AP Calculus AB/IB Math SL2 Unit 1: Limits and Continuity. Name:

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1 AP Calculus AB/IB Math SL Unit : Limits and Continuity Name: Block: Date:. A bungee jumper dives from a tower at time t = 0. Her height h (in feet) at time t (in seconds) is given by the graph below. In this problem, you may base your answers on estimates from the graph or use the fact that the jumper s height function is given by s(t) = 00 cos(0.75t) e 0.t (a) What is the change in vertical position of the bungee jumper between t = 0 and t = 5? (b) Estimate the jumper s average velocity on each of the following time intervals: [0, 5], [0, ], [, 6], and [8, 0]. Include units on your answers. (c) On what time intervals do you think the bungee jumper achieves her greatest average velocity? Why? (d) Estimate the jumper s instantaneous velocity at t = 5. Show your work and eplain your reasoning, and include units on your answer. (e) Among the average and instantaneous velocities you computed in earlier questions, which are positive and which are negative? What does negative velocity indicate?. A diver leaps from a meter springboard. His feet leave the board at time t = 0, he reaches his maimum height of 4.5 m at t =. seconds, and enters the water at t =.45. Once in the water, the diver coasts to the bottom of the pool (depth.5 m), touches bottom at t = 7, rests for one second, and then pushes off the bottom. From there he coasts to the surface, and takes his first breath at t =. (a) Let s(t) denote the function that gives the height of the diver s feet (in meters) above the water at time t. (Note that the height of the bottom of the pool is -.5 meters. Sketch a carefully labeled graph of s(t) on the provided aes below. Include scale and units on the vertical ais. Be as detailed as possible. (b) Based on your graph in (a), what is the average velocity of the diver between t =.45 and t = 7? Is his average velocity the same on every interval within [.45, 7]? (c) Let the function v(t) represent the instantaneous vertical velocity of the diver at time t (note that velocity in the upward direction is positive, while the velocity of a falling object is negative). Based on your understanding of the diver s behavior, as well as your graph of the position function, sketch a carefully labeled graph of v(t) on the aes provided above. Include scale and units on the vertical ais. Write several sentences that eplain how you constructed your graph, discussing when you epect v(t) to be zero, positive, negative, relatively large, and relatively small. (d) Is there a connection between the two graphs that you can describe? What can you say about the velocity graphy when the height function is increasing? decreasing? Make as many observations as you can.

2 AP Calculus AB/IB Math SL Unit : Limits and Continuity. According to the U.S. census, the population of the city of Grand Rapids, MI, was 8,84 in 980; 89,6 in 990; and 97,800 in 000. (a) Between 980 and 000, by how many people did the population of Grand Rapids grow? (b) In an average year between 980 and 000, by how many people did the population of Grand Rapids grow? (c) Just like we can find the average velocity of a moving body by computing change in position over change in time, we can compute the average rate of change of any function f. In particular, the average rate of change of a function f over an interval [a, b] is the quotient f(b) f(a). b a What does the quantity f(b) f(a) b a measure on the graph of y = f() over the interval [a, b]? (d) Let P (t) represent the population of Grand Rapids at time t, where t is measured in years from January, 980. What is the average rate of change of P on the interval t = 0 to t = 0? What are the units on this quantity? (e) If we assume the population of Grand Rapids is growing at a rate of approimately 4% per decade, we can model the poulation function with the formula P (t) = 884(.04) t/0. Use this formula to compute the average rate of change of the population on the intervals [5, 0], [5, 9], [5, 8], [5, 7], and [5, 6]. (f) How fast do you think the population of Grand Rapids was changing on January, 985? Said differently, at what rate do you think people were being added to the population of Grand Rapids as of January, 985? How many additional people should the city have epected in the following year? Why? 4. Evaluate each of the following limits using the graph pictured below. (a) (b) (c) 5 + (d) lim 4 f() = (e) (f) 5. Evaluate each of the following limits using the graph pictured below. (a) (b) (c) 4 4 lim + f() = (d) lim f() = (e) (f) 4

3 AP Calculus AB/IB Math SL Unit : Limits and Continuity 6. Consider the function whose formula is f() = (a) What is the domain of f? (b) Use a table of values of near a = to estimate the value of lim f(), if you think the limit eists. If you think the limit doesn t eist, eplain why. (c) Use algebra to simplify and hence work to evaluate lim f() eactly, if it eists, or to eplain how your work shows the limit fails to eist. Discuss how your findings compare to your results in (b). (d) True or false: f() = 8. Why? 6 (e) True or false: 4 4 = 4. Why? How is this connected to your work above with the function f? (f) Based on all of your work above, construct an accurate, labeled graph of y = f() on the interval [, ], and 6 write a sentence that eplains what you now know about lim Let g() = +. (a) What is the domain of g? (b) Use a sequence of values near a = to estimate the value of lim g(), if you think the limit eists. If you think the limit doesn t eist, eplain why. + (c) Use algebra to simplify the epression and hence work to evaluate + lim g() eactly, if it eists, or to eplain how your work shows the limit fails to eist. Discuss how your findings compare to your results in (b). (Hint: a = a whenever a 0, but a = a whenever a < 0.) (d) True or false: g( ) =. Why? (e) True or false: + + =. Why? How is this equality connected to your work above with the function g? (f) Based on all of your work above, construct an accurate, labeled graph of y = g() on the interval [ 4, ], and write a sentence that eplains what you now know about lim g(). 8. For each of the following prompts, sketch a graph on the provided aes of a function that has the stated properties. (a) y = f() such that f( ) = and f( ) = and f() is not defined and 0 f() = and lim f() does not eist (b) y = g() such that g( ) =, g( ) =, g() =, and g() = At =,, and, g has a limit, and its limit equals the value of the function at that point. g(0) is not defined and lim 0 g() does not eist.

4 AP Calculus AB/IB Math SL 9. lim 5 ( 5 + ) ln 0. lim e. lim α 5π tan α α 6. lim + 4. lim θ π (sin θ + cos θ) 4. lim lim lim + 7. lim 8. lim lim lim lim + 5. lim 0 sin. lim 0 + sin 4. lim 0 sin 5 sin 6 5. Assume that lim f() = 7 and lim g() =. (a) lim (f() + ) = (b) lim (f() g()) = (c) lim f () = g() (d) lim f() = (e) lim 5f() g() = (f) lim 8g() = 6. At what points c does lim f() eist? c, < (a) f() = +, >, (b) f() =, = Unit : Limits and Continuity Evaluate each limit. What do the results show about the eistence of a vertical or horizontal asymptote? 7. lim + 8. lim lim + 0. lim. lim ln( + ). lim 0 ln( + ). lim ln( + ) + 4. lim Evaluate lim + 6. Evaluate lim Evaluate lim Evaluate lim + 9. lim [ (0.4) 4] 40. lim [ (0.4) 4] [ ( 4. lim ) ] [ ) ] lim 4. lim 44. lim ( ( + ) ( e + ) ( 45. lim e + ) 46. lim [ ( ) ] + 4

5 AP Calculus AB/IB Math SL 47. Use the graphs of f and g to answer the following questions. Graph of f Graph of g 4 (a) lim (f() + g()) 0 (b) lim (f()g()) (c) lim (f() + g()) (d) lim (f() + g()) + (e) lim ( + (ln ) g()) (f) lim (f() g()) (g) lim g() f() (h) f() lim + g() (i) lim f() g() (j) lim + f() + g() (k) lim (f() + g()) 48. Find the average rate of change of the function over each interval. (a) f() = e ; [, 0] (b) f() = ln ; [, 4] (c) f() = cot ; [ π 4, π 4 ] 49. Use the difference quotient to find the slope of y = at the point = a. 50. y = ; a = (a) find the slope of the curve at = a (b) find an equation of the tangent line at = a (c) find an equation of the normal line at = a Unit : Limits and Continuity (d) sketch the curve, tangent line, and normal line 5. y = ; a = (a) find the slope of the curve at = a (b) find an equation of the tangent line at = a (c) find an equation of the normal line at = a (d) sketch the curve, tangent line, and normal line 5. y = ; a = 0 (a) find the slope of the curve at = a (b) find an equation of the tangent line at = a (c) find an equation of the normal line at = a (d) sketch the curve, tangent line, and normal line 5. The equation for free fall at the surface of Jupiter is s =.44t m with t in seconds. Assume a rock is dropped from the top of a 500-m cliff. Find the speed of the rock at t = sec. 54. At what point is the tangent to f() = + 4 horizontal? 55. A bungee jumper dives from a tower at time t = 0. Her height s in feet at time t in seconds is given by s(t) = 00 cos(0.75t) e 0.t (a) Write an epression for the average velocity of the bungee jumper on the interval [, + h]. (b) Use technology to estimate the value of the limit as h 0 of the quantity you found in (a). (c) What is the meaning of the value of the limit in b? What are its units? 56. Find any points of discontinuity and identify them as removable, jump, infinite, or oscillating. (a) y = ( + ) (b) y = + (c) y = (d) y = e /, < (e) f() = +, >, < (f) f() =, =, > (g) f() =, < + 5,, (h) f() =, = 5

6 AP Calculus AB/IB Math SL 57. Use the function f defined and graphed below to answer the following questions., < 0, 0 < < f() =, = + 4, < < 0, < < 60. Find a value a so that the function is continuous. f() = Unit : Limits and Continuity 4, < a, 6. For what values of k and m is the function g() everywhere continuous? Use limits to set up your work. k + m, < g() = e ln(+), k m, > (a) Is f continuous at =? Eplain. (b) Is f continuous at =? Eplain. (c) Is f continuous at =? Eplain. 58. Sketch a possible graph for a function f that has the stated properties. (a) f() eists but lim f() does not. (b) f( ) eists, lim lim f() + = f( ), but f() does not eist. (c) f(4) eistis, lim f() eists, but f is not continuous at = Find a value a so that the function, < f() = a, is continuous. 6. Show, using the intermediate value theorem, that the function f() = + has a zero on the interval [0, ]. 6. Show, using the intermediate value theorem, whether or not the function g(θ) = θ cos θ has a zero on the interval [0, π]. 64. Determine whether the IVT can be used to show = 0 on the interval [0, ]. If so, find the value of c guaranteed by the theorem. 65. Determine whether the IVT can be used to show + = 4 on the interval [0, ]. If so, find the value of c guaranteed by the theorem. 66. Determine whether the IVT can be used to show + = 6 on the interval [ 5, 4]. If so, find the value of c guaranteed by the theorem. 6