Calculus 2 Final Exam

Group X CALCULUS II Final Exam π/2 Z Student ID : .................................................................. 14/06/2022 2. Name-Surname 1 + sin4 x 0 A. : .................................................................. sin(2x) d x =? π 4 B. 1 INSTRUCTIONS C. 0 D. • Duration of the exam is 100 minutes. π 2 E. − π2 • Wrong answers aren’t counted against you. Z 3. Which one of the following may be obtained from the integral • Use pencil for marking your choices. a trigonometric substitution? R A. 25 tan2 θ sec θd θ R B. 5 tan2 θ sec θd θ • Usage of calculators is not allowed. • Write down your name-surname and id by pen. • No one is allowed to leave the classroom in the first 30 minutes of the exam. R tan2 θ dθ sec θ R D. 25 sin θd θ R tan θ E. 5 dθ sec2 θ Z ∞ 1 4. d x =? 4 (x − 2)(x − 3) C. 5 • Turn off your mobile devices. Otherwise, your exam is going to be invalidated. • Use the empty space in your exam paper for your computations. • Exam papers are attached. Do not attemp to split the papers. • At the end of the exam, deliver the exam paper and the optic form. • There are 20 questions in the exam. A. ln 2 QUESTIONS B. ln 3 1. Which one of the following is the area of the region enclosed by the parabola y 2 = − 12 x and the line 2y = x + 4 ? A. 9 B. 6 C. 3 D. 9/5 E. 5 2021–2022 Spring Page 1/4 C. ln 12 D. 3 ln 2 E. The given integral is divergent. p x2 x 2 + 25 d x using Group X MAT806 CALCULUS II Final Exam 5. The region bounded by y = 0 and y = sin x for 0 ≤ x ≤ π is rotated about the line y = −3. Which of the following integrals gives the volume of the generated solid? π Z A. 0 π Z B. 0 π Z C. 0 π Z D. 0 ¡ ¢ π (sin x + 3)2 − 32 d x π(sin x + 3)2 d x π(sin2 x + 32 ) d x π(sin2 x − 32 ) d x π Z E. 0 7. Which of the following integral expressions gives the surface area of a sphere of radius a? s Z ap x2 A. 2π dx a2 − x2 1 + 2 a − x2 −a s Z ap x2 B. 4π a2 − x2 1 + 2 dx a − x2 −a s Z ap x2 dx C. 2π a2 + x2 1 + 2 a − x2 −a s Z pa p x2 D. 2π p dx a2 − x2 1 + 2 a − x2 − a s Z pa p x2 E. 4π p a2 + x2 1 + 2 dx a − x2 − a 8. 2πx(sin x + 3) d x ∞ 4n+2 X =? n−1 n=1 6 A. 192 B. 288 3 6. The region bounded by y = 0 and y = x for 0 ≤ x ≤ 1 is rotated about the line x = 2. Find the volume of the generated solid. C. 18 D. 3 E. ∞ A. 3π 5 B. 2π 5 C. π 2 D. 3π 7 E. π 7 2021–2022 Spring 14/06/2022 9. Which of the following series converges absolutely ? ∞ cos nπ X 2n n=1 ∞ (−1)n X B. p n=1 n + 1 p ∞ (−1)n n + 1 X C. p n+ n n=1 ∞ (−1)n X D. n=2 ln n ∞ n! X E. n n=1 2 A. Page 2/4 MAT806 CALCULUS II Final Exam Group X 10. Which of the followings is the interval of convergence for the power series ∞ X 1 (x − 2)n ? n n2 n=1 14/06/2022 13. Given the function f (x, y) = D. (0, 5] 1 A. − p 2 2 1 B. p 2 2 1 C. 3/4 2 D. 0 E. [0, 5] E. −1 A. [0, 4) B. (0, 4) C. [0, 4] 11. Which of the followings is the Taylor series expansion for the function f (x) = e x about x = −5 ? A. B. 14. Given that z = ln(x 2 + y 2 ) and x = u − v, y = u + v, evaluate A. ∞ e −5 X (x + 5)n n! n=0 B. ∞ (x + 5)n X n! n=0 C. ∞ e −5 X C. (x − 5)n n! n=0 ∞ xn X D. n=0 n! ∞ X x 5n+1 E. n=0 (5n + 1)! D. E. ∂z ∂u 2u u2 + v 2 2v u2 + v 2 1 u2 + v 2 2u u2 − v 2 2v u2 − v 2 15. Which one of the followings is the directional derivative of the function f (x, y) = x 2 + x y 2 − y at (1,1) in the direction of u = i − 2j ? B. Limit does not exist. 1 A. p 5 −1 B. p 5 p C. 5 p D. − 5 C. 0 E. 1 p p x − y +2 x −2 y =? 12. lim p p (x,y)→(0,0) x− y A. 2 D. 1 E. 3 2021–2022 Spring p 4 − x 2 − y 2 , f x y (1, 1) =? Page 3/4 MAT806 CALCULUS II Final Exam Ï 16. Calculate yd A where D is bounded by y = x − 2 and x = y 2 . D Group X 14/06/2022 18. Which one of the following integral is obtained ¶when the order of integration is Z 2 µZ 4 f (x, y)d y d x ? reversed in the iterated integral 0 p Z 4 µZ y A. A. 0 9 4 B. 0 B. 0 Z 2 µZ 15 4 Z 4 ÃZ f (x, y)d x d y 0 p Z 4 ÃZ ! x f (x, y)d x d y E. 3 4 ! y2 D. 0 E. f (x, y)d x d y ! 0 Z 2 ÃZ 7 4 ¶ f (x, y)d x d y 0 D. y2 C. 39 4 ¶ f (x, y)d x d y y y2 0 C. p x2 1 ¯ p ¯ 19. Given that D = {(x, y)¯0 ≤ x ≤ 1, x 2 ≤ y ≤ x}, Ï D Z 1Z 2¡ 17. 0 A. 1 3 2 1 B. 2 C. 1 A. ¢ x y 2 + x y d xd y =? D. 2 5 E. 3 5 4 Ï B. 5 3 C. 11 3 D. 7 4 E. 2 3 1 d A =? x 2021–2022 Spring 20. Evaluate the integral T A. 18 16 B. 3 8 C. 3 D. 9 E. 3 Page 4/4 ¯ π ¯ (x + y)d A over the domain T = {(r, θ)¯0 ≤ r ≤ 3, 0 ≤ θ ≤ } 2