FINAL EXAM, MATH 131 FRIDAY, DECEMBER 20,2002

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1 FINAL EXAM, MATH 131 FRIDAY, DECEMBER 20,2002 This examination has 18 multiple choice questions and 5 true-false questions. Please check it over and if you find it to be incomplete, notify the proctor. Do all your supporting calculations in this booklet. In case of a doubtful mark on your answer card, we can then check here. When you mark your card, use a soft lead pencil (#2). Erase fully any answers you want to change. Problems 1 through 18 are worth 5 points apiece. Problems 19 through 23 are worth 2 points apiece. There is a total of 100 points for the whole examination. You may use a Mathemati~ Department approved scientific calculator. 1. Find the inverse function for the function f(x) = (A).:-=-! x+2 (B)!-=..:: x+2 (C)..!~ 2x-l x+1 2x+1 (D)~ x-i (E) ~! x+l (F).!.~ 1+2.1: (G) ~~ x+l (H)!!.~ 3x+ 1 (I) ~~ x-3 ( J) ~..:!:.2 x+3

2 2 FINAL EXAM, MATH 131 FRIDAY, DECEMBER 20, Eliminate the parameter to find the Cartesian equation of the curve 1r 7r X = 2 sec 9, tl = 3 tan < 9 < -. 11, 2--2 (A) X2 + y2 = 1 (B) 2z2-3y2 = 6 (C) 4z2 + g,jl = 36 (D) 9z2 + 4y2 = 25 (E) 9x2-4y2 = 36 (F) 4%2-9y2 = 25 (G) 2X2 + 3y2 = 25 (H) 2z2-3y2 = 16 (1) 36x2-25y2 = 200 (J) 36x2 + 25y2 = 200

3 FINAL EXAM, MATH 131 FRIDAY, DECEMBER 20, Suppose we know that a bacteria population triples every 4 hours. Assuming exponential growth, if we start with 100 bacteria, how many bacteria will there be 32 hours later? (A) 656,100 (B) 646,100 (C) 636,100 (D) 626,100 (E) 616, 100 (F) 606, 100 (G) 596,100 (H) 586, 100 (I) 576, 100 (J) 566,100

4 4 FINAL EXAM, MATH 131 FRIDAY, DECEMBER 20, In(..fX"+1.. ~+1) = (A) (X2 + 1)..fi+I (B) (X2 + 1)(x + 1) (C) i(x2 + 1)(x + 1) (D) X2 + x + 1 (E) x2ln(x + 1) (F) x2 + In(x + 1) (G)! In(x + 1) + 2x (H)!In(x + 1) + X2 + 1 (I) 2ln(x + 1) + X2 + 1 (J) In(X2 + 1)

5 FINAL EXAM, MAm 131 FRIDAY, DECEMBER 20, (A) 0 (B) ~ (C) 1 (D) ~ (E) 2 (F) ~ (G) 3 (H) ~ (I) 4 (J) limit doffi not exist

6 6. Let y = tan x FINAL EXAM, MATH FRIDAY, DECEMBER 20,2002 ( A ) limh-+oo tanh (B) limh-+oo Ii 7r tan 12 'If tanh tan 12" (C) limh-+o h tan(fi_t~) - tann (D) limh-+o h (E) limh-+o tan ~ - tan h wh 12 tan(b + h) (F) limh-+fi (G) limh~i2 (H) limh-+fi 11. Ii tanh tan(fi +h) -~Ji h 7fh tan 12 (I) limh-+o h (J) limb-"*,o tanh h 11" tan 12

7 FINAL EXAM, MATH 131 PRlDAY, DECEMBER 20, Given that f(x) is continuous on the interval [1,4], with f(l) = 2 and f(4) = -3 we can conclude that f(x) has a root between 1 and 4. This conclusion follows from (A) the Extreme Value Theorem. (B) the Squeeze Theorem. (0) Newton's Method. (D) the Mean Value Theorem. (E) the First Derivative Test. (F) the Intermediate Value Theorem. (G) L'Hospital's Rule. (H) Linear Approximation. (I) the Chain Rule. (J) the Uncertainty Principle.

8 8 FINAL EXAM, MATH 131 FRIDAY, DECEMBER 20, Suppose we know that 1(7) 1(6.98). (A) (B) (C) (D) (E) (F) (G) (H) (I) (J) l 2 and /,(7) Use linear approximation to estimate

9 FINAL EXAM, MATH 131 FRIDAY, DECEMBER 20, A ladder 12 feet long leans against a wall. The top of the ladder slides down the wall at the rate of 1 ft/sec. How fast (in ft/sec) is the base of the ladder sliding away from the wall when it is 2 feet from the wall? (A) 6 (B) M (C) 20 (D).f35 (E) {fi5 (F) 9 (G).fi;f (H) 12 (1) {f56 (J)

10 10 FINAL EXAM, MAm 131 x 10. Let f(x) Then f'(x) is sin x + cosx FRIDAY, DECEMBER 20,2002 (F) (x -l)sinx + (x + l)~x (sin x + ~X)2 (G) %~% - (z:!)~% (sin X + CO8X)2 (I) x(~x - CO8X) (smx + CO8X)2 (J) x(~x - sinx) (smx +~X)2

11 FINAL EXAM, MATH Find the equation to the tangent line to the curve y (A) y = x - 1 (B) 2y = x - 1 (0) y = 2(x - 1) (D) 3y = 4(x - 1) (E) 4y = 3(x - 1) (F) ~y = x - 1 (0) V'Jy = x - 1 (H) 3y = x - 1 (I) 2y = 3(x - 1) (J) 7(y-1)=x- 8 FRIDAY, DECEMBER 20, x-l x+l at the point (1,0) I

12 12 FINAL EXAM, MATH 131 FRIDAY, DECEMBER 20, The derivative of y eza.z is: (A) ezc08x-l (B) ~C08XX COB x (C) _~C08Xxsinx (D) ezc08x(cosx - x sin x) (E) ~C08X(XCOSx - sin x) (F) ex~z-l(cosx - x sin x) (G) ~OO8Z(x + cosx) (H) ~OO8Z(sinx + xcosx) (I) e(x-l)oo8z(cosx - x sin x) (J) ~(C08z-1)(sinx + x c~x)

13 FINAL EXAM, MATH 131 FRIDAY. DECEMBER Find the derivative of y with respect to x, given that x2y + xy2 3x z,2 + 2zy (B) 2z,y + y2 3z (C) Z2 + 2xy X2+2xy (F) 3-2zy - Y2 x+2y 3-2x-y 3-2x-y x+2y (1) 2x +7/ x + 27/ (J).x + 2y 2x + 11

14 14 FINAL EXAM, MATH 131 FRIDAY, DECEMBER 20, The graph of 11 = 2z3 3X2 12x is concave up precisely on the interval(s): (A) (1,2) (B) (-2, (C) (-«>, j) -2) and (1,00) (D) (-1,2) (E) (2,00) (F) (~, 00) (G) (-00,00) (H) ( -00, 1) and (2,00) (I) (l,2) (J) ( '00, 1)

15 FINAL EXAM, MATH z~1ijo LXlOO fa) (2) (C) (D) (E) (F) (G) (H) (1) (J) e roof e 100 e IOOi! e eloo IOOi FRIDAY, DECEMBER 20,

16 16 FINAL EXAM, MATH 131 FRIDAY, DECEMBER 20, The absolute maximum ~ of the function f(x) = sinx + coax on the interval 0 < x <.!: is: - -3 (A) (B) (C) (D) (E) (F) (0) (H) (I) (3) 1 V2 V :1 4 3".Jl Ii There is no absolute maximum value.

17 FINAL EXAM, MATH 131 FRIDAY. DECEMBER 20, A box with a square base and a ~ top must have a volume of C cubic inches. The minimal surface area of such a box is: (A) C! (B) 8C (C) va (D) 2va (E) Q2 (F) C- i (G) 4C3 (H) va~ 1 (I) v'~i (J) 6C~ 11

18 18 FINAL EXAM, MAm 131 FRIDAY, DECEMBER 20, We use Newton's Method to search for a root of the equation X4 + x the first ~timate to be Xl = 1, the value of the third ~timate X3 is: (A) 1.3 (B) (C) (D) (E) (F) (G) (H) (1) (J) = o. Taking

19 FINAL EXAM, MATH 131 FRIDAY. DECEMBER 20,2002 It True-False Section 19. Hy (A) True (B) False e2x, then y' 2xe2:t-l 20. If f(x) is a one-to-one function, then f-1($) (A) True (B) False MO 21. If x > 0 and a > 0 then In (~ lnx lna. (A) True (B) False

20 20 FINAL EXAM, MAm 131 FRIDAY, DECEMBER 20, If f is continuous on an open interval (a, b), then f attains an absolute maximum value at some number c in (a, b). (A) Thue (B) False (T) True (F) False lnx x