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Lawrence Perkins
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1 Math 155 FINAL EXAM A May 5, 2003 NAME: Section # SSN: X X X X Question Grade 1 5 (out of 25) 6 10 (out of 25) 11 (out of 20) 12 (out of 20) 13 (out of 10) 14 (out of 10) 15 (out of 16) 16 (out of 24) 17 (out of 12) 18 (out of 12) 19 (out of 26) TOTAL (out of 200)

2 No Calculators Permitted. PART 1.A This portion of the exam is to test basic knowledge and calculation skills to a correct solution. The questions are multiple choice with None of these as a possible valid choice. No partial credit is given in this section, so work very carefully. Value: 5 points each x 2 x 2 1. Compute lim x 1 x 2. The answer is: + 3x + 2 A) 0; B) 6 5 ; C) 3 ; D) 3; 8 E) 2; F) 6 ; G) 3; H) The limit does not exist Find the indefinite integral (8x 3 + 4x x 2 ) dx: A) 4x 4 + 2x 2 x 3 + C; B) 8x 4 + 2x 2 x 1 + C; C) 2x 4 + 2x 2 + x 1 + C; D) 8 3 x3 + 2x 2 x C; E) 2x 4 + 4x 2 x 1 + C; F) 8x 4 + 4x 2 2x 1 + C; G) None of these. 3. The function f(x) is continuous on the interval [3, 6] with f(5) = 4 and f(4) = 2. The limit lim x 2 f(4x 2 11) equals: A) 16; B) 5; C) 4; D) 2; E) 11; F) None of these. 4. For a function f(x) = 4x 5 3x 3 + x 2, the derivative f (x) is, after simplification: A) 4x 4 3x 2 + x 3 ; B) 20x 4 9x 2 2x 1 ; C) 20x 4 9x 2 + 2x 1 ; D) 20x 4 9x 2 x 3 ; E) 20x 4 + 9x 2 2x 3 ; F) 20x 4 9x 2 + 2x 1 ; G) 20x 4 9x 2 2x 3 ; H) None of these. 2

3 5. The derivative, f (x), of the function f(x) = sin(x 2 + 2) after simplification is A) cos(x 2 + x); B) cos(x 2 + x); C) cos(x 2 + x)(x 2 + x) + (2x + 1) sin(x 2 + x); D) (2x + 1) sin(x 2 + x); E) cos(2x + 1); F) (2x + 1) cos(x 2 + x); G) None of these. 6. The slope of the line tangent to the curve y 3 + xy 15x + 5 = 0 at the point (1, 2) is: A) 1; B) 3y 2 + y 15; C) 0; D) 1; E) 14; F) ; G) ; H) None of these Give the x-coordinate of any local maxima or minima of the function f(x) = 2x 3 +3x 2 +4: A) Max: x = 0, x = 1, Min: None; B) Max: x = 1, Min: x = 1; C) Max: x = 4, Min: x = 2; D) Max: x = 0, Min: x = 1; E) Max: x = 0, x = 1, Min: None; F) Max: x = 0, Min: x = 1; G) Max: x = 2, Min: x = 2; H) None of these 8. The line tangent to the curve y = x at the point (1, 2) is: A) y 2 = 3(x 1); B) y = 3x + 1; C) y 2 = 3x 2 ; D) y = 3x 2 1; E) y = 3x 3; F) x 1 = 3(y 2); G) y + 2 = 3(x + 1); H) None of these. 3

4 9. Let F (x) = x 0 2(t 1)(t 7) dt be a function defined for 0 x 8. From Part I of the Fundamental Theorem of Calculus, we know that the function F (x) has two local extreme points in the open interval (0, 8), which are A) Max at x = 2, Min at x = 7; B) Max at x = 7, Min at x = 1; C) Max at x = 1, Min at x = 2; D) Max at x = 8, Min at x = 0; E) Max at x = 1, Min at x = 7; F) Max: None, Min at x = 1 and x = 7; G) Max at x = 0 and x = 7, Min: None; H) None of these. 10. The graph of the function f(x) = x2 + 1 x 2 1 has A) no asymptotes of any kind; B) two vertical asymptotes and no horizontal asymptotes; C) exactly one horizontal and one vertical asymptotes D) two horizontal and one vertical asymptotes E) just one vertical asymptote and no horizontal asymptotes; F) two vertical and one horizontal asymptotes G) None of these. 4

5 PART 2: This portion of the exam will be graded on a partial credit basis. Answers without supporting work shown on the paper will receive NO credit. No calculators permitted. 11. Differentiate the following functions (DO NOT SIMPLIFY) (a) (10 points) f(x) = (2 x 3 ) 3 (x 2 5) 5/2 (b) (10 points) f(x) = cos(1 + 3t2 ) (x 1) Evaluate the following integrals: (a) (10 points) 1 0 5x 2 3x + 1 dx. x (b) (10 points) 3x 1 x 2 dx 5

6 13. (10 points) Find the average value of the function f(x) = sin(x) cos 2 (x) on the closed interval [0, π]. 14. (10 points) Let F (x) = critical points of F (x). x t 1 + t 4 dt be a function defined on (, ). Find the 6

7 15. Set up an integral in the following problems. Do not evaluate the integral. (a) (8 points) Set up an integral that gives the surface area of revolution generated by rotating the smooth arc with equation y = x x 3 and with 0 x 1 around the x-axis. (Do not evaluate the integral). (b) (8 points) Set up an integral that gives the surface area of revolution generated by rotating the smooth arc with equation y = x x 3 and with 0 x 1 around the y-axis. (Do not evaluate the integral). 7

8 16. Follow the instruction for these problems. (a) (16 points) First set up an integral that gives the volume of the solid generated by rotating the region bounded by the curves y = 2x x 2 and y = 0 around the x-axis. Then evaluate the integral to find the volume. (b) (8 points) Set up an integral that gives the volume of the solid generated by rotating the region bounded by the curves y = 2x x 2 and y = 0 around the line x = 2. (Do not evaluate the integral). 8

9 17. (12 points) A spherical balloon is filled with water. If the water is leaking out at 2 cm 3 /min, find the rate at which the radius is changing when the volume of the balloon is V = 4000π cm 3. (Hint: If the radius of the sphere is r, then the volume of the sphere 3 is V = 4π 3 r3 ). 18. (12 points) The sum of two nonnegative real numbers is 30. Find the minimum possible value of the sum of their cubes. 9

10 19. (26 points total) Consider the function g(x) = 2x 3 + 5x 2 4x. (a) (6 points) Find all open intervals on which g is increasing and those on which g is decreasing. (b) (5 points) Determine the (x, y) coordinates of any local maxima and minima. (c) (6 points) Find all open intervals on which g is concave up, and those on which it is concave down. (d) (3 points) Determine the (x, y) coordinate of the inflection point. (e) (6 points) Sketch the graph of y = g(x). In your graph, you should accurately present the following features of the curve: local max/min, x and y intercepts, increasing/decreasing, concavity and inflection points. 10

1) Find. a) b) c) d) e) 2) The function g is defined by the formula. Find the slope of the tangent line at x = 1. a) b) c) e) 3) Find.

1) Find. a) b) c) d) e) 2) The function g is defined by the formula. Find the slope of the tangent line at x = 1. a) b) c) e) 3) Find. 1 of 7 1) Find 2) The function g is defined by the formula Find the slope of the tangent line at x = 1. 3) Find 5 1 The limit does not exist. 4) The given function f has a removable discontinuity at x