Name AP Calculus BC Summer Review Packet (Limits & Derivatives) Limits 1. Answer the following questions using the graph of ƒ() given below. (a) Find ƒ(0) (b) Find ƒ() (c) Find f( ) 5 (d) Find f( ) 0 (e) Find f( ) (f) Find f( ) (g) List all -values for which ƒ() has a removable discontinuity. Eplain what section(s) of the definition of continuity is (are) violated at these points.
(h) List all -values for which ƒ() has a nonremovable discontinuity. Eplain what section(s) of the definition of continuity is (are) violated at these points. In problems -10, find the it (if it eists) using analytic methods (i.e. without using a calculator).. 10 8. 1 cos 4 /6 4. 4 1 4 5. 1/ ( 1) 1 0
6. 1/ 1 1 0 7. sin 6 0 7 8. sin t t 0 t 9. 6 6 6 6 10. sin(( / 6) ) (1/ ) 0 Hint: sin( ) sin cos cos sin
11. Suppose 1, 0 f( ) 1. 4 k, 0 (a) For what value of k will f be piecewise continuous at = 0? Eplain why this is true using one-sided its. (Hint: A function is continuous at =c if (1) f(c) eists, () f( ) eists, and () f ( ) f ( c ).) c c (b) Using the value of k that you found in part (a), accurately graph f below. Approimate the value of f( ) 1 f( ) 1 (c) Rationalize the numerator to find f( ) analytically. 1
1. Analytically determine the values of b and c such that the function f is continuous on the entire real number line. See the hint given in problem 11. f( ) 1,1 b c, 1 or In problem 1, circle the correct answer and eplain why the answer is the correct one. 1. If f ( ), and if c is the only real number such that f(c) = 0, then by the Intermediate Value Theorem, c is necessarily between (A) - and -1 (B) -1 and 0 (C) 0 and 1 (D) 1 and (E) and Hint: The Intermediate Value Theorem states that if f is a continuous function on the interval [a, b] and k is any number between f(a) and f(b), then there must eist at least one number c [ a, b] such that f(c) = k.
Derivatives In problems 1 &, find the derivative of the function by using the it definition of the derivative. 1. f ( ). f( ) 1 1
In problems -14, find the derivative of the given function using the power, product, quotient, and/or chain rules.. f ( ) ( 7)( ) 4. f ( ) sin 5. f ( t) t cos t 6. f( ) 1 1 7. 4 f( ) 8. tan f ( ) sec 9. f ( ) csc cot 10. 5 f( ) 6
11. f ( ) ( ) (5 1) 5 1. f 4 ( ) cot (7 ) 1. f 4 ( ) 5sin ( 1) Problems continue on the net page.
In problems 14 & 15, find an equation of the tangent line to the graph of f at the indicated point P. 14. 1 cos f ( ), P,1 1 cos 15. f ( ) 1, P(,4) In problems 16 & 17, find the second derivative of the given function. 16. f ( ) (4 ) 17. h ( ) cos( )
In problem 18, use the position function s( t) 16t v t s for free-falling objects. 0 0 18. A ball is thrown straight down from the top of a 0-foot tall building with an initial velocity of - feet per second. (a) Determine the average velocity of the ball on the interval [1, ]. (b) Determine the instantaneous velocity of the ball at t =. (c) Determine the time t at which the average velocity on [0, ] equals the instantaneous velocity. (d) What is the velocity of the ball when it strikes the ground?
In problem 19-4, circle the correct answer and eplain why the answer is the correct one. 19. cos h cos 6 6 h0 h (A) (B) (C) (D) (E) Does not eist 1 1
0. Let f and g be differentiable functions with values for f(), g(), f (), and g () shown below for = 1 and =. f() g() f () g () 1 4-4 1-8 5 1-6 4 Find the value of the derivative of f ( ) g( ) at = 1. (A) -96 (B) -80 (C) -48 (D) - (E) 0 1. Let 4, 1 f( ). Which of the following is true?, 1 I. f() is continuous at = 1 II. f() is differentiable at = 1 III. f ( ) f ( ) 1 1 (A) (B) (C) (D) (E) I only II only III only I and III only II and III only
. The equation of the line tangent to the curve the value of k? (A) - (B) -1 (C) 1 (D) (E) 4 k 8 f( ) k at = - is y = + 4. What is. An equation of the line normal to the curve y 1 at the point where = is (A) y1 8 (B) y4 10 (C) y 4 (D) y 8 (E) y 4 Hint: A normal line to a curve at a point is perpendicular to the tangent line to the curve at the same point.
( n) 4. If the nth derivative of y is denoted as y and y sin, then (14) y is the same as (A) (B) (C) (D) (E) y dy d d y d d y d None of the above
Answers Limits: 1. (a) -1 (b) (c) 0 (d) -1 (e) 1 (f) (g) = -5, 5 (h) = -, 0,. /4. /(8π) 4. 1/ 5. -1 6. -1/ 7. 0 8. 0 9. 6 10.
11. (a) k 1 (b).577 (c) 1 1. b = -, c = 4 1. D Derivatives: 1. f ( ). f( ) ( 1). 4. 5. 6. 7. 8. 9. f ( ) 1 18 14 sin f ( ) cos f ( t) t sin t t cos t 1 f( ) ( 1) 4 tan tan sec sec f( ) tan 4 f ( ) 9 sec tan 6sec f ( ) csc cot csc csc cot
10. 11. 1. 1. f( ) ( 10)( 10 0) ( 6 ) 9 f ( ) 5 (( )(5 1)) (5 1) ( ) 5 1 f 4 ( ) 8 cot (7 )csc (7 ) cot (7 ) f( ) 4 4 60 sin 1cos 1 4 1 14. y1 ( ) 15. y 4 ( ) 16. 17. f ( ) 1 4 (8 ) 4 4 h ( ) cos 6 sin 6cos 18. (a) -70 ft./s. (b) -118 ft./s. (c) t = 1 s. (d) 10.688 ft./s. 19. C 0. B 1. B. D. D 4. C