point (0,4) is: The equation of this tangent line can be written in the form y = mx + b where m is: and where b is:. ( pt) rochesterlibrary/setlimitsrates6rates/s 6.pg The slope of the tangent line to the curve y = 3x 3 at the point (, 3) is: The equation of this tangent line can be written in the form y = mx + b where m is: and where b is: 3. ( pt) rochesterlibrary/setlimitsrates6rates/s 6 3.pg The slope of the tangent line to the curve y = x at the point (,.884) is: The equation of this tangent line can be written in the form y = mx + b where m is: and where b is: 4. ( pt) rochesterlibrary/setlimitsrates6rates/s 6 4.pg The slope of the tangent line to the curve y = 3 x at the point (7,0.486) is: The equation of this tangent line can be written in the form y = mx + b where m is: and where b is: 5. ( pt) rochesterlibrary/setlimitsrates6rates/s 6 8.pg The slope of the tangent line to the parabola y = 4x 7x + 5 at the point where x = 0 is: The equation of this tangent line can be written in the form y = mx + b where m is: and where b is: 6. ( pt) rochesterlibrary/setlimitsrates6rates/s 6.pg 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 = ( pt) rochesterlibrary/setlimitsrates6rates/s 6.pg If an arrow is shot straight upward on the moon with a velocity of 79 m/s, its height (in meters) after t seconds is given by s(t) = 79t 0.83t. What is the velocity of the arrow (in m/s) after 6 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) rochesterlibrary/setlimitsrates6rates/s 6 4.pg The displacement (in meters) of a particle moving in a straight line is given by s = t 3 where t is measured in seconds. Find the average velocity of the particle over the time interval [6, 9]. Find the (instantaneous) velocity of the particle when t = ( pt) rochesterlibrary/setlimitsrates6rates/csp.pg Let p(x) = 5.7x Use a calculator or a graphing program to find the slope of the tangent line to the point (x, p(x)) when x =.8. Give the answer to 3 places. 0. ( pt) rochesterlibrary/setlimitsrates6rates/ur lr 6.pg A rock is thrown off of a 00 foot cliff with an upward velocity of 35 m/s. As a result its height after t seconds is given by the formula: h(t) = t 5t What is its height after 4 seconds? What is its velocity after 4 seconds? (Positive velocity means it is on the way up, negative velocity means it is on the way down.). ( pt) rochesterlibrary/setlimitsrates6rates/ur lr 6.pg The following chart shows living wage jobs in Rochester per 000 working age adults over a 5 year period. Year Jobs What is the average rate of change in the number of living wage jobs from 997 to 999? Jobs/Year What is the average rate of change in the number of living wage jobs from 999 to 00? Jobs/Year Based on these two answers, should the mayor from the last two years be reelected? (These numbers are made up. Please do not actually hold the mayor accountable.) Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester