1.18 Multiple Choice Questions on Limits

Transcription

1 24 The AP CALCULUS PROBLEM BOOK 3x 4 2x lim x 7x 8x 5 =.8 Multiple Choice Questions on Limits A) B) C) 0 D) 3 7 E) lim x 0 x = A) B) C) 0 D) E) does not exist 9x lim x /3 3x = A) B) C) 0 D) 2 E) lim x 0 x 3 8 x 2 4 = A) 4 B) 0 C) D) 3 E) In order for the line y = a to be a horizontal asymptote of h(x), which of the following must be true? A) lim h(x) = x a + B) lim h(x) = x a C) lim h(x) = x D) lim h(x) =a x E) lim h(x) = x x 3 x>2 38. The function G(x) = 5 x =2 3x 7 x<2 is not continuous at x =2because A) G(2) is not defined B) lim x 2 G(x) doesnotexist C) lim x 2 G(x) G(2) D) G(2) 5 E) All of the above 39. lim x 0 3x 2 +2x 2x + = A) B) C) 0 D) E) 3 2

2 CHAPTER. LIMITS 25 2x 2 3x lim = x /2 2x + A) B) C) D) 3 2 E) 5 2 2x lim = x 2 x +2 A) B) 0 C) D) E) does not exist 3x 2 +2x lim x x 4 +7x 2 3 = A) 0 B) 2 C) 3 7 D) E) 323. lim x 0 x 2 +4 x 2 = A) B) 0 C) 4 D) E) { 324. The function G(x) = x 2 x>2 is not continuous at x =2because 4 2x x < 2 A) G(2) does not exist B) lim x 2 G(x) doesnotexist C) lim x 2 G(x) =G(2) D) All three statements A, B, and C E) None of the above 325. The domain of the function f(x) = 4 x 2 is A) x< 2 orx>2 B) x 2orx 2 C) 2 <x<2 D) 2 x 2 E) x lim x 5 x 2 25 x 5 = A) 0 B) 0 C) 0 D) 5 E) does not exist x 2 6 x Find k so that f(x) = x 4 is continuous for all x. k x =4 A) any value B) 0 C) 8 D) 6 E) no value Insanity means we keep trying the same thing and hope it comes out differently. Albert Einstein

3 56 The AP CALCULUS PROBLEM BOOK 2.9 Multiple-Choice Problems on Derivatives x 2 + x x Let F (x) = x Which of the following statements are true of F? x =0. I. F is defined at x =0. II. lim F (x) exists. x 0 III. F is continuous at x =0. A) I only B) II only C) I, II only D) II, III only E) I, II, and III 659. If P (x) =(3x +2) 3 then the third derivative of P at x =0is A) 0 B) 9 C) 54 D) 62 E) If F (x) =3x then F (5) = A) 0 B) 5 C) 5 D) 3 E) The slope of the curve y 3 xy 2 =4atthepointwherey =2is A) 2 B) 4 C) 2 D) 2 E) If F (x) =x/(x ) 2 then the set of all x for which F (x) existsis A) all real numbers B) {x x } C) {x x 3 } D) {x x ±} E) {x x } 663. If lim x b G(x) =K, thenwhichofthefollowingmustbetrue? A) G (b) exists. B) G(x) iscontinuousatx = b. C) G(x) isdefinedatx = b. D) G(b) =K. E) None of the above must be true Which of the following functions are continuous for all real numbers x? I. y = x 4/3 II. y = 3 3x III. y = 3x 4x 2 +5 A) None of these B) I only C) II only D) I, II only E) I, II, and III 665. The equation of the tangent line to the curve y = x 2 4x at the point where the curve crosses the y-axis is A) y =8x 4 B) y = 4x C) y = 4 D) y =4x E) y =4x The tangent to the curve y =2xe x is horizontal when x = A) 2 B) C) D) e E) None of the above If you have an unpleasant nature and dislike people, that is no obstacletowork. J. G. Bennett

4 CHAPTER 2. DERIVATIVES 57 ( ) e x 667. If y =ln e x,then dy 0 dx = A) x ex e x If y =ln(x x 2 +),then dy dx = A) + x x If y = e x ln x then dy dx B) e x C) B) + x x 2 + when x =is C) 0 0 e x D) 0 E) ex 20 e x 0 2x 2 + x x 2 + D) 2x 2 + x(x 2 +) E) x2 + x + x(x 2 +) A) 0 B) Does not exist C) 2 e D) e E) e 670. The slope of the line tangent to the graph of y =lnx 2 at x = e 2 is A) e 2 B) 2 e 2 C) 4 e 2 D) e 4 E) 4 e If y =ln(x 2 + y 2 )thenthevalueof dy dx at (, 0) is A) 0 B) C) D) 2 E) undefined 672. If z = 3w dz,then cos w dw = A) 3 sin w D) 3cosw +3w sin w cos 2 w 673. Find the derivative of y = B) 2sin2x. 3cosw 3w sin w cos 2 w E) None of the above A) csc 2x cot 2x B) csc 2 2x C) 4csc2x cot 2x D) 674. If y =sec 2 x then dy dx = A) sec x tan x x D) sec2 x tan x x B) tan x x 675. If y =sinx cos x, thenthederivativeofy is cos 2x 2 sin 2x E) 2 sec 2 x tan x C) 3 sin w E) 4 sec 2x C) 2 sec x tan 2 x A) cos x B) cos 22x C) sin 2 x cos 2 x D) 2 sin 2 x E) 2 sin x cos x

5 CHAPTER 3. APPLICATIONS OF DERIVATIVES Multiple-Choice Problems on Applications of Derivatives 820. The value of c guaranteed to exist by the Mean Value Theorem for V (x) =x 2 in the interval [0, 3] is A) B) 2 C) 3 2 D) If P (x) is continuous in[k, m] and differentiable in (k, m), then the Mean Value Theorem states that there is a point a between k and m such that A) P (k) P (m) m k = P (a) B) P (a)(k m) =P (k) P (m) C) D) m k P (m) P (k) = a m k P (m) P (k) = P (a) 822. The Mean Value Theorem does not apply to f(x) = x 3 on [, 4] because A) f(x) isnotcontinuouson[, 4] B) f(x) is not differentiable on (, 4) C) f() f(4) D) f() >f(4) 823. Which of the following function fails to satisfy the conclusion of the Mean Value Theorem on the given interval? A) 3x 2/3 ; [, 2] B) 3x 2 ; [, 2] C) 4x 3 2x +3; [0, 2] D) x 2; [3, 6] Success and failure have much in common that is good. Both mean you retrying. Frank Tyger

6 90 The AP CALCULUS PROBLEM BOOK 824. If a function F is differentiable on [ 4, 4], then which of the following statements is true? A) F is not continuous on [ 5, 5] B) F is not differentiable on [ 5, 5] C) F (c) =0forsomec in the interval [ 4, 4] D) The conclusion of the Mean Value Theorem applies to F (x 2)(x 3) 825. The function G(x) = x on the interval [ 3, 2] because does not satisfy the hypothesis of Rolle s Theorem A) G( 3) = G(2) = 0 B) G(x) is not differentiable on [ 3, 2] C) G(x) isnotcontinuouson[ 3, 2] D) G(0) The function F below satisfies the conclusion of Rolle s Theorem in the interval [a, b] because A) F is continuous on [a, b] B) F is differentiable on (a, b) C) F (a) =F (b) =0 D) All three statements A, B and C a b 827. The intervals for which the function F (x) =x 4 4x 3 +4x 2 +6increasesare A) x<0, <x<2 B) only x>2 C) 0 <x<, x>2 D) only 0 <x< E) only <x< If Q(x) =(3x +2) 3,thenthethirdderivativeofQ at x =0is A) 0 B) 9 C) 54 D) 62 E) 224

7 CHAPTER 3. APPLICATIONS OF DERIVATIVES The function M(x) =x 4 4x 2 has A) one relative minimum and two relative maxima B) one relative minimum and one relative maximum C) no relative minima and two relative maxima D) two relative minima and no relative maxima E) two relative minima and one relative maximum 830. The total number of all relative extrema of the function F whose derivative is F (x) = x(x 3) 2 (x ) 4 is A) 0 B) C) 2 D) The function F (x) =x 2/3 on [ 8, 8] does not satisfy the conditions of the Mean Value Theorem because A) F (0) does not exist B) F is not continuous on [ 8, 8] C) F () does not exist D) F is not defined for x<0 E) F (0) does not exist 832. If c is the number defined by Rolle s Theorem, then for R(x) =2x 3 6x on the interval 0 x 3, c must be A) B) C) ± D) 0 E) Find the sum of the values of a and b such that F (x) =2ax 2 + bx +3 has a relative extremum at (, 2). A) 3 2 B) 5 2 C) D) 834. Which of the following statements are true of the graph of F (x) shownbelow? A) I only B) I, II only C) I, III only D) II, III only E) None are true I. There is a horizontal asymptote at y =0. II. There are three inflection points. III. There are no absolute extrema. It is not enough to have a good mind. The main thing is to use it well. Rene Descartes

8 CHAPTER 4. INTEGRALS π/2 π/6 cot xdx= A) ln 2 B) ln 2 C) 2 D) ln( 3 ) x + dx = A) 5 2 B) 7 2 C) 9 2 D) 2 E) (3x 2) 3 dx = A) 6 3 B) 63 4 C) 3 3 D) π/2 π/4 sin 3 θ cos θ dθ = A) 3 6 B) 8 C) 8 D) 3 6 E) e x (3 e x ) 2 dx = A) 3 ln(e 3) B) C) 3 e D) e 2(3 e) E) e 2 3 e 2. 0 e x dx = A) e B) e e C) e D) e E) e x x 2 + dx = A) π 4 B) ln 2 C) 2 (ln 2 ) D) 3 2 E) ln 2 Anyone who cannot cope with mathematics is not fully human. At bestheisatolerablesubhumanwhohas learned to wear shoes, bathe, and not make messes in the house. Robert A. Heinlein