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HW02-solutions.pdf

HW02-solutions.pdf - young(toy68 HW02 villafuerte...

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young (toy68) – HW02 – villafuerte altu – (53615) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Evaluate the integral I = integraldisplay π/ 3 0 2 sin 2 x cos x dx . 1. I = 2 correct 2. I = 1 3. I = 4 4. I = 2 - 2 5. I = 2(2 - 2) 6. I = 4(2 - 3) Explanation: Since sin 2 x = 2 sin x cos x , the integrand can be rewritten as 2 sin 2 x cos x = 4 sin x . But then I = integraldisplay π/ 3 0 4 sin x dx = bracketleftBig - 4 cos x bracketrightBig π/ 3 0 . Consequently, I = 2 . 002 10.0points Evaluate the integral I = integraldisplay 4 1 x parenleftBig 3 - 2 x parenrightBig dx . 1. I = 11 2. I = 8 3. I = 7 4. I = 9 5. I = 10 correct Explanation: We first expand x parenleftBig 3 - 2 x parenrightBig = 3 x - 2 x , and then integrate term by term. This gives I = bracketleftBig 2 x 3 / 2 - 4 x 1 / 2 bracketrightBig 4 1 . Consequently, I = 10 . 003 10.0points Evaluate the integral I = integraldisplay π/ 4 0 (7 sec 2 θ - 8 sin θ ) dθ. 1. I = - 1 + 8 2 correct 2. I = - 1 + 7 2 3. I = 15 - 8 2 4. I = 15 + 8 2 5. I = - 1 - 7 2 Explanation:
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young (toy68) – HW02 – villafuerte altu – (53615) 2 By the Fundamental Theorem of Calculus, I = bracketleftBig 7 tan θ + 8 cos θ bracketrightBig π/ 4 0 = parenleftBig 7 + 8 2 parenrightBig - 8 = - 1 + 8 2 . keywords: integral, FTC, trig function 004 10.0points Determine the indefinite integral I = integraldisplay 3 - 2 cos 2 ( θ ) cos 2 ( θ ) dθ . 1. I = 3 4 tan( θ ) + 1 2 θ + C 2. I = 3 4 tan( θ ) - 1 2 θ + C 3. I = 3 tan( θ ) + 2 θ + C 4. I = 3 2 tan( θ ) - θ + C 5. I = 3 2 tan( θ ) + θ + C 6. I = 3 tan( θ ) - 2 θ + C correct Explanation: After division, 3 - 2 cos 2 ( θ ) cos 2 ( θ ) = 3 sec 2 ( θ ) - 2 . Thus I = integraldisplay braceleftbig 3 sec 2 ( θ ) - 2 bracerightbig dθ . Consequently, I = 3 tan( θ ) - 2 θ + C with C an arbitrary constant. keywords: integral, trig function 005 10.0points Evaluate the definite integral I = integraldisplay 7 2 | x - 4 | dx. 1. I = - 7 2. I = 15 2 3. I = - 13 2 4. I = 13 2 correct 5. I = - 15 2 6. I = 8 7. I = 7 Explanation: Since | x - 4 | = braceleftbigg 4 - x, x < 4, x - 4 , x 4, we split the integral I into two parts I = integraldisplay 4 2 (4 - x ) dx + integraldisplay 7 4 ( x - 4) dx = I 1 + I 2 . Then I 1 = bracketleftBig 4 x - 1 2 x 2 bracketrightBig 4 2 = 2 . Similarly, I 2 = bracketleftBig 1 2 x 2 - 4 x bracketrightBig 7 4 = 9 2 . Consequently, I = 13 2 . 006 10.0points
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young (toy68) – HW02 – villafuerte altu – (53615) 3 The projected rate of increase in enrollment at a new branch of the UT-system is estimated by E ( t ) = 24000(2 t + 25) 3 / 2 where E ( t ) is the projected enrollment in t years. If the inital enrollment is 2000, de-
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