AP Calculus AB Chapter 1 Limits

Transcription

1 AP Calculus AB Chapter Limits SY: Mr. Kunihiro

2 . Limits Numerical & Graphical Show all of your work on ANOTHER SHEET of FOLDER PAPER. In Exercises and 2, a stone is tossed vertically into the air from ground level with an initial velocity of 5 m/s. Its height at time t is h t m. = 5t 4.9t 2. Compute the stone s average velocity over the time interval 0.5,2.5 and indicate the corresponding secant line on a sketch of the graph of h t. [ ] 2. Compute the stone s average velocity over the time intervals,.0,, [ ] [ 0.99,] [ 0.999,] [ ,],.000 and,,, and then estimate the instantaneous velocity at t =. (Round your answers to four decimal places.) In Exercise 3, use the following graph provided to help you answer the following questions. NOTE: PAY ATTENTION to the VOCABULARY! 3. The figure below shows the estimated number N of internet users in Chile, based on the data from the United Nations Statistics Division. (a) Estimate the rate of change of N at t = [ ] [,.00] (b) Does the rate of change increase or decrease as t increases? Explain graphically. [ ] (c) Let R be the average rate of change over 200,2005. Compute R. (d) Is the rate of change at t = 2002 greater than or less than the average rate R? Explain graphically.

3 In Exercises 4 and 5, find the following limits, or explain why they do not exist. 4 y y g(x) x 2 3. (a) lim g x ( x) (b) lim g x 2 ( x) (c) lim g x x 3 5 s s f(t) 2 0 t. (a) lim t 2 (b) t (c) t 0 In Exercise 6, use the graph below to answer the following limit questions. 6. Which of the following statements about the function y = f ( x) graphed here are true, and which are false? y y f(x) 2 x following statements about the fun (a) lim exists (b) (c) lim = 0 lim = (d) lim (e) x = lim x = 0 (f) lim exists at every point in the interval x x 0 x 0 (,)

4 .2 One Sided Limits Show all of your work on ANOTHER SHEET of FOLDER PAPER. In Exercises through 4, use the graph below to answer the following limit questions.. Which of the following statements about the function y = f ( x) graphed here are true, and which are false? y y f(x) 2 3 x 2 (a) lim x 2 does not exist. (b) lim x 2 = 2 (c) lim (d) lim f ( x) exists at every point x 0 in the interval (,). x x 0 (e) lim x x0 exists at every point x 0 in the interval (,3). x does not exist 2. Which of the following statements about the function y = f ( x) graphed here are true, and which are false? s 2 y y f(x) (a) lim x + (d) lim x = (b) lim = 2 (e) lim (g) lim f ( x) = lim + x 2 x + (h) lim does not exist (c) lim = (f) lim x c x x 2 = 2 does not exist exists at every c in the open interval (, ) (i) lim x c exists at every c in the open interval (,3) (j) lim f ( x) = 0 x (k) lim x 3 + f ( x ) does not exist x

5 3. Let f ( x) = 3 x, x < 2 x 2 +, x > 2 (a) Find lim x 2 + (b) Does lim and lim f ( x). x 2 exist? If so, what is it? If not, why not? and lim f ( x) x 4 + exist? If so, what is it? If not, why not? x 2 (c) Find lim x 4 (d) Does lim x 4 4. Let f ( x) = 0, x 0 sin x, x > 0 (a) Does lim + (b) Does lim (c) Does lim exist? If so, what is it? If not, why not? exist? If so, what is it? If not, why not? exist? If so, what is it? If not, why not?

6 5. (Calculator Use) Let f ( x) = x x (a) Graph f ( x) on the interval 3 x 3. Is f ( x) undefined within this interval? (b) Now find lim f ( x) & lim + 6. (Multiple Choice) f ( x ). (c) What is lim The graph of the function f is shown in the figure above. Which of the following statements about f is true? (A) lim f ( x) = lim f ( x) (B) lim f ( x) = 2 (C) lim f ( x) = 2 x a x a (D) lim = (E) lim x a does not exist

7 .3 Finding Limits Analytically (Part ) Show all of your work on ANOTHER SHEET of FOLDER PAPER. Multiple Choice x 2 + x 6. lim is x 2 2 x (A) 5 (B) 3 (C) 3 (D) 5 (E) DNE 2. lim x 9 x 5 2 x 9 is (A) 4 (B) 4 (C) (D) 0 (E) DNE 3. lim x 2 x 2 x 2 is (A) 4 (B) 4 (C) (D) (E) DNE 4. lim x tan x sin x + is (A) 0 (B) 4 (C) 2 (D) π 2 (E) π 2π + 4 For problems 5 & 6, use the table provided below. 5. Given the following selected values for continuous functions table below: ( ) f g x lim x 3 g is x g x and g( x) in the (A) 4 (B) 3 (C) (D) 3 (E) 4

8 6. lim x 2 ( ( x) ) is f g ( x) g f (A) 4 3 (B) (C) 3 4 (D) 3 (E) 4 Free Response Find the limits for each of the following. 7. lim 7 + sec 2 x 8. lim + x + sin x 3cos x x 4 9. lim x x 3 0. lim x 2 x + 2 x x 2 5. lim x 3 x + 3 4x x 2 2. lim x 4 2 x 3. Suppose lim (a) lim = 7 and lim ( + g( x) ) (b) lim g( x) = 3. Find g x (c) lim 4g( x) (d) lim f ( x) g( x) 4. The graphs of f ( x) = x, g( x) = x, and h( x) = x cos 50π x on the interval x are given at the right. Use the Squeeze Theorem to find lim x cos 50π x. Justify your answer. 5. If f ( x) x 2 + 2x + 2 for all x, find lim x 6. If 3cos( π x) f ( x) x 3 + 2, evaluate lim. Justify your answer. f ( x ). Justify your answer. x

9 .4 Finding Limits Analytically (Part 2) Show all of your work on ANOTHER SHEET of FOLDER PAPER. Multiple Choice sin2x. lim x cos x is (A) 0 (B) (C) 2 (D) 2 (E) DNE cos 2 x 2. lim 2xsin x is (A) (B) 2 (C) (D) 2 (E) 0 cot 6x 3. lim csc 3x is (A) 2 (B) 0 (C) 2 (D) 2 (E) DNE sinα cos x 4. lim x cosα sin x is (A) (B) cosα (C) sinα (D) sinα (E) DNE 5. lim x e x + x is (A) 0 (B) (C) 2 (D) DNE (E) None of the above 6. lim x 3 + x 2 x + 3 x 2 9 is (A) 0 (B) (C).5 (D).5 (E) DNE 7. lim tan 3x 2x is (A) 0 (B) 2 (C) 2 3 (D) 3 2 (E) None of the above

10 Free Response Evaluate the following limits. Note: the symbol x is the greatest integer function. 8. lim ( x + 3) x + 2 x 2 x + 2. lim x 4 2. lim x x 2x x 2 9. lim x + x 3 6x2 ( cot x) csc2x 0. lim x x 3 + x 3. lim tan x cos x 2x sin x 4. lim tan 3x csc8x 5. lim 5x 2 x 6. lim sin 3 x x 3 + cos x 7. Explain why lim x x does not exist.

11 .5 Limits Involving Infinity Show all of your work on ANOTHER SHEET of FOLDER PAPER. Multiple Choice. The graph of y = x2 9 3x 9 has (A) a vertical asymptote at x = 3 (B) a horizontal asymptote at y = 3 (C) a removable discontinuity at x = 3 (D) an infinite discontinuity at x = 3 (E) None of the above 2. Which statement is true about the curve y = (A) The line x = is a vertical asymptote. 4 (B) The line x = is a vertical asymptote. 2x x 4x 2? (C) The line y = 4 is a horizontal asymptote. (D) The graph has no vertical or horizontal asymptote. (E) The line y = 2 is a horizontal asymptote. 2x lim x 2 x ( 2 + x) is (A) 4 (B) 2 (C) (D) 2 (E) DNE 2 x 4. lim is x 2 x (A) (B) (C) 0 (D) (E) DNE 2 x 5. lim is x 2 x (A) (B) (C) 0 (D) (E) DNE 6. lim x 5x x 2 +0x + 9 is (A) (B) (C) 0 (D) 3 (E)

12 Free Response Evaluate the following limits. 2x lim 8. lim x x 3 x 2 + x + 7 x 9x 4 + x 2x 4 + 5x 2 x + 6 0x 5 + x lim x x 6 0. lim x 8x 2 3 2x 2 + x. lim x x 2 5x x 3 + x 2 x + x 4 2. lim x x 2 x 3 3x 3. lim x 5 2x lim sec x 5. lim arctan x x ( π 2) x 6. lim ( + csc x) 7. lim x lim ( x 7 x 5 ) x 5 Sketch the graph of a function y = f ( x) that satisfies the given conditions. No formulas are required just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so there are multiple solutions) 9. f ( 0) = 0, lim x ± = 0, lim + = 2, and lim f ( x) = f ( 2) =, f ( ) = 0, lim x = 0, lim f ( x) =, lim + =, and lim x =

13 .6 Continutiy & IVT Show all of your work on ANOTHER SHEET of FOLDER PAPER. Multiple Choice. Let g x be a continuous function. Selected values of g are given in the table below. What is the fewest number of times g x [ 3,0]? will intersect y = on the closed interval (A) None (B) One (C) Two (D) Three (E) Four 2. Let h( x) be a continuous function. Selected values of h are given in the table below. For which value of k will the equation h( x) = 2 3 closed interval [ 2, 7]? have at least two solutions on the (A) (B) 3 4 (C) 7 9 (D) 2 3 (E) 8 3. If = x +, x 3+ ax 2, x >, then f ( x) is continuous for all x if a =? (A) (B) (C) 2 (D) 0 (E) 2 4. If f ( x) = 2x + 5 x + 7, x 2 x 2, and if f is continuous at x = 2, then k =? k, x = 2 (A) 0 (B) 6 (C) 3 (D) (E) 7 5

14 5. Let f be the function defined by the following: sin x, x < 0 x f ( x) 2, 0 x < = 2 x, x < 2 x 3, x 2 For what values o is f NOT continuous? (A) 0 only (B) only (C) 2 only (D) 0 and 2 only (E) 0,, and 2 [ ]. If f ( 3) = and = 3, then the Intermediate Value Theorem guarantees that 6. Let f be a continuous function on the closed interval 3,6 f 6 (A) f ( 0) = 0 (B) The slope of the graph of f is 4 9 somewhere between 3 and 6 3 for all x between 3 and 6 = for at least one c between 3 and 6 = 0 for at least one c between and 3 (C) (D) f c (E) f c 7. Let f be the function given by a is f continuous for all real numbers x? ( = x ) x2 4 x 2 a. For what positive values of (A) None (B) only (C) 2 only (D) 4 only (E) and 4 only 8. If f is continuous on 4, 4 must be true? (A) f 0 = 0 = 8 (B) lim x 2 (C) There is at least one c 4,4 = lim [ ] such that f ( 4) = and f ( 4) =, then which [ ] such that f ( c) = 8 (D) lim x 3 x 3 (E) It is possible that f is not defined at x = 0 Free Response 9. A toy car travels on a straight path. During the time interval 0 t 60 seconds, the toy car s velocity v, measured in feet per second, is a continuous function. Selected values are given below:

15 For 0 < t < 60, must there be a time when v( t) = 2? Justify. ( x 2) 0. For the function f ( x) 2, x = 4 = 5, 4 < x 0 IVT guarantee a y -value u on 4 x 0 such that f 4 not? Sketch the graph of. Find f ( 4) and f ( 0). Does the < u < f 0 for added visual proof.? Why or why. The functions f and g are continuous for all real numbers. The table below gives values of the functions at selected values o. The function h is given by h x + 2. = g f ( x) Explain why there must be a value w for < w < 5 such that h( w) = The functions f and g are continuous for all real numbers. The function h is x. The table below gives values of the functions at given by h( x) = f g( x) selected values o. Explain why there must be a value u for < u < 4 such that h( u) =.