Greetings AP Calculus AB Class,

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1 Greetings AP Calculus AB Class, I am ecstatic to have the opportunity to teach you all next school year. In fact, I am so excited, I have already compiled a homework assignment for you. Enclosed is said assignment. It consists of a few review problems to keep you fresh for next school year. They should be fairly familiar to you; however, do not fret if there is something you do not understand. I will be using this to gauge our starting point for the course. Since this is primarily for feedback purposes, I have included the answers at the end of the document. By all means, check your work, but please do not look at the answers before attempting the problems. You may check your graphs using Desmos (also available as an app). Feel free to work together if you would like, but everyone make their own answers. Note that the section in green is advanced content (for BC Calculus) so you do not have to attempt them unless you are interested in the challenge. Please attempt all the other problems showing your work and have ready a list of any problems that gave you any issue, and I will have some homework credit ready for you come August. I look forward to our epic mathematic journey and hope you do as well. Enjoy your summer, and feel free to contact me with any questions, comments, or concerns. Sincerely, --Dr. Hamilton mhamilton@parkerschoolhawaii.org

6 Chapter 1 1. (AB/BC, non-calculator) The function f is defined as follows: f( x) x 5x 6. x 7x 3 (a) State the value(s) of x for which f is not continuous. (b) Evaluate lim f ( x). Show the work that leads to your answer. x3 (c) State the equation(s) for the vertical asymptote(s) for the graph of y f( x). (d) State the equation(s) for the horizontal asymptote(s) for the graph of y f( x). Show the work that leads to your answer.

7 . (AB/BC, calculator neutral) y y x x Graph of f Graph of g The graphs of functions f and g are shown above. Evaluate each limit using the graphs provided. Show the computations that lead to your answer. (a) lim f( x) 4 (c) lim f ( x) g( x) x1 x (b) 5 gx lim x3 ( ) (d) f ( x) lim x3 gx ( ) 1 (Assume that f and g are linear on the interval [,3]).

8 3. (AB/BC, calculator neutral) A hot cup of tea is placed on a counter and left to cool. The temperature of the tea, in degrees Fahrenheit (correct to the nearest degree), x minutes after the cup is placed on the counter is modeled by a continuous function T( x) for 0 x 10. Values of T( x) at various times x are shown in the table below. x T( x ) (a) Evaluate: lim T( x). Justify your answer. x4 (b) Using the data in the table, find the average rate of change in the temperature of the tea for 3 x 8. Include units on your final answer. (c) Identify, using the times listed in the table, the shortest interval during which there must exist a time x for which the temperature of the tea is166.5? Justify your answer. (d) Use the data in the table to find the best estimate of the slope of the line tangent to the graph of T at 8 x.

9 4. (AB/BC, non-calculator) Find the value of each of these limits, or else explain why the limit does not exist. Show the computations which lead to your answers. (a) x x 4 3 3x 6x x x 1 lim x (b) lim x0 x 5 5 5x (c) cos x lim x0 xsin x (d) 5 5 lim x x x

10 5. (AB/BC, non-calculator) The position function st () 4.9t fallen from a height of meters after t seconds., gives the height (in meters) of an object that has (a) Explain why there must exist a time t, 1 t, at which the height of the object must be 38 meters above the ground. (b) Find the time at which the object hits the ground. (c) Find the average rate of change in s over the intervalt 8,9. Include units of measure. Explain why this is a good estimate of the velocity at which the object hits the ground. How can this estimate be improved? (d) Evaluate: st () s(3) lim. Show the work that leads to your answer. Include units. t3 t 3

11 6. (AB/BC, calculator neutral) Let a and b represent real numbers. Define ax x b x if f( x) axb if x5. ax 7 if x 5 (a) Find the values of a and b such that f is continuous everywhere. (b) Evaluate: lim f ( x). x3 (c) Let f ( x) gx ( ). x 1 Evaluate: lim gx ( ). x1

12 7. (AB/BC, calculator neutral) y x The graph of function g is shown above. Which of the following is true? I. lim gx ( ) 1 x II. lim gx ( ) g() x III. g is continuous at x 3. (a) I only (b) I and II only (c) I and III only (d) III only (e) I, II and III

13 8. (AB/BC, calculator neutral) y x The graph of the function f is shown above. The line x 1is a vertical asymptote. Which of the following statements about f is true? (a) lim x1 (b) lim 1 x3 (c) f x x3 x3 lim ( ) lim f( x) (d) lim f ( x) does not exist x4 (e) lim f ( x) lim f( x) x0 x3

14 9. (AB/BC, non-calculator) Define x 4x 3 if x, 4 x x8 f( x). 8 if x 4 Which of the following statements about f are true? I. f is not continuous at x 4. II. lim f( x) 4 x III. x 4 is a vertical asymptote of the graph of y f( x). (a) None (b) I only (c) I and II only (d) I and III only (e) I, II and III

15 10. (AB/BC, calculator neutral) y x The figure above shows three rectangles each with a vertex on the graph of of the areas of these rectangles is y 16 x. The sum (a) 4 sq. units (b) 40 sq. units (c) 34 sq. units (d) 33 sq. units (e) 9 sq. units

18 Chapter 1 (Solutions) Question 1 The function f is defined as follows: f( x) x 5x6. x x x (a) State the value(s) of x at which f is not continuous. (b) Evaluate lim f ( x). Show the work that leads to your answer. x3 (c) State the equation(s) for the vertical asymptote(s) for the graph of y f( x). (d) State the equation(s) for the horizontal asymptote(s) for the graph of y f( x). Show the work that leads to your answer. (a) f( x) x3x 1 3 x x x (b) f is discontinuous at x 1 lim x3 x ( x 1) 15 1 x ; x 3 and x 0 3: 1 per answer : 1:reduced fraction 1: answer (c) 1 x ; x 0 : 1 per answer

19 Question y y x x Graph of f Graph of g The graphs of functions f and g are shown above. Evaluate each limit using the graphs provided. Show the computations that lead to your answer. (a) lim f( x) 4 x1 (b) 5 lim x3 gx ( ) (c) lim f ( x) g( x) x (d) f ( x) lim x3 gx ( ) 1 (Assume that f and g are linear on the interval [,3]). (a) lim f( x) 4 x1 = 34 7 : 1:breaking up limit 1: answer (b) 5 lim x3 gx ( ) = : 1:breaking up limit 1: answer

20 Question 3 A hot cup of tea is placed on a counter and left to cool. The temperature of the tea, in degrees Fahrenheit (correct to the nearest degree), x minutes after the cup is placed on the counter is modeled by a continuous function T( x) for 0 x 10. Values of T( x) at various times x are shown in the table below. x T( x ) (a) Evaluate: lim T( x). Justify your answer. x4 (b) Using the data in the table, find the average rate of change in the temperature of the tea for 3 x 8. Include units on your final answer. (c) Identify, using the times listed in the table, the shortest interval during which there must exist a time x for which the temperature of the tea is 166.5? Justify your answer. (d) Use the data in the table to find the best estimate of the slope of the line tangent to the graph of T at x 8. (a) Since T is continuous for 0 x 10, lim T( x) T(4). x4 lim T( x) 17 : 1: justification x4 1: answer (b) T(8) T(3) F 8 3 min 1: setup 3: 1: answer 1: units

21 Question 4 Find the value of each of these limits, or else explain why the limit does not exist. Show the computations which lead to your answers. (a) 4 3 3x 6x x x1 lim x 3 4 x 9x 5 (b) lim x0 x 5 5 5x (c) cos x lim x0 xsin x (d) 5 5 lim x x x (a) 1 1: answer 3 (b) (c) (d) x 1 1 lim lim 3: x0 0 :simplify 5x x5 5 5 x : answer x sin x lim : 1: simplify x0 x 1: answer x lim lim x xx ( ) x x 4 3: : simplify 1: answer

22 The position function st () 4.9t Question 5 fallen from a height of meters after t seconds., gives the height (in meters) of an object that has (a) Explain why there must exist a time t, 1 t, at which the height of the object must be 38 meters above the ground. (b) Find the time at which the object hits the ground. (c) Find the average rate of change in s over the intervalt 8,9. Include units of measure. Explain why this is a good estimate of the velocity at which the object hits the ground. How can this estimate be improved? (d) Evaluate: st () s(3) lim. Show the work that leads to your answer. Include units. t3 t 3 (a) s(1) 38 s(). The function st ( ) is a polynomial, and is therefore continuous. Therefore by the Intermediate Value Theorem, 1: explanation there exists a value t,1t, for which st () 38. (b) (c) 0 4.9t t 9 s 1: answer s(9) s(8) m This is the average velocity on 98 sec the interval 8t 9. Since the object hits the ground at t 9, 1: answer 4: 1: units : 1 per explanation this average velocity is close to the instantaneous velocity s at t = 9. The estimate (9) s ( t ) can be improved by allowing the value of t to approach 9. 9 t

23 Question 6 Let a and b represent real numbers. Define ax x b if x f( x) axb if x5. ax 7 if x 5 (a) Find the values of a and b such that f is continuous everywhere. (b) Evaluate: lim f ( x). x3 (c) Let f ( x) gx ( ). x 1 Evaluate: lim gx ( ). x1 (a) 4 a b a b a ; b 3 5ab10a7 1:limits 4: 1:equations : answers (b) 9 : 1: correct interval 1: answer (c) x x3 lim 5 x1 x 1 : 1: correct interval 1: answer

24 Questions c g() 3 lim g( x) 1 x 8. e both of these limits equal 9. b lim f( x) x4 lim f ( x) 1 so the horizontal asymptote is y 1 x 10. c = 34

. Show the work that leads to your answer. (c) State the equation(s) for the vertical asymptote(s) for the graph of y f( x).

Chapter 1 1. (AB/BC, non-calculator) The function f is defined as follows: f( ) 5 6. 7 3 (a) State the value(s) of for which f is not continuous. (b) Evaluate f ( ). Show the work that leads to your answer.