gray (tmg2549) - HW14 - hamrick - (53665) This print-out should have 23 questions. Multiple-choice questions may continue on the next column or page - find all choices before answering. 001 5. f (x) = 3 sin x + 7 cos x2 6. f (x) = 3 cos x + 7 sin x + 8 10.0 points Find the most general antiderivative, F , of the function f (x) = 9x2 10x + 8 . 1. F (x) = 9x3 10x2 + 8x + C 3 004 g (x) = 1. g(x) = 2 3. F (x) = 3x3 + 5x2 + 8x 2. g(x) = 4. F (x) = 3x3 5x2 + 8x 3. g(x) = 5. F (x) = 3x3 + 5x2 + 8x + C 4. g(x) = 10.0 points Find the value of f (1) when f (t) = 9t + 2 5. g(x) = f (1) = 6, 003 f (1) = 5 . 10.0 points Find f (x) when f (x) = 3 cos x 7 sin x and f (0) = 5. 1. f (x) = 3 cos x + 7 sin x + 2 2. f (x) = 3 cos x + 7 sin x + 8 3. f (x) = 3 sin x + 7 cos x2 4. f (x) = 3 sin x + 7 cos x + 2 5x2 + 4x + 1 . x \u0013 \u0012 4 2 x x + x+1 +C 3 \u0013 \u0012 4 2 2 x x + x+1 +C 3 \u0013 \u0012 4 2 2 x x + x1 +C 3 \u0001 2 x 5x2 + 4x 1 + C \u0001 x 5x2 + 4x + 1 + C \u0001 6. g(x) = 2 x 5x2 + 4x + 1 + C 005 and 10.0 points Find all functions g such that 2. F (x) = 3x 5x + 8x + C 002 1 10.0 points Find the most general function f such that f (x) = 18 cos(3x) . 1. f (x) = 3 cos(3x) + Cx2 + D 2. f (x) = 3 sin(x) + Cx2 + D 3. f (x) = 2 sin(x) + Cx + D 4. f (x) = 2 cos(3x) + Cx + D 5. f (x) = 2 cos(x) + Cx + D 6. f (x) = 3 sin(3x) + Cx + D 006 10.0 points gray (tmg2549) - HW14 - hamrick - (53665) If F = F (x) is the unique anti-derivative of f (x) = (2 x)2 6 (2 x)2 which satisfies F (0) = 0, find F (3). 5 9 1. f () = + 3 2 2 3 9 3 2 2 2. f () = 3 9 3 3. f () = + 2 2 1. F (3) = 11 2. F (3) = 13 4. f () = 5 7 3 2 2 5. f () = 3 7 3 2 2 3. F (3) = 12 4. F (3) = 25 2 5. F (3) = 23 2 007 5 7 6. f () = + 3 2 2 009 10.0 points Find the unique anti-derivative F of f (x) = 2e4x + 3e2x + 5e2x e2x for which F (0) = 0. 5 1 1. F (x) = e2x + 3x e4x + 4 4 2. F (x) = 2 3 1 4x e 3x + e2x 2 4 3. F (x) = e2x + 3x e2x 10.0 points \u0010 \u0011 when Find f (x) on , 2 2 f (x) = 5 + 4 tan2 x and f (0) = 3. 1. f (x) = 3 + x + 4 tan2 x 2. f (x) = 3 x 4 tan x 3. f (x) = 7 x 4 sec x 5 1 4. F (x) = e2x 3x + e2x 4 4 4. f (x) = 3 + x + 4 tan x 5 9 5. F (x) = e2x 3x + e4x 4 4 5. f (x) = 1 + 5x + 4 sec x 5 3 1 6. F (x) = e4x + 3x e4x 2 4 4 6. f (x) = 1 + 5x + 4 sec2 x 010 008 10.0 points Find the value of f () when f (t) = and f \u0010 \u0011 2 = 4. 1 2 7 cos t 2 sin t 3 3 3 10.0 points If the graph of f passes through the point (1, 4) and the slope of the tangent line at (x, f (x)) is 6x 5, find the value of f (2). 1. f (2) = 11 2. f (2) = 9 gray (tmg2549) - HW14 - hamrick - (53665) 3 3. 3. f (2) = 10 4. f (2) = 12 5. f (2) = 8 011 10.0 points If the graph of f is 4. 5. which one of the following contains only graphs of anti-derivatives of f ? 1. 2. 6. 012 10.0 points Find the value of y(1) when dy = 5e5x + 4x, dx 1. y(1) = e5 + 1 2. y(1) = e5 + 5 y(0) = 2. gray (tmg2549) - HW14 - hamrick - (53665) 4 using sigma notation. 5 3. y(1) = e +1 4. y(1) = e5 1 5. y(1) = e5 1 1. S = 7 X 4 4 X 7 28 X k 4 X 7k 7 X 4k 28 X 7 k=1 2. S = k=1 6. y(1) = e5 + 5 3. S = 013 10.0 points A stone is dropped off a cliff and falls under gravity with a constant acceleration of 32 ft/sec2 . If it hits the ground with a speed of 160 ft/sec, determine the height of the cliff. k=1 4. S = k=1 5. S = k=1 1. height = 408 ft 2. height = 412 ft 6. S = k=1 3. height = 416 ft 4. height = 404 ft 5. height = 400 ft 014 10.0 points To avoid an accident a car brakes with a constant deceleration of 4 ft/sec2 , producing skid marks measuring 50 ft before coming to a stop. How fast was the car traveling when the brakes were first applied? 1. speed = 18 ft/sec 2. speed = 20 ft/sec 016 10.0 points Rewrite the sum n \u0010 8 \u00112 o n \u0010 1 \u00112 o n \u0010 2 \u00112 o + 6+ +. . .+ 24+ 3+ 9 9 9 using sigma notation. 1. 9 n X 3i + i=1 \u0010 i \u00112 o 9 9 n \u0010 3i \u00112 o X 2. 3 i+ 9 i=1 8 n \u0010 i \u00112 o X 3. 3 i+ 9 i=1 3. speed = 14 ft/sec 4. 4. speed = 16 ft/sec 5. speed = 12 ft/sec 015 10.0 points Rewrite the sum S = 4 + 8 + 12 + . . . + 28 8 n X i=1 i+ \u0010 3i \u00112 o 9 9 n \u0010 i \u00112 o X 5. 3 i+ 9 i=1 6. 8 n X i=1 3i + \u0010 i \u00112 o 9 gray (tmg2549) - HW14 - hamrick - (53665) 017 3. 43 ft < distance < 59 ft 10.0 points Estimate the area under the graph of f (x) = 20 x 5 4. 43 ft < distance < 61 ft 2 5. 45 ft < distance < 61 ft on [0, 4] by dividing [0, 4] into four equal subintervals and using right endpoints as sample points. 6. 41 ft < distance < 61 ft 7. 45 ft < distance < 63 ft 1. area 52 8. 43 ft < distance < 63 ft 2. area 50 9. 41 ft < distance < 59 ft 3. area 51 019 10.0 points 4. area 49 5. area 53 018 Estimate the area, A, under the graph of 10.0 points Cyclist Joe brakes as he approaches a stop sign. His velocity graph over a 5 second period (in units of feet/sec) is shown in f (x) = 3 x 20 on [1, 5] by dividing [1, 5] into four equal subintervals and using right endpoints. 16 12 1. A 8 2. A 4 3. A 4. A 1 2 3 4 5 Compute best possible upper and lower estimates for the distance he travels over this period by dividing [0, 5] into 5 equal subintervals and using endpoint sample points. 5. A 15 4 37 10 77 20 19 5 73 20 020 10.0 points 1. 45 ft < distance < 59 ft 2. 41 ft < distance < 63 ft The graph of a function f on the interval [0, 10] is shown in gray (tmg2549) - HW14 - hamrick - (53665) 6 10 Find an expression for the area of the region under the graph of 9 8 7 6 5 4 3 2 1 8 f (x) = x4 6 on the interval [4, 7]. (Use right hand endpoints as sample points.) 4 2 1. area = lim n 4+ 6i \u00114 3 n n n \u0010 X 4+ 4i \u00114 4 n n n \u0010 X i=1 6i \u00114 4 4+ n n n \u0010 X i=1 3i \u00114 3 4+ n n n \u0010 X 3i \u00114 4 4+ n n n \u0010 X 4+ i=1 0 -1 n \u0010 X 2 4 6 8 10 2. area = lim n i=1 Estimate the area under the graph of f by dividing [0, 10] into 10 equal subintervals and using right endpoints as sample points. 1. area 54 n 5. area = lim n 3. area 56 i=1 4. area 53 6. area = lim n i=1 5. area 55 023 10.0 points Estimate the area under the graph of f (x) = 4 sin x between x = 0 and x = using five approx2 imating rectangles of equal widths and right endpoints. 1. area 4.635 2. area 4.595 3. area 4.655 4. area 4.615 5. area 4.575 022 n 4. area = lim 2. area 52 021 3. area = lim 10.0 points 4i \u00114 3 n n 10.0 points Decide which of the following regions has \u0012 \u0013 n X i area = lim tan n 3n 3n i=1 without evaluating the limit. n o 1. (x, y) : 0 y tan(3x), 0 x 3 n o 2. (x, y) : 0 y tan(2x), 0 x 3 n o 3. (x, y) : 0 y tan(4x), 0 x 6 n o 4. (x, y) : 0 y tan(x), 0 x 3 n o 5. (x, y) : 0 y tan(3x), 0 x 6 n o 6. (x, y) : 0 y tan(x), 0 x 6