CALCULUS 2 WORKSHEET #2. The asymptotes of the graph of the parametric equations x = t t, y = t + are A) x = 0, y = 0 B) x = 0 only C) x =, y = 0 D) x = only E) x= 0, y = 2. What are the coordinates of the inflection point on the graph of y = (x + ) Arctan x? A) (,0) B) (0,0) C) (0,) D), π E). The Mean Value Theorem guarantees the existence of a special point on the graph of y = x etween (0,0) and (,2). What are the coordinates of this point? A) (2,) B) (,) C) (2, 2 ) D) 2, E) None of the aove 2, π 2. 8 0 + x = A) B) 2 C) 2 D) E) 6 5. If x 2 + 2xy + y 2 = 2, then the value of dy at x = is A) 2 B) 0 C) 2 D) E) not defined lim 6. What is h!0 8 8 " $ # 2 + h % " ' 8 % $ ' & # 2& h 8? A) 0 B) 2 E) It cannot e determined from the information given. C) D) The limit does not exist. 7. For what value of k will x + k x have a relative maximum at x = 2? A) B) 2 C) 2 D) E) None of these 8. If h(x) = ƒ 2 (x) g 2 (x), ƒ ' (x) = g(x), and g ' (x) = ƒ(x), then h' (x) = A) 0 B) C) f(x)g(x) D) [ g(x)] 2 = [f(x)] 2 E) 2[ g(x) + f(x)] 9. The area of the closed region ounded y the polar graph of r = + cos q is given y the integral 2π π π/2 ( + cos q) dq A) 0 + cos q dq B) 0 + cos q dq C) 2 0 π D) 0 π/2 ( + cos q) dq E) 2 0 + cos q dq 0. 0 x 2 x 2 + = A) π B) ln 2 C) 0 D) 2 ln 2 E) + π. The point on the curve x 2 + 2y = 0 that is nearest the point 0, 2 occurs where y is A) 2 B) 0 C) 2 D) E) none of the aove
CALCULUS 2 WORKSHEET #2 x e t2 dt, then F ' (x) = 2. If F(x) = 0 A) 2xe x2 B) 2xe x2 C) e x2 + x 2 + e D) e x2 E) e x2. The region ounded y the x axis and the part of the graph of y = cos x etween x = π 2 and x = π 2 is separated into two regions y the line x = k. If the area of the region for π 2 x k is three times the area of the region for k x π 2, then k = A) arc sin B) arc sin C) π 6 D) π E) π. If y = x 2 + 2 and u = 2x, then dy du = A) 2x2 2x + (2x ) 2 B) 6x 2 2x + C) x 2 D) x E) x 5. If ƒ ' (x) and g ' (x) exist and ƒ ' (x) > g ' (x) for all real x, then the graph of y = ƒ(x) and the graph of y = g(x) A) intersect exactly once B) intersect no more than once C) do not intersect D) could intersect more than once E) have a common tangent at each point of intersection 6. If y is a function of x such that y ' > 0 for all x and y " < 0 for all x, which of the following could e part of the graph of y = ƒ(x)? Answer is B 7. The graph of y = 5x x 5 has a point of inflection at A) (0,0) only B) (,62) only C) (, 256) only D) (0,0), and (, 62) E) (0,0) and (, 256) 8. ƒ(x) = 2 + x for all x and the value of the derivative ƒ ' (x) at x = is A) B) 0 C) D) 2 E) nonexistent 9. A point moves on the X axis in such a way that its velocity at time t (t > 0) is given y v = ln t t. At what value of t does v attain its maximum? A) B) e /2 C) e D) e /2 E) There is no maximum value for v.
CALCULUS 2 WORKSHEET #2 20. An equation for a tangent to the graph of y = arc sin x 2 at the origin is A) x 2y = 0 B) x y = 0 C) x = 0 D) y = 0 E) 2x 2y = 0 2. At x = 0, which of the following is true of the function f defined y ƒ(x) = x 2 + e 2x? A) ƒ is increasing B) ƒ is decreasing C) ƒ is discontinuous. D) ƒ has a relative minimum E) ƒ has a relative maximum. x t dt, which of the following is FALSE? + 2 22. If ƒ(x) = 0 A) ƒ(0) = 0 B) ƒ is continuous at x for all x 0. C) ƒ() > 0 D) ƒ'() = E) ƒ( ) > 0 2. If the graph of y = ƒ(x) contains the point (0,2), dy = x yex2, and ƒ(x) > 0 for all x, then ƒ(x) = A) + e x2 B) + e x C) + e x D) + e x2 E) + e x2 2. If sin x = e y, 0 < x < π, what is dy is terms of x? A) tan x B) cot x C) cot x D) tan x E) csc x 25. A region in the plane is ounded y the graph of y = x x = 2m, m>0. The area of this region, the x axis, the line x = m, and the line A) is independent of m B) increases as m increases C) decreases as m increases D) decreases as m increases when m < 2 ; increases as m increases when m > 2 E) increases as m increases when m < 2 ; decreases as m increases when m > 2 26. 0 x 2 2x + is A) B) 2 C) 2 D) E) none of the aove 27. If dy = tan x, then y = A) 2 tan 2 x + C B) sec 2 x + C C) ln sec x + C D) ln cos x + C E) sec x tan x + C 28. What is x!0 lim e 2x tan x? A) B) 0 C) D) 2 E) The limit does not exist. 29. 0 ( x 2 ) /2 = A) 2 B) 2 C) 2 D) E) 2 2
CALCULUS 2 WORKSHEET #2 0. " (!) n x n # is the Taylor series aout zero for which of the following functions? n! n = 0 A) sin x B) cos x C) e x D) e x E) ln ( + x). If ƒ ' (x) = ƒ(x) and ƒ() =, then ƒ(x) = A) 2 e 2x + 2 B) e x C) e x D) e x E) e x 2. For what values of x does the series + 2 x + x + x +... + n x +... converge? A) No values of x B) x < C) x ³ D) x > E) All values of x. What is the average (mean) value of t t 2 over the interval t 2? A) B) 7 2 C) 8 D) E) 6. Which of the following is an equation of a curve that intersects at right angles every curve of the family y = x + k (where k takes all real values)? A) y = x B) y = x 2 C) y = x D) y = x E) y = ln x 5. At t = 0 a particle starts at rest and moves along a line in such a way that at time t tis acceleration is 2t 2 feet per second per second. Through how many feet does the particle move during the first 2 seconds? A) 2 B) 8 C) 6 D) 96 E) 92 6. The approximate value of y = + sin x at x = 0.2, otained from the tangent to the graph at x = 0, is A) 2.00 B) 2.0 C) 2.06 D) 2.2 E) 2.2 7. Of the following choices of d, which is he largest that could e used successfully with an aritrary e in an epsilon delta proof of lim x!2(x) = 5? A)! = e B)! = e C)! = e 2 D)! = e E)! = e 5 8. If ƒ(x) = (x 2 + ) (2 x), then ƒ ' () = A) 2 ln (8e) B) ln (8e) C) 2 ln 2 D) 2 9. If y = tan u, u = v v A) 0 B) e, and v = ln x, what is the value of dy C) D) 2 e E) sec 2 e E) 8 at x = e? 0. If n is a non negative integer, then 0 x n = 0 ( x) n for A) no n B) n even, only C) n odd, only D) nonzero n, only E) all n. If ƒ(x) = 8 x 2 for 2 x 2 ƒ(x) = x 2 elsewhere, then ƒ(x) is a numer etween A) 0 and 8 B) 8 and 6 C) 6 and 2 D) 2 and 2 E) 2 and 0
CALCULUS 2 WORKSHEET #2 2. If x2 cos x = ƒ(x) 2x sin x, then ƒ(x) = A) 2 sin x + 2x cox x + C B) x 2 sin x + C C) 2x cos x x 2 sin x + C D) cos x 2x sin x + C E) (2 x 2 ) cos x sin x + C. Which of the following integrals gives the length of the graph of y = tan x etween x = a and x =, where 0 < a < < π 2? x 2 + tan 2 x x + tan x + sec 2 x A) a B) a C) a D) a + tan 2 x E) a + sec x. If ƒ " (x) ƒ ' (x) 2ƒ(x) = 0, ƒ ' (x) = 2e x, and ƒ(0) = 2, then ƒ() = A) e 2 + e B) C) 0 D) e 2 E) 2e (x + ) 5. The complete interval of convergence of the series k k 2 is k= A) 0 < x < 2 B) 0 x 2 C) 2 < x 0 D) 2 x < 0 E) 2 x 0