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1 Math 30 - Final Exam Version Student Name: Instructor: INSTRUCTIONS READ THIS NOW Print your name in CAPITAL letters. It is your responsibility that your test paper is submitted to your professor. Please turn off all cellphones, audio recording/playing devices and any other electronic devices, and put them in your backpack. For each problem clearly put a circle around ONE and ONLY ONE correct answer, both on the problem page and on the front page; if your answers on the problem page and on the front page do not match, the answer on the front page will be taken as your final answer. Do not separate the front page from the rest of your exam; if you do so, your exam will not be graded! OFFICIAL USE ONLY VERSION T F B Total: Over: 00 For each problem, a correct answer is worth 4 points; selecting a wrong answer or more than one answer will result in - point; not selecting any answer gets 0 point. P. a b c d e P. a b c d e P3. a b c d e P4. a b c d e P5. a b c d e P6. a b c d e P7. a b c d e P8. a b c d e P9. a b c d e P0. a b c d e P. a b c d e P. a b c d e P4. a b c d e P5. a b c d e P6. a b c d e P7. a b c d e P8. a b c d e P9. a b c d e P0. a b c d e P. a b c d e P. a b c d e P3. a b c d e P4. a b c d e P5. a b c d e P3. a b c d e

2 Tulane University Mathematics Department FORMULAS x n dx = xn+ + C (n ) n + e x dx = e x + C sin x dx = cos x + C sec x dx = tan x + C sec x tan x dx = sec x + C tan x dx = ln sec x + C x + a dx = ( x ) a tan + C a ( x ) a x dx = sin + C a [ ] cos(a b)+cos(a+b) = cos a cos b [ ] sin(a b)+sin(a+b) = sin a cos b dx = ln x + C x a x dx = ax ln a + C cos x dx = sin x + C sec x dx = ln sec x + tan x + C csc x dx = cot x + C csc x cot x dx = csc x + C cot x dx = ln sin x + C + cos(θ) = cos θ cos(θ) = sin θ [ ] cos(a b) cos(a + b) = sin a sin b tan x + = sec x PROBLEMS START ON NEXT PAGE

3 Tulane University Mathematics Department 3 P. The domain of the function f(x) = x 3 4x x is (a) x and < x 0 and < x (b) x 0 and x (c) < x < and < x < 0 and 0 < x < and < x < (d) x < and < x 0 and x P. The limit below represents the derivative of a certain function f(x), evaluated at a particular point: ( 5 + h) 5 lim. h 0 h (a) This is the derivative of f(x) = 5x evaluated at x = 0 (b) This is the derivative of f(x) = (x 5) 5 evaluated at x = 5 (c) This is the derivative of f(x) = x evaluated at x = 5 (d) This is the derivative of f(x) = 5x evaluated at x = (e) none of the above statements is correct. P3. For what value of x does the curve y = e x x have a horizontal tangent line? (a) ln (b) x (c) e (d) 0 P4. What values of a and b make the piecewise function { ax + b x f(x) = x x < continuous and differentiable for all values of x? (a) a = and b = 3 (b) a = 3 and b = (c) a = and b = 3 (d) a = 3 and b =

4 Tulane University Mathematics Department 4 P5. For what value of x is the corresponding point on the graph of y = 4 x closest to the point (, )? (a) x = 0 (c) x = (b) x = (d) x = P6. Given the equation y 3 + xy x 3 = 6. Then dy dx is (a) dy dx = 3x + y 3y + x (b) dy dx = 3x y 3y + x (c) dy dx = 3x + y 3y x (d) dy dx = 3x y 3y x ( P7. The limit lim x 0 x sin x ) equals to x 3 (a) 6 (c) (b) 3 (d) P8. The value of the definite integral (a) (b) 9 (c) 5 (d) 7 6 (x + x ) dx is P9. The area of the region enclosed by y = x and y = 3 x is (a) 3 (c) (b) 9 (d) 3 5

5 Tulane University Mathematics Department 5 P0. A cable that weighs lb/ft is used to lift 800 lb of coal up a mineshaft 500 ft deep. The total work done is (a) 50,000 (ft-lb) (c) 650,000 (ft-lb) (b) 400,000 (ft-lb) (d) 800,000 (ft-lb) x 3 + 4x x + P. In computing the integral dx, which of the following gives the x 4 + x correct partial fraction decomposition of the rational function x3 + 4x x +? x 4 + x (a) x (c) x + x x + 3x x + (b) x + x x + (d) x + x x + P. Which of the following improper integrals diverges? (a) (b) 0 0 x dx x dx P3. In order to evaluate the integral (c) (d) x dx e x dx x x x + 4 dx a trigonometric substitution is made. Which of the following is the integral that results after making the appropriate substitution? (a) sec θ tan θ( tan θ ) dθ (b) (c) (d) tan θ( tan θ ) dθ sec θ sec θ( sec θ ) dθ tan θ sec θ( sec θ ) dθ

6 Tulane University Mathematics Department 6 P4. A solid is made by rotating the region that is below the curve y = sin x and above the x-axis for 0 x π about the line x =. The volume of the resulting solid is (a) π (b) π / (c) π + 4π (d) π / + 4π P5. Which of the following integrals represents the area of the surface obtained by rotating the curve y = x + 4, for x 5, about the y-axis? (a) (b) (c) (d) π(y + 4) + (y) dy π y 4 + 4(y 4) dy πx + 4(y 4) dy π(x + 4) + 4(y 4) dy P6. In solving the linear differential equation y xy = ( x)e x, the integrating factor is (a) e x (c) e x3 (b) e x (d) e x3 { y P7. Given the initial-valued problem xy = ( x)e x y(0) = (a) e e 4 (b) e + e 4 (c) e e 4 (d) e + e 4. Then y() is P8. The population P of wolves in a certain forest is modeled by the logistic equation ( dp dt = P P ), where t refers to time and is measured in years. 500 Initially, there are 50 wolves in the forest. Then lim P (t) is equal to t (a) 0 (c) 5 (b) 45 (d) 500

7 Tulane University Mathematics Department 7 P9. At which of the following points on the parametric curve { x = t 3 y = + 4t t does the tangent line have slope? (a) (, 4) (c) (0, ) (b) (, 4) (d) (, /3) P0. The arc length of the polar curve r = 4 cos θ, for 0 θ π, is (a) π/ (c) π (b) π (d) 4π P. Find the area enclosed by the limaçon given in polar coordinates by the equation r = + sin θ. (a) 5π/8 (c) 9π/8 (b) π (d) π P. If the series n= a n converges to, and s n = a + a + + a n is its n-th partial sum, then (a) lim a n = and lim s n = 0 n n (b) lim a n = 0 and lim s n = n n (c) lim a n = and lim s n = n n (d) lim a n = 0 and lim s n = 0 n n (e) none of the above statements is correct. { P3. The sequence a n = ( )n (n + 3) } is 3n + n= (a) convergent to 0 (c) convergent to 3/ (b) convergent to /3 (d) divergent

8 Tulane University Mathematics Department 8 P4. Consider the following series: () n= n () n= cos n n (3) ne n. n= Then (a) only () and () are convergent (b) only () and (3) are convergent (c) only () and (3) are convergent (d) all of these series are convergent (e) none of these series are convergent. P5. The series n= ( ) n n + is (a) absolutely convergent (b) conditionally convergent (c) divergent (d) convergent on weekdays, but divergent on weekends STOP. THIS IS THE END.