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Find the most general antiderivative. 1) 1 x5 - x5 - 1 6 ∫ dx A) -5x4 - 5x5 + C B) 1 6x6 - x4 4 + 1 36 + C C) 1 5x6 - x6 6 - 1 6x + C D) -1 4x4 - x6 6 - x 6 + C 2) 9t2 + t 5 ∫ dt A) 18t + 1 5 + C B) 3t3 + t + C C) 3t3 + t 2 10 + C D) 27t3 + 2 5 t2 + C 3) y 7 + 2 y ∫ dy A) 2 21 y3/2  - 4 y + C B) 3 14 y3/2 + 1 4 y + C C) 1 14 y - 1 4 y+C D) 2 21 y3/2 + 4 y + C 4) sin θ(cot θ + csc θ) ∫ dθ A) sin θ + θ + C B) sin θ + C C) csc θ + cos θ + C D) cos θ  + C Solve the initial value problem. 5) dy dx = 3x-3/4,  y(1) = 4 A) y = 12x1/4 - 8 B) y = 3x1/4 + 1 C) y =- 3 4 x-7/4 - 2 D) y = 12x1/4 + 48 Solve the problem. 6) Suppose that h is continuous and that 4 -1 h(x) dx ∫ = 7 and 8 4 h(x) dx = -11. ∫   Find 8 -1 h(t) dt ∫ and -1 8 h(t) dt. ∫ A) 18; -18 B) 4; -4 C) -4; 4 D) -18; 18 7) Suppose that f is continuous and that 3 -3 f(z) dz = 0 ∫ and 5 -3 f(z) dz = 7. ∫   Find - 5 3 4f(x) dx ∫ . A) 28 B) -4 C) -7 D) -28 Find the area of the shaded region. 8) A) 26 3 B) 23 3 C) 25 3 D) 22 3 Evaluate the integral. 9) 16 0 5 x ∫ dx A) 40 B) 320 C) 640 3 D) 480 10) 6 -2 6x5 ∫ dx A) 46,592 B) 1280 C) 279,552 D) -46,592 Evaluate the integral. 1) 3π/4 -π/4 8 sec θ tan θ dθ ∫ A) -8 2 B) -16 2 C) 8 2 D) 0 Find the derivative. 2) y = x6 0 cos t dt ∫ A) 6x5 cos (x3) B) cos (x3) C) sin (x3) D) cos (x3) - 1 Find the average value over the given interval. 3) f(x) = sec2 x; [0, π 4 ] A) 4 π B) 0 C) 1 D) π 4 Find the point(s) at which the given function equals its average value on the given interval. 4) f(x) = 1 - x2; [-4, 5] A) 5 B) ±3 C) ± 6 D) ± 7 Evaluate the integral. 5) x dx (7x2 + 3)5 ∫ A) - 7 3 (7x2 + 3)-6 + C B) - 1 14 (7x2 + 3)-6 + C C) - 7 3 (7x2 + 3)-4 + C D) - 1 56 (7x2 + 3)-4 + C 6) 3x2 ∫ 4 10 + 2x3 dx A) - 2 10 + 2x3 -3/4 + C B) 3 10 + 2x3 5/4 + C C) 12 5 10 + 2x3 5/4 + C D) 2 5 10 + 2x3 5/4 + C 7) csc2 (6θ + 8) dθ ∫ A) - 1 6 cot (6θ + 8) + C B) -cot (6θ + 8) + C C) 6 cot (6θ + 8) + C D) 12 csc (6θ + 8) cot (6θ + 8) + C 8) sin t (4 + cos t)6 dt ∫ A) 1 (4 + cos t)5 + C B) 1 5(4 + cos t)5 + C C) 5 (4 + cos t)5 + C D) 1 7(4 + cos t)7 + C Use the substitution formula to evaluate the integral. 9) 1 0 4 r dr 4 + 2r2 ∫ A) 6 - 2 B) 2 6 - 4 C) 6 2 - 1 D) - 2 6 + 4 10) 2π π/3 4 cos3 x sin x dx ∫ A) - 15 4 B) - 32769 524288 C) 15 16 D) - 15 16 MATH 2413 - Quiz 8 Sections 5.1, 5.2, and 5.4 Find the derivative of y with respect to x, t, or θ, as appropriate. 1) y = x6ln x - 1 3 x3 A) x5 - x2 + 6x5ln x B) x6ln x - x2 + 6x5 C) 6x5 - x2 D) 7x5 - x2 2) y = ln(ln 8x) A) 1 8x B) 1 x ln 8x C) 1 ln 8x D) 1 x Evaluate the integral. 3) cos x dx 1 + 5 sin x ∫ A) ln 1 + 5 sin x + C B) 5 ln 1 + 5 sin x + C C) 5 sin x + C D) 1 5 ln 1 + 5 sin x + C Find the integral. 4) 5x4 1 + 4x5 ∫ dx A) 1 4 ln |1 + 4x4| + C B) 1 4 e4x + C C) 1 4 e5x + C D) 1 4 ln |1 + 4x5| + C 5) 2x3 + x2 - 4x + 2 x + 4 ∫ dx A) (2x3/3) - 7x2/2 + 24x - 7 ln 1 x + 4 + C B) (2x3/3) + 9x2/2 + 8x - 7 ln |x + 4| + C C) (2x3/3) - 7x2/2 + 24x - 94 ln 1 x + 4 + C D) (2x3/3) - 7x2/2 + 24x - 94 ln |x + 4| + C Find Dxy. 6) y = e(4x2 + 2x) A) e(8x + 2) B) e(4x2 + 2x) C) (8x + 2) e(4x2 + 2x) D) (4x2 + 2x) e(4x2 + 2x) Find the derivative. 7) g(x) = e3x cos(2x) A) 3e3x cos(2x) + e3x sin(2x) B) -3e3x sin(2x) C) 3e3x cos(2x) - 2e3x sin(2x) D) e3x cos(2x) - 2e3x sin(2x) Evaluate the integral. 8) x4e-x5 ∫ dx A) - 1 5 e-x5 + C B) e-x5 + C C) -5e-x6 + C D) - 1 5 e-x6 + C 9) e1/x 4x2 ∫ dx A) e1/x 4 + C B) e -1/x 4 + C C) - e1/x 4 + C D) - 4 e1/x + C 10) e 5 θ 1 + e 5 θ ∫ d θ A) ln (1 + e 5 θ ) + C B) ln (1 + e 5 θ ) 5 + C C) 5 ln (1 + e 5 θ ) + C D) ln (1 + 5 e θ ) 5 + C MATH 2413 - Quiz 9 Sections 5.3 and 5.5 Find the value of the derivative of the inverse of the function. 1) Let f(x) = x2 + 4x + 9 be defined for x ≥ - 2. Find (f-1)'(14). A) 1 6 B) 1 32 C) 32 D) 1 14 2) Given that f(x) = 6x - cos x, find (f-1)'(-1). A) - 1 6 B) - 1 5 C) 1 7 D) 1 6 Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 3) y = (sin x)cos x A) (sin x)cos x(cos x cot x - sin x ln (sin x)) B) cos x cot x - ln (sin x) C) cos x ln ( sin x) D) cos x cot x - sin x ln(sin x) 4) y = (x + 8)x A) (x + 8)x ln(x + 8) + x x + 8 B) x + (8)x-1 C) ln(x + 8) + x x + 8 D) x ln(x + 8) Find the derivative of y with respect to the independent variable. 5) y = 5cos πθ A) 5cos πθ B) -5cos πθ ln 5 sin πθ C) π5cos πθ ln 5 D) -π5cos πθ ln 5 sin πθ Find dy dx . 6) y = 8 4x2 + 9x A) (8x + 9) 8 4x2 + 9x B) 88x + 9 ∙ ln 8 C) (8x + 9) 8 4x2 + 9x ∙ log8 x D) (8x + 9) 8 4x2 + 9x ∙ ln 8 Find the derivative of the function. 7) f(x) = log9 (x6 + 1) A) 6(ln 9) x5 (x6 + 1) B) 6x5 (ln 9) (x6 + 1) C) 6x5 (x6 + 1) D) 1 (ln 9) (x6 + 1) + 6x5 8) y = log 5 7x + 9 A) 7 2(ln 5)(7x + 9) B) 7 ln 5 C) 7 ln 5 (7x + 9) D) 7 ln5 7x + 9 Evaluate the integral. 9) 6 xx ∫ dx A) x 6 x ln 6 + C B) 6 x ln 6 + C C) 2 6 x + C D) 2 6 x ln 6 + C 10) 8 2 x + 6 ∫ dx A) ( 8 2 x + 6 )ln 8 2 + C B) 8 2 x + 6 2 + C C) 2 ∙ 8 2 x + 6 ln 8 + C D) 8 2 x + 6 2 ln 8 + C Quiz Note: It is recommended that you save your response as you complete each question. Question 1 (1 point) Find the limit: Question 1 options: a) 0 b) c) –1 d) e) 1 Question 2 (1 point) Find the x-values (if any) at which the function is not continuous. Which of the discontinuitites are removable? Question 2 options: a) (Removable), (Not removable) b) (Not removable), (Not removable) c) No points of continuity. d) (Not removable), (Removable) e) No points of discontinuity. Question 3 (1 point) Determine the limit (if it exists): Question 3 options: a) 0 b) 2 c) Does not exist. d) 1 e) Question 4 (1 point) Use the product rule to differentiate. Question 4 options: a) b) c) d) e) Both A and B Question 5 (1 point) Use the quotient rule to differentiate. Question 5 options: a) b) c) d) e) Question 6 (1 point) Find the derivative of the function. Question 6 options: a) b) c) d) e) Question 7 (1 point) Find the indefinite integral. Question 7 options: a) b) c) d) e) Question 8 (1 point) Find the indefinite integral. Question 8 options: a) b) c) d) e) Question 9 (1 point) Find the indefinite integral. Question 9 options: a) +C b) c) d) e) Question 10 (1 point) Find the indefinite integral. Question 10 options: a) b) c) d) e)