• CHAPTER 2 REVIEW QUESTIONS Complete the following review questions using the techniques outlined in this chapter. Then, see Chapter 8 for answers and explanations. 1. Consider the sequence (x) whose terms are given by the formula . (cosmtXsin2 n) X= ([n n e n for each integer n > 1. Given that this sequence converges, what is its limit? (A) 0 (B) 1 (C) log 2 (D) 42 (E) :Fe 2. Let (xn) be the sequence with Xl = 2 and xn = ~5xn_l + 6 for every integer n > 2. Given that this sequence converges, what is its limit? (A) 4. (B) 6 (C) 8 (D) 10 (E) 16 3. Let [xl denote the greatest integer $; x. If n is a positive integer, then x~Jlxl-[xl) -~(lxl-[xl) = ? (A) -2 (B) 0 (C) 2 4. Evaluate the following limit: • lim arcsm x -X x-->o x3 (A) 0 5. The curve whose equation is ~ + 3x -2xy -y = 6 has two asymptotes. Identify these lines. (A) x = -1 and y = -2 1 (D) x = --and y = x + 1 2 ; (B) x = -2 and y = 1 1 (E) x = -and y = 1 -x 2 (D) 2n-1 (E) 2n (E) 1 1 (C) x = --and y = x 2 CALCU LUS I 6. If the function f (x) = X3 -2X2 + 2x -4 k is continuous everywhere, what is the value of k? (A) 1 (B) ! 2 7. Evaluate the following limit: (A) 1 21t (B) ! 1t lim 1 rx t + t2 dt x->O x2 Jo 1 + sint 8. Determine the domain of the following function: 1 (A) [0, 2] e f (x) = arcsin (log Jx" ) 1 (B) [ 2' 1] e 9. Evaluate the derivative of the following function at x = e: f(x) = arcsin (log Jx") 1 (A) ~ e,,3 e (B) J2 1te (C) 2 ifx=2 1 (D) --3 (D) 1 1 (D) [ 2';] e (D) £e . (E) -1 (E) ~ 2 3e (E) J2 10. For what values of m and b will the following function have a derivative for every x? (A) m = 3, b = -2 (D) m = -2, b = 1 (B) m = -2, b = -3 (E) m = 3, b = -4 GRE MATHEMATICS SUBJECT TEST • ifxl (C) . m = I, b = -4 11. If j(x) is a function that's differentiable everywhere, what is the value of this limit? lim j(x+3h2)-f(x-h2) h-->O 2h2 (A) 4f'(x) (B) 2f'(x) (C) f'(x) (0) t f'(x) (E) The limit does not exist. 12. What is the equation of the tangent line to the curve y = X3 -3x2 + 4x at the curve's inflection point? (A) Y = 2x-3 (B) Y = i-I (C) y == x + 1 (0) y = 3x-2 (E) x + Y = 1 13. What is the slope of the tangent line to the curve xy(x + y) = x + y4 at the point (I, I)? (A) 2 (B) 1 (C) 0 (0) -1 14. If f(x) = 21x -11 +(x _1)2, what is the value of f'(O)? (A) 4 (B) 2 (C) 0 (0) -2 , 15. If r f(x) = e arccosx cos x then the slope of the line tangent to the graph of j at its y-intercept is (B) -1 (A) 0.08 (B) 0.04 1t (C) --1 2 (C) -D.02 (0) 1 (0) -D.06 d f(x 3 ) 17. Ifj(l)= landf'(l)=-l,thenthevalueof d f 2) atx=lisequalto x x (x (A) 1 (B) 0 (C) -1 (0) -2 (E) ...:2 (E) -4 1t (E) -+1 2 • (E) -D.09 (E) -3 • • CALCU LUS I • 1 1 18. If n is a positive integer, what is the value of the nth derivative of j(x) == at x = --? 1-2x 2 (B) l(n') 2 . (0) n nn (E) n! 19. Let f(x) be continuous on a bounded interval, [a, b], where a "# b, such that f(a) = 1 and f(b) = 3, and j'(x) exists for every x in (a, b). What does the Mean-Value theorem say aboutf? , (A) There exists a number c in the interval (a, b) such that j'(c) == O. (B) There exists a number c in the interval (a, b) such that j(c) == o. (C) There exists a number c in the interval (a, b) such that j'(c) == 2. (0) There exists a number c in the interval (a, b) such that j'(c) == 2(b -a) . . (E) There exists a number c in the interval (a, b) such that (b -a)j'(c) = 2. 20. What is the maximum area of a rectangle inscribed in a semicircle of radius a? 21. The following function is defined for all positive x: f(x) = 2x sint dt x t 1t . (0) J2 a2 2 2 At what value of x on the interval (0, 31t) does this function attain a local maximum? 2 (A) ~ (B) ~ (C) 1t (0) 1t (E) 21t 632 3 22. Let j(x) = Xk e-x ,where k is a positive constant. For x > 0, what is the maximum value attained by f? , e (A) -k k (B) k e e (C) Qogkl k (0) e logk k (E) k k -e 23. The radius of a circle is decreasing at a rate of 0.5 cm per second. At what rate, in cm2/sec, is the circle's area decreasing when the radius is 4 cm? (A) 41t (B) 21t (C) 1t 1 (0) -1t 2 1 (E) -1t 4 e2x . 24. The function f(x) = t t log tdt has an absolute minimum at x = 0, and a local maximum at x = (A) -log4 (B) -log 2 (C) log 2 (0) 1 (E) log 4 GRE MATHEMATICS SUBJECT TEST • , 25. Evaluate the following integral: (A) _ 7 20 (B) _ 1 60 (C) 2 15 7 . 26. If [x] denotes the greatest integer < x, then f02 [x]dx = (A) ~ 2 27. If (B) "-2 (C) . ~ 2 -2(x+l) ifxO then the value of k for which t f(x)dx = 1 is (A) -1 (B) 0 (C) 1 x2 dx 28. Integrate I 2 • \l'l-x (0) 1 60 (0) 17 2 , (0) 2 • (E) 7 20 (E) 37 2 (E) 3 (A) ! arcsinx-~r-l--x-2 +c ' (B) ! arcsinx+x~l-x2 +c . (C)! xarcsinx-.Jl-x2 +c (0) ! arcsinx-x.Jl-x2 +c (E) J xarcsinx+.Jl-x2 +c 29. What is the area of the region in the first quadrant bounded by the curve y = x arctan x and the . line x = I? (A) 1t-4 4 (B) 1t-2 4 30. Simplify the following: 1t (C) 4 5 dx exp 3 x2 -3x+2 1t+2 (0) 4 [Note: Recall that exp x is a standard, alternate notation for If.] (A) ~ 8 2 (B) -3 (C) i 3 (0) 3 2 1t+4 (E) 4 . (E) j CALCU LUS I , 31. Calculate the area of the region in the first quadrant bounded by the graphs of y = 8x, Y = ~, and y = 8. (A) 12 (B) 8 (C) 6 (0) 16 3 (E) 4 32. Which of the following expressions gives the area of the region bounded by the two circles pic-, tured below? 11 , y r =-J3 sin e (A) , fo2 ~[(~SinOf -(3cos9)2 ]dO 11 11 (B) fo6 !(3COSO)2 dO + f; ~ (~sinO)2 dO 6 11 11 (C) fo6 ~(~Sin9)2d9+ J; ~(3cosWite 6 11 11 (0) r31(3cosO)2d9+f21(~sin9)2d9 Jo 2 11 2 3 11 11 (E) J0 3 ~(~SinO)2d9+ J; ~(3COS9)2d9 3 r = 3cos e --,-/, x • • 33. Let a and 'b be positive numbers. The region in the second quadrant bounded by the graphs of y = ax2 and y = -bx is revolved around the x-axis. Which of the following relationships between a and b would imply that the volume of this solid of revolution is a constant, independent of a and b? i (E) b2 = 2a3 34. The region bounded by the graphs of y ;" rand y = 6 -I xl "is revolved around the y-axis. What is the volume of the generated solid? (A) 32n 3 (B) 9n GRE MATHEMATICS SUBJECT TEST (C) 8n (0) 20 n 3 (E) 16n 3 35. Calculate the length of the portion of the hypocycloid y:2/3 + '1/3 = 1 in the first quadrant from the 1 3J3 point 8' 8 ' to the point (1, 0). (A) 9 . 8 (8) 3J2 4 (C) 1 36. What positive value of a satisfies the following equation? (A) ! (8) {[e e 37. Evaluate the following limit: (A) ! 2 (8) 1 .,fe dx --=1 fQX~ e x a Y • (C) .,fe 2 lim (COSx)cot x x--->o (C) .,fe 2 38. Let n be a number for which the improper integral ~ dx e xQogx)" converges. Determine the value of the integral. (A) 1 n+1 (8) ! n (C) 1 n-l 39. Find the positive value of a that satisfies the equation: a (A) 2J2 (B) 1 1t dx (C) 1t 2J2 a xdx (D) sJ2 8 (D) e (D) 1 (D) logn n+1 (D) J2 . (E) J3 2 (E) t? (E) .,fe (E) logn n-l (E) ~ 2 CALCU LUS I 40. Which of the following improper integrals converge? ~ dx I. ~ (x2 + 1)2 II. r xe-x dx 2 III. o (2-xf dx (A) I only (0) I and III only (B) I and II only (E) II and III only 41. Which of the following infinite series converge? ~ cos 4 (arctan n) I. ~ ~ n=1 n"n ~ 1 II. L-l-n=2 n ogn ~ . (n + 1)3 III. L ----'---------'---n=O 5(n + 2Xn + 3Xn + 4) (A) I only (0) I and III only (B) I and II only (E) II and III only • (C) II only (C) II only 42. Find the smallest value of b that makes the following statement true: ~ (n 1)2 an If 0::; a < b, then the series L' converges. . n=1 (2n)! . (A) 1 (B) 2 log 2 43. Evaluate the following limit: (A) ~ 1 (B) -2 GRE MATHEMATICS SUBJECT TEST (C) 2 (0) J2 1 (0) 6 (E) 4 1 (E) 12 • 44. Which of the following statements are true? ,~ ~ I. If an ~ 0 for every n, then: ~>n converges "LF. converges. n=l n=l ~ ~ II. Han ~o foreveryn, fte1: Lnan converges => Lan converges. n=l n=1 ~ ~ III. If an ~ 0 ani an+1 < an for every n, thn ~>~ converges => L(-I)"an converges. (A) I and II only (0) II and III only ~ 45. If -1 < x < I, then Inx2n = n=l x3 (A) (l-xf (0) x3 (1 + X)2 n=l • (B) I and III only (E) III only x2 (B) (1-x2 f (E) x2 (1 + X2)2 n=l (C) II only , (C) x (1 +x2f Th 11 ... f hihh . ~n!(2n)!xnd t . 46. e sma est posItive mteger x or w c t e power senes f:t (3n)! oes no converge IS (A) 4 (B) 6 (C) 7 (0) 8 (1;:) 9 47. In the Taylor series expansion (in powers of x) of the function f(x) = eX'-x, what is the coefficient of :x3? ' (A) -7 , 7 (C) --6 (0) Z. 6 (E) ~ 2 48. If kj (i = 0, 1,2,3,4) are constants such that X4 = ko + k1 (x + 1) + k2(x + 1)2 + k3(x + II + k4 (x + 1)4 is an identity in x, what is the value of k3? (A) -4 (B) -3 (C) -2 (0) 3 (E) 4 CALCULUS I • 49. If the function f(x) = If is expanded in powers of x, what is the minimum number of terms of the Taylor series that must be used to ensure that the resulting polynomial will approximate ~ to within 1O--6? • (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 50. There is exactly one value of the constant k such that lim (e X2 -x2 -1Xcosx-l) x->O Xk is finite and nonzero. What is this value of k, and what is the limit, L? 1 (A) k = 4, L = -2 1 (D) k = 6, L = --4 · , 1 (B) k = 6, L = "4 1 (E) k = 4, L = 2 GRE MATHEMATICS SUBJECT TEST (C) k = 4, L = 1 • •