AP Calculus AB/BC ilearnmath.net

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1 CALCULUS AB AP CHAPTER 1 TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-8: Calculator, 5 minutes. Choose the letter that best completes the statement or answers the question..159t 1. A population grows according to the equation P() t = 6 55e for t. This population will approach a limiting value as time goes on. During which year will the population reach half its limiting value? (a) second (b) third (c) fourth (d) eighth (e) twenty-ninth sin,. Consider the function f ( ) = k, = f to be continuous, the value of k must be. In order for ( ) (a) (b) 1 (c) -1 (d) π (e) any number greater than 1 +. lim = (a) - (b) -1 (c) 1 (d) (e) noneistent. For all >, if f ( ln ) f =. =, then ( ) (a) e (b) ln (c) e (d) ln (e) = have for < 1? (a) One (b) Two (c) Three (d) Four (e) More than four 6. The function f ( ) = tan ( ) has one zero in the interval [,1. ]. The derivative at this point is. (a).11 (b) 1. (c).51 (d).76 (e) undefined 7. The table below gives the values of three functions f, g, and h near =. Based on the values given, for which of the functions does it appear that the limit as approaches zero is? f() g() h() undefined (a) f only (b) g only (c) h only (d) f and h only (e) f, g,and h 5. How many zeros does the function y sin ( ln ) 8. The graph of a function f whose domain is the closed interval [ 1, 7 ] is shown below. Which of the following statements about f ( ) is true? e (a) lim f ( ) = 1 (b) lim f ( ) = (c) ( ) (d) f ( ) is continuous at = 5 (e) lim f ( ) = f ( 6) 6 f is continuous at =

2 Numbers 9-15: NO calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. ( ) + h 9. The lim h h at the point = is. (a) 6 (b) 1 (c) 8 (d) (e) b 1. lim =. b b (a) b (b) b (c) b (d) b (e) b 11. If f ( ) = + and if c is the only real number such that f ( c ) =, then c is between. (a) - and -1 (b) -1 and (c) and 1 (d) 1 and (e) and 1. Suppose that f is a continuous function defined for all real numbers and f ( 5) = and f ( 1) =. If f ( ) = for one and only one value of, then which of the following could be? (a) -7 (b) - (c) (d) 1 (e) ( ) 1. lim =. ( ) (a) (b) - (c) 1 (d) -1 (e) noneistent 9 1. If f ( ) = is continuous at =, then f ( ) =. + (a) (b) - (c) (d) 6 (e) A function f ( ) has a vertical asymptote at =. The derivative of f ( ) is positive for all. Which statement is true? I. lim f ( ) =+ II. f ( ) = + III. f ( ) = + + (a) I only (b) II only (c) III only (d) I and II only (e) I, II, and III

3 1. C I-. B II-8. E I-.E III-1 PC 5. E III-9 PC (more TI synta zeroes) 6. C II- PC (s s have to use nderiv, so teach them beforehand) 7. D I-7 8. D IV- 9. B VI-1 1. A IV A V- 1. B II- 1. A III- 1. E I C III test review probs.:

4 CALCULUS AB AP CHAPTER.1-. QUIZ Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-5: Calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 1. A particle moves along the -ais so that its position at any time t = t 1t + 9t 6t+. For t > is given by ( ) which value of t is the speed the greatest? (a) t = 1 (b) t = (c) t = (d) t = (e) t = 5 g( ) g( ). If lim =.68, then at the point =, the graph of g( ). (a) is decreasing (b) is increasing (c) is concave upwards (d) is concave downwards (e) attains a relative minimum point f?. The table below gives values of a differentiable function f. What is the approimate value of ( ) f() f cannot be determined from the information given 5. At how many points on the curve y = will the line tangent to the curve pass through the origin? (a) One (b) Two (c) Three (d) Four (e) Five 5. Let f and g be differentiable functions such that f ( 1) =, g( 1) =, f ( ) = 5, f ( 1) =, g ( 1) =, g ( ) =. If h f g h 1 =. (a). (b).89 (c).7 (d). (e) ( ) ( ) = ( ( )), then () (a) -9 (b) 15 (c) (d) -5 (e) -1 Numbers 6-1: NO calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 6. A particle moves along the -ais so that its position at any time t is given by ( t) = t 18t + t. At which time t is its average velocity zero? (a) Never (b) only (c) only (d) and only (e),, and 7. If f ( ) = ( 1) cos, then f ( ) =. (a) - (b) -1 (c) (d) 1 (e) tan( ( + h) ) tan( ) 8. lim =. h h sec (e) noneistent (a) (b) cot( ) (c) sec ( ) (d) ( ) =, then f ( ) =. 9. If f ( ) 7 (a) (b) 7 (c) 1 (d) 1 1 (e) The equation of the tangent line to the curve y = at the point ( 1, 7 ) is. (a) y+ 5= (b) y 1= (c) y 7= (d) y+ 5= 1 (e) y 5= If f ( ) (a) 1 +, 1 = + 1, > 1, then f () 1 =. (b) 1 (c) (d) (e) noneistent 1. What is the th derivative of y = sin ( )? 19 1 (a) sin( ) (b) sin( ) (c) cos( ) (d) cos( ) (e) cos( )

5 1. D IV-. A III-8. D II-7. A I- 5. B I- 6. E V-9 7. A VI- 8. D IV-6 9. C V A VI- 11. C V-5 1. B V quiz review probs.: AP Calculus AB/BC ilearnmath.net

6 CALCULUS AB AP CHAPTER TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-5: Calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 1. If + y = y, then dy d =. 1 1 y 1 1 (c) 1 y 1 (a) (b) (d) + y 1 (e) y y. A missile rises vertically from a point on the ground 75, feet from a radar station. If the missile is rising at the rate of 16,5 feet per minute at the instant when it is 8, feet high, what is the rate of change, in radians per minute, of the missile s angle of elevation from the radar station at this instant? (a).175 (b).19 (c).7 (d).69 (e).57. Two cars start at the same place and at the same time. One car travels west at a constant speed of 5 miles per hour and a second car travels south at a constant speed of 6 miles per hour. Approimately how fast is the distance between them changing one-half hour later? (a) 7 miles per hour (b) 7 miles per hour (c) 76 miles per hour (d) 78 miles per hour (e) 8 miles per hour k + 8. The equation of the line tangent to the curve y = at = is y = +. What is the value of k? k+ (a) - (b) -1 (c) 1 (d) (e) 5. If f ( ) = ( + ), then the fourth derivative of f is. (a) (b) ( ) (c) 6.: NO calculator, 1 minutes. 5!! ( ) (d) ( )! (e)! ( + ) Numbers 7-1: NO calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. dy 7. If y y = 1, then at the point (, ) is. d 6 (a) (b) (c) (d) (e) The volume of a cube is increasing at the rate of cubic centimeters per second. How fast, in square centimeters per second, is the surface area of the cube increasing at the instant when each edge of the cube is 1 centimeters long? (a) (b) (c) (d) 6 (e) 8 9. If sin ( y) =, then dy d =. sec( y) sec( y) sec( y) y (a) sec( y ) (b) (c) sec ( y) y (d) (e) y 1. The volume of an epanding sphere is increasing at a rate of 1 cubic feet per second. When the volume of the sphere is 6π cubic feet, how fast, in square feet per second, is the surface area increasing? Note: V = π r, S = π r. (a) 8 (b) 6 (c) 8π (d) 8 π (e) 1

7 d y 11. If y+ y = 6, then d (a) -18 (b) -6 (c) 6 (d) 1 (e) If the radius of a sphere is increasing at the rate of inches per second, how fast, in cubic inches per second, is the volume increasing when the radius is 1 inches? (a) 8π (b) 8 (c) π (d) π (e) 8π at the point ( 1, ) is. 1. The equation of the tangent line to the curve y 169 at the point 5, 1 is. (a) 5 y 1= 1 (b) 5 1y = 119 (c) 5 1y = 169 (d) 1 + 5y = (e) 1+ 5y = : NO calculator, 15 minutes. + = ( )

8 1. C II-1. A VI-5. D IV-6. D III-1 5. C II- 6. AB/BC (no calc): omitted part c requires section.; reduced time to 15 minutes too 7. E VI E V-7 9. E V-8 1. A IV E III-5 1. A III- 1. C I B AB test review probs.:

9 CALCULUS AB AP CHAPTER.1-. QUIZ Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-5: Calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 1. Let f ( ) be a differentiable function defined for all real numbers. The table below gives the value of f ( ) f ( ) for several values of f() f'() Which of the following statements is true? I. At =, the function is increasing. II. There is a relative minimum in the interval 1 1, but not necessarily at =. III. There is a relative maimum in the interval 1 1. and its derivative (a) I only (b) II only (c) III only (d) I and II only (e) I, II, and III =, then how many critical points does the function f ( ) the interval [.,.6 ]? (a) (b) 1 (c) (d) (e). If the derivative of a function f is given by f ( ) sin ( ) have on. The equation of the tangent line to the curve y = 6 at its point of inflection is. (a) y = 1+ 8 (b) y = 1+ (c) y = 1 8 (d) y = 1+ 1 (e) y = 1. If f ( ) ( 1)( ) = +, how many numbers in the interval satisfy the conclusion of the Mean Value Theorem? (a) (b) 1 (c) (d) (e) 5. Suppose that f ( ), f ( ), and f ( ) (i) f is negative on (,6) and positive on ( 6, ) (ii) f is increasing on (,8) and decreasing on ( 8, ) (iii) f is concave up on (,) and concave down on (, ) are continuous for all real numbers, and that f has the following properties. Of the following, which has the smallest numerical value (i.e. -5 is smaller numerically than )? (a) ( (b) f (c) f ) ( 1 ) f ( ) (d) f ( 1) (e) f ( 7)

10 Numbers 6-1: NO calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 6. What are all values of for which the graph of 6 y = is concave downward? < 1 < (c) 1< < (d) < < 1 (e) > 1 (a) (b) 7. If f is a continuous function on the closed interval [ ab, ], which of the following is NOT necessarily true? I. f has a minimum on [ ab, ] II. f has a maimum on [, ] f ( c) = a< c< b ab III. for (a) I only (b) II only (c) III only (d) I and II only (e) I, II, and III 8 + k 8. For what value of k will have a relative maimum at =? (a) - (b) -16 (c) (d) 16 (e) 9. Suppose that f ( ) is a twice differentiable function on the closed interval [ ab, ]. If f ( c) = for a< c< b, which of the following statements must be true? f a = f b II. f has a relative etremum at = c III. f has a point of inflection at = c I. ( ) ( ) (a) None (b) I only (c) II only (d) I and II only (e) II and III only How many points of inflection does the graph of y = have? (a) None (b) One (c) Two (d) Three (e) Four f =? (a) (b) 1 (c) (d) (e) 11. What is the maimum value of the derivative of ( ) 1. At which of the three points on the graph of y = f ( ) in the figure below is f ( ) f ( ) <? (a) A only (b) B only (c) C only (d) A and B (e) A and C

11 1. E V-. D III-. A III-9. D I- 5. B V-8 6. C VI C V-5 8. B V- 9. A V-1 1. C V D V-6 1. A IV quiz review probs.: AP Calculus AB/BC ilearnmath.net

12 CALCULUS AB AP CHAPTER.-.6 TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-5: Calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. ( ) ( ) ( ) = + < < 6? 1. How many etrema does the function f 5 have on the open interval (a) None (b) One (c) Two (d) Three (e) Four. If f is a continuous odd function and the f ( ) lim f ( ) lim =, which of the following is true? I. = II. There are no vertical asymptotes. III. The lines y = and y = are horizontal asymptotes. (a) I only (b) II only (c) III only (d) II and III only (e) I, II, and III P( ) ( ), ) P( ) is. (a) None (b) One (c) Two (d) Three (e) Four. The derivative of a polynomial function has a relative minimum at critical points. The maimum number of real zeros of. The figure below shows the graph of a function f ( ) which has horizontal asymptotes of following statements is true? 1, and a relative maimum at ( and no other y = and y =. Which of the ( ) ( ) = III. f ( ) I. f < for all II. lim f lim = (a) I only (b) II only (c) III only (d) I and II only (e) I, II, and III

13 5. The figure below shows the graph of f ( ), the second derivative of a function f ( ). The function f ( ) all. Which of the following statements about f is true? is continuous for I. f is concave up for < and b< < c II. f has a relative minimum in the open interval b< <c = = b III. f has points of inflection at and (a) I only (b) II only (c) III only (d) I and III only (e) I, II, and III 6.: Calculator, 15 minutes.

14 Numbers 7-1: NO calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. f ( ) 7. If lim = f ( ) =, which of the following must be true? I. f ( ) = II. f ( ) is continuous at = III. f ( ) has a horizontal tangent line at = (a) I only (b) II only (c) I and II only (d) II and III only (e) I, II, and III 8. 1 lim (a) - 9. The graph of is? (b) y = has. 1 1 (c) (a) no horizontal asymptotes and one vertical asymptote (c) two horizontal asymptotes and one vertical asymptote (e) two horizontal asymptotes and two vertical asymptote 1 (d) 1 (e) 1. Which of the following are the equations of all horizontal and vertical asymptotes for the curve Horizontal Asymptote Vertical Asymptote (a) y = 1 =, = (b) y = =, =, = (c) y = =, = (d) y = 1 =, =, = (b) one horizontal asymptotes and one vertical asymptote (d) one horizontal asymptotes and two vertical asymptote y = ( ) (e) None =, = y = + b What must be b 11. The function has a horizontal tangent and a point of inflection for the same value of the value of? (a) -1 (b) (c) 1 (d) 6 (e) If y = 7 is a horizontal asymptote of a rational function f, then which of the following must be true? (a) lim f ( ) = (b) lim f (c) ( ) = 7 lim f ( ) = 7 (d) lim f ( ) = (e) lim f ( ) = lim = (a) (b) 1 (c) -1 (d) 1 1 (e) : NO calculator, 15 minutes.?

15 1. C VI-. E IV-. C IV-9. D II- 5. D VI B AB 7. E VI D VI-6 9. E V-6 1. C IV E II-8 1. B II-9 1. A I AB/BC test review probs.: AP Calculus AB/BC ilearnmath.net

16 CALCULUS AB AP CHAPTER.1-. QUIZ Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-5: Calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. d t dt d =. (a) e + (b) e + 1 (c) t 1. ( e + ) e + 1 (d) e + (e) e + 8. The present price of a new car is $1,5. The price of a new car is changing at a rate of t dollars per year. How much will a new car cost 5 years from now? (a) $15, (b) $15, (c) $16, (d) $18,1 (e) $,6. The average value of the function f ( ) = e sin on the closed interval [ ] 1,π is. (a).19 (b).15 (c).155 (d).76 (e).1. If ( ) b b ( ) = +1 a b ( ) < ( ) f = + 1 and g, for what values of and is f d g d a? a I. a= 1, b= II. a=, b= 1 III. a= 1, b= (a) I only (b) II only (c) I and II only (d) I and III only (e) I, II, and III 6 5. For the function whose values are given in the table below, ( ) intervals of equal width. The approimation is. f d is approimated by a Midpoint Riemann Sum with three f() (a).6 (b).6 (c).7 (d).76 (e).6

17 Numbers 6-1: NO calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 8 ( ) ( ) ( ) 6. If f d= 1 and f d= 6, then f d=. 8 (a) -16 (b) -6 (c) - (d) (e) If ( 6+ 9) d is approimated by inscribed rectangles of equal width on the -ais, then the approimation is. (a) 1 (b) 1 (c) 6 (d) 5 (e) 8. Let f be a differentiable function for all. Which of the following must be true? d d d d (a) II only (b) III only (c) I and II only (d) II and III only (e) I, II, and III I. f ( ) d= f ( ) II. f ( ) d= f ( ) III. f ( ) d= f ( ) 9. ( 1) d dt t + t dt =. (a) 1 (b) (c) 6 (d) 55 (e) d =. (a) (b) 8 (c) 1 (d) 17 (e) The velocity of a particle moving along the -ais is given by a third-degree polynomial P( t ). The roots of P( t ) are all in the open interval < t < a. Which of the following statements must be true? I. The velocity of the particle will be zero at least once and at most three times for < t < a. II. In the interval < t < a, the particle moves both left and right. III. The total distance traveled by the particle from t = to t = a is given by a P () t dt. (a) I only (b) II only (c) III only (d) I and II only (e) I, II, and III 1. Which of the following properties of the definite integral is true? b b ( ) ( ) I. kf d = k f d, k is constant a b b II. ( ) = ( ) a f d f d a b c b III. ( ) = ( ) + ( ) a a c a f d f d f d (a) I only (b) I and II only (c) I and III only (d) II and III only (e) I, II, and III

18 1. E VI-1. C V-9. A V-7 (on BC assessment too). C III-7 5. D II-1 6. E V- 7. D V-11 (on BC assessment too) 8. B V-7 9. B IV C IV-9 (on BC assessment too) 11. D IV C III quiz review probs.: AP Calculus AB/BC ilearnmath.net

19 CALCULUS AB AP CHAPTER TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-5: Calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 1. If three subdivisions of [ ], are used, what is the trapezoidal approimation of ( 6 + 9) d? (a) (b) 9 (c) 9½ (d) 1 (e) 19 f d 8 6 =. What is ( ) f d=? (a) (b) 8 (c) 1 (d) 16 (e) f and its first and second derivatives all be positive on a closed interval [ ab, ]. The interval n equal length subintervals and these are used to compute an Upper Sum (U), a Lower Sum (L), and a. Suppose that f is a continuous function and ( ). Let f ( ) be a function and let ( ) [ ab, ] is partitioned into Trapezoidal approimation (T) to the eact area b a ( ) I = f d. Which statement below is true? (a) L<U<T<I (b) L<U<I<T (c) L<T<I<U (d) L<I<T<U (e) L<I<U<T. In the interval 5 the graphs of y = cos and y = sin intersect four times. Let A, B, C, and D be the -coordinates of these points so that <A<B<C<D<5. Which of the definite integrals below represents the largest number? A B C cos sin d sin cos d sin cos d (a) ( ) (b) ( ) (c) ( ) A B D D (d) ( cos sin ) d (e) ( sin cos ) C d C 1 5.Some values of a continuous function are given in the table below. The Trapezoidal Rule approimation for f ( ). d is F() (a).85 (b).5 (c).5 (d). (e) : Calculator, 15 minutes.

20 Numbers 7-1: NO calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. g 7. If f and are continuously differentiable functions for all real numbers, which of the following definite integrals is equal to 8. ( ( ) ) ( ( ) )? (a) f ( g ( )) d (b) f ( g ( )) f ( ) d (c) ( ( )) ( ) (d) ( ( )) ( ) (e) ( ( )) ( ) f g f g f g g d d = f g g d f g g d (a) 1 (b) (c) (d) (e) 5 9. csc d =. csc (a) sec + C (b) + C (c) tan + C (d) csc cot+ C (e) cot + C 1. If the substitution u = 5 is made, the integral 5 d= (a) udu (b) udu (c) udu 5 (d) udu (e) 16 udu ( + 5) d =. (a) 1 ( + 5 ) + C (b) ( + 5) + C (c) 6 ( + 5) + C (d) 1 ( ) C (e) 1 ( + 5 ) + C 9 k 1 1. If < k < π, then cos( ) d=, when k =. π π π (a) (b) (c) (d) π (e) 5 π The figure below shows the graph of velocity of an object moving on the -ais as a function of time. At which of the marked points is the object farthest to the right? (a) A (b) B (c) C (d) D (e) E

21 1.: NO calculator, 15 minutes. AP Calculus AB/BC ilearnmath.net

22 1. C VI-6. D VI-9. D IV-. D II-9 (on BC quiz too) 5. B I- (on BC test too) 6. 5AB 7. D II-1 8. A VI-8 9. E VI- 1. D IV- 11. D IV-5 1. A III- 1. C VI- 1. B AB/BC test review probs.: AP Calculus AB/BC ilearnmath.net

23 CALCULUS AB AP CHAPTER QUIZ Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-5: Calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 1. If y = ln ( + 5), then (a) + 5 d y d =. (b) ( + 5 ) 9 (c) ( + 5 ) 9 (d) ( ) + 5 (e) ( ) + 5 ( ) = ( + ) k ( ). Let f be the function defined by f ln for some positive constant k. If f =, what is the value of k? (a) ln ln 8 (b) ln 8 (c) (d) 8 (e) 16. Which statement is true for the function f ( ) = ln ( tan ) on the open interval (a) f ( ) is increasing at an increasing rate (b) f ( ) (c) f ( ) has an absolute maimum in the open interval (d) f ( ) (e) f ( ) has a point of symmetry in the open interval 5π π < <? is increasing at an decreasing rate has a point of inflection in the open interval. If b f () t dt = ln f ( b) a f () t, then ( ) f a =. (a) (b) 1 e (c) 1 (d) e (e) undefined tt>,, ln t t e 5. The position of a particle on the -ais at time is. The average velocity of the particle for 1 is. (a) 1 (b) 1 1 e (c) 1 e 1 (d) e (e) e 1

24 Numbers 6-1: NO calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 6. d ln = (a) ln e. e (b) 1 (c) 1 (d) (e) e 6 d 7. If k > and = ln k, then k =. k + (a) 1 (b) (c) (d) (e) 5 8. If f ( ) = ln, then f ( ) (a) + ln( ) (b) 1 ln( ) =. + (c) ln + (d) (e) 1 π 9. For, an antiderivative of tan is. ln sec (b) (a) ( ) sec (c) ( ) ln sec (d) ln( cos ) (e) ln ( sec ) 1. d = (a) ln 5 (b) ln 1 (c) l n (d) 1 ln 5 (e) 5 ln 11. d = + (a) l n (b) 6l n (c) ln (d) + ln (e) + ln. 1 ( ln ) y = 1. If, then dy d =. ln ln ) (a) ln ( ) (b) ( )( + (c) ( ln )( + ln ) (d) ( ln )( + ln ) (e) ( ln )( 1+ ln )

25 1. D VI-9. D VI-6. B VI-. C VI-8 5. C IV-6 6. A V B V A IV-8 9. C III-1 1. D III C I- 1. C I quiz review probs.: AP Calculus AB/BC ilearnmath.net

26 CALCULUS AB AP CHAPTER (5.1),5. TEST NO FREE-RESPONSE Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-7: Calculator, 1 minutes. Choose the letter that best completes the statement or answers the question. () 1. A particle with velocity at any time t given by v t = e t moves in a straight line. How far does the particle travel during the time interval when its velocity increases from to? (a) 1 (b) (c) (d) e (e) e e 8 y. If e =, then at the point ( 1, ln ), dy d =. (a) ln (b) l n (c) ln (d) e (e) ln.. The graphs of the functions f ( ) = 5 + and g( ) 5e = intersect. (a) Never (b) Once (c) Twice (d) Three times (e) Four times e +, <. If f ( ) = is differentiable at, then a+ b=. a + b, (a) (b) 1 (c) (d) (e) = e ( ) 5. The tangent line to the graph of at the point formed by this tangent line and the coordinate aes? e y e 1 (a) (b) (c) (d) 1, e intersects both coordinate aes. What is the area of the triangle e e e (e) e ( ) ( ) f = e + 5. At what value of is ( ) 6. The derivative of f is given by for f an absolute minimum? (a) For no value of (b) (c).618 (d) 1.6 (e) 5 ().159 t 7. A population grows according the equation P t = 6 55e for t. This population will approach a limiting value as time goes on. During which year will the population reach half of this limiting value? (a) second (b) third (c) fourth (d) eighth (e) twenty-ninth

27 Numbers 8-17: NO calculator, minutes. Choose the letter that best completes the statement or answers the question. = 8. If f ( ) = e ln, f ( e) e e 1 + e e. e 1 e + + e (a) (b) (c) e e e + e (d) e 1 e + e (e) e 1 e 1 e, 5 9. Let f ( ) = +. Which of the following statements is true? e, 5 1 I. f ( ) is continuous for all values of in the interval [,1 ]. II. f ( ), the derivative of f ( ), is continuous for all values of in the interval [ ] III. The graph of f ( ) is concave upwards for all values of in the interval [,1 ].,1. (a) I only (b) II only (c) III only (d) I and III only (e) I, II, and III 1. The minimum value of f ( ) e = is. (a) ln (b) e (c) 1 e (d) 1 ( ln) (e) f = e + 1, then f =. 11. If ( ) ( ) (a) (b) (c) (d) 1 (e) 1. If y = e, then y has a point of inflection at =. (a) (b) 1 (c) (d) - (e) n d y 1. If y = e, then n d =. n (a) e (b) (c) e ( n) e + (d) n e (e) ( + n ) e 1 1. Suppose that f ( ) = ln and f ( ) (a) e C (b) 1 + e d=. + (d) 1 1 denotes the inverse of f. Then f ( ) + C (c) 1 C + C (e) 1 e + C d 15. ( e ln ) d (a) ln e =. (b) ln e (c) (d) (e) y = 1 y The graph of which function has as an asymptote? (a) y = e (b) = (c) y ln ( 1) = + (d) sin 17. If f ( ) = e, how many zeros does f ( ) have on the closed interval [ ] y = (e) + 1, π? (a) 1 (b) (c) (d) (e) 5 y = 1

28 1. A IV-. A IV-9 (on BC 5bt). D III-. C III- 5. A II-5 6. E II-6 7. C I- 8. A VI D IV- 1. D IV C III- 1. C III- 1. C II- 1. B I D I E I B I test review probs.: AP Calculus AB/BC ilearnmath.net

29 CALCULUS AB AP CHAPTER 5.6, TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-5: Calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. dy 1. If = y, and if d y = when =, then when y = e, =. (a).7 (b).55 (c).69 (d) 1. (e).718. At any time t, in days, the rate of growth of bacteria population is given by y = ky, where y is the number of bacteria present and k is a constant. The initial population is 15 and the population is quadrupled during the first days. By what factor will the population have increased during the first days. (a) (b) 5 (c) 6 (d) 8 (e) 1. If dy d = y and y () (a) 1 = 1, then y =. (b) (c) (d) +1. If f is a continuous function on the closed interval [, ] I. f has a minimum on [ ab, ] II. f has a maimum on [ ab, ] III. f ( c) ab, which of the following is NOT necessarily true? = for a< c< b (a) I only (b) II only (c) III only (d) I and II only (e) I, II, and III 5. A function whose derivative is a constant multiple of itself must be. (a) periodic (b) linear (c) eponential (d) quadratic (e) logarithmic (e) 6.: Calculator, 15 minutes.

30 Numbers 7-1: NO calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 7. If dy d y = and ( ) (a) y =, then. y = 7 (b) + y = 7 (c) y = 7 (d) y = 5 (e) y = 7 8. At each point (, y ) on a certain curve, the slope of the curve is y. If the curve contains the point (, ), then its equation is. (a) y = e + (b) y = e + (c) y = e (d) y = + (e) y = + 9. The rate of decay of a radioactive substance is proportional to the amount of substance present. Four years ago there were 1 grams of substance. Now there are 8 grams. How many grams will there be 8 years from now? (a) (b) 8 (c) 9 (d) (e) If y = arcsin (a) 16 9, then dy d =. (b) (c) 16 9 (d) (e) d du d 1+ u =. (a) 1 +1 (b) (c) + 1 (d) + 1 (e) arctan 1. If f ( ) 1 = arctan, then f ( ) 1 (a) (b) + =. 1 (c) (d) +1 1 (e) + 1 d 1. =. (a) arcsin + C 1 arcsin + C (b) C + (c) arcsin ( ) + C (d) + C (e)

31 1.: NO calculator, 15 minutes. Copy the aes and points for Part a, and draw the slope field on your own paper.

32 1. B IV-1. D III-6. B I-8. C [V-5]; MVT/Rolle s 5. C II AB/BC 7. A V-8 8. C III- 9. C II- 1. E IV- 11. B III E III-9 1. A I-1 1. B AB test review probs.: AP Calculus AB/BC ilearnmath.net

33 CALCULUS AB AP CHAPTER 7.1,7. TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-5: Calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 1. The base of a solid is the region enclosed by the graph of ( ) y = and the coordinate aes. If every cross section perpendicular to the -ais is a square, then the volume of the solid is. (a) 8. (b) 19. (c). (d) 5.6 (e) The region enclosed by the line y 1 solid generated? (a) 17 π (b) 17 π (c) π. The region in the first quadrant enclosed by the -ais, the line π + = and the coordinate aes is rotated about the line y = 1. What is the volume of the =, and the curve ( ) (d) π (e) π y = cos cos is rotated about the -ais. What is the volume of the solid generated? (a) 1.9 (b).78 (c) 6. (d) 8.1 (e).7. The region in the first quadrant enclosed by the graphs of y = and y = sin is evolved about the -ais. The volume of the resulting figure is. (a) (b).16 (c) (d) (e) The volume of the solid formed by revolving the region bounded by the graph of y = ( ) and the coordinate aes about the -ais is given by which of the following integrals? π d π d π d (a) ( ) (b) ( ) (c) ( ) (d) π ( ) d (e) π ( ) d 6.: Calculator, 15 minutes.

34 Numbers 7-1: NO calculator, 15 minutes. Choose the letter that best completes the statement or answers the question. 7. The total area between the curves y = and y = is. (a) 1 (b) 1 (c) 1 (d) (e) b 8. If the definite integral ( ) represents the area of the region bounded by y f ( ) f a d = b, which of the following must be true? (a) a > b and f ( ) > (b) a b (d) a > and ( ) < b and f ( ) < (e) None of the above =, the -ais, and the lines = a and f < (c) a b < and f ( ) > 9. A solid has a circular base of radius. If every plane cross section perpendicular to the -ais is an equilateral triangle, then its volume is. (a) 6 (b) 1 (c) 18 (d) (e) 6 1. Let R be the region in the fourth quadrant enclosed by the -ais and the curve y = k, where k >. If the area of the region R is 6, then the value of k is. (a) (b) (c) (d) 6 (e) The base of a solid is the region in the first quadrant bounded by the line + y = and the coordinate aes. What is the volume of the solid if every cross section perpendicular to the -ais is a semicircle? (a) π (b) π (c) 8 π (d) π (e) 6 π 1. The area of the region between the graph of y = + and the -ais from = 1 to = is. (a) 6 (b) (c) 1 (d) 6 (e) 1 1. For the region below, the area of the shaded region is. (a) 1 (b) 16 (c) 8 (d) (e) 65

35 1.: Calculator, 15 minutes. AP Calculus AB/BC ilearnmath.net

36 1. E V-. E IV-. C III-. D II- 5. B I B AB/BC1 7. B VI-1 8. E V-1 9. E IV-1 1. B IV A III-6 1. B III-1 1. B I AB test review probs.: AP Calculus AB/BC ilearnmath.net