MATH 152, Fall 2017 COMMON EXAM II - VERSION A

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1 MATH 15, Fall 17 COMMON EXAM II - VERSION A LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited.. TURN OFF cell phones and put them away. If a cell phone is seen during the exam, your exam will be collected and you will receive a zero. 3. In Part 1 (Problems 1-16), mark the correct choice on your ScanTron using a No. pencil. The scantrons will not be returned, therefore for your own records, also record your choices on your exam! 4. In Part (Problems 17-19), present your solutions in the space provided. Show all your work neatly and concisely and clearly indicate your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. 5. Be sure to write your name, section number and version letter of the exam on the ScanTron form. THE AGGIE HONOR CODE An Aggie does not lie, cheat or steal, or tolerate those who do. Signature: 1

3 1. After an appropriate substitution, the integral π/4 (a) sec θ tan θ dθ π/6 (b) 4 (c) (d) 4 (e) π/ π/6 π/3 π/ π/ π/3 π/ π/6 cos θ dθ cos θ dθ cos θ dθ cos θ dθ Part 1: Multiple Choice (5 points each) 3 4 x dx can be written as. Find the arc length of the curve parameterized by x = 3 sin(t), y = 3 cos(t), t π 4 (a) 3 π (b) 3 π (c) 3π (d) 3π (e) 3π 4 3. Which of the following is the correct partial fraction decomposition for (a) A x + B x + (b) Ax + B x C x D x 1 + Ex + F x Cx + D x 1 + Ex + F x + 1 (c) A x + B x + Cx + D x 1 + Ex + F x + 1 (d) A x + B x 1 + C x + 1 (e) A x + B x + C x D (x + 1) + E x 1 + F (x 1) 1 x (x 1)(x + 1)? 3

4 4. Which of the following integrals correctly calculates the area enclosed by one loop of the rose given by r = cos(3θ)? (a) (b) (c) 1 (d) (e) 1 π/6 π/6 π/6 π/6 π/ π/ π/ cos (3θ) dθ cos (3θ) dθ cos (3θ) dθ cos (3θ) dθ cos (3θ) dθ 5. Find the arc length for the parameterized curve given by x = e t t, y = 4e t/, t 3. (a) e 3 (b) e 3 1 (c) e (d) e 3 + (e) e What is the equivalent Cartesian equation for the polar curve r = 4 sin θ? (a) (x ) + (y ) = 4 (b) x + y = 4 (c) (x ) + y = 4 (d) x + y = (e) x + (y ) = 4 4

5 4x Evaluate x + 1 dx. (a) ln (b) 3 ln 5 + (c) 6 ln 5 + (d) 4 ln (e) 3 ln Find the area of the region inside the circle r = 6 sin θ and outside the circle r = 3. (a) 9 3 (b) 9 3 (c) (d) π (e) + 9π + 3π + 3π 3π 9 3 5

6 1 9. Evaluate ln x dx. 1 (a). (b) ln(). (c) 4 ln() 4. (d) 4 ln(4). (e). 1. What would be an appropriate substitution in order to evaluate (a) x + = 3 tan θ (b) x + 3 = tan θ (c) x + = 3 sec θ (d) x + = 3 sin θ (e) x + 3 = sec θ dx (x + 6x + 13) 3/? 11. The improper integral (a) diverges by comparison to (b) converges by comparison to (c) diverges by comparison to (d) diverges by comparison to (e) converges by comparison to e x + x dx x dx. x dx. (e x + x) dx. e x dx. e x dx. 6

7 1. Rewrite the parametric curve given by x = sin t, y = tan(t), t π/6, in Cartesian form. (a) y = x 1 x, x 1. (b) y = x, x 3. (c) y = x 1 x 1 x, x 1. (d) y = x 1 x, x 1. (e) y = 4x 1 x 1 x, x Which of the following pairs of polar coordinates are correct alternate representations of the point with polar coordinates (, 5π 6 )? (a) (, 5π 6 ) and (, π 6 ). (b) (, π 6 ) and (, 7π 6 ). (c) (, 5π 6 ) and (, π 6 ). (d) (, π 6 ) and (, 7π 6 ). (e) (, 11π 6 ) and (, π 6 ). 14. Which of the following correctly represents the surface area obtained when the parametric curve given by x = t sin t, y = t cos t, t π, is rotated about the y axis? (a) π (b) π (c) π (d) π (e) π π/ π/ π/ π/ π/ (t sin t) 1 + t dt. (t cos t) 1 + t dt. (t sin t) 1 + t + 4t sin t cos t dt. t sin t dt. t cos t dt. 7

8 15. What is the area inside the cardioid r = + cos θ? (a) 9π (b) 9π (c) 9π + (d) 4π + (e) 4π 16. The improper integral (a) converges to 5 7. (b) converges to (c) converges to 1 8. (d) converges to (e) diverges. 3 1 x 3 dx 8

9 Part : Work Out Directions: Present your solutions in the space provided. Show all your work neatly and concisely and box your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. 17. (8 points) Evaluate 3 1 dx. (x 1) 3/ 9

10 x x (8 points) Evaluate x 3 + 3x dx. 1

11 19. (8 points) Find the equations of the tangent lines to the curve x = 3t + 1, y = t + 1 that pass through the point (13, 9). FOR INSTRUCTOR USE ONLY Question Points Awarded Points TOTAL 14 11

MATH 151, FALL 2017 COMMON EXAM III - VERSION B

MATH 151, FALL 2017 COMMON EXAM III - VERSION B LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited. 2. TURN OFF cell