Homework #8 - jiwani(amj566 – Homework08 – Fouli...

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Unformatted text preview: jiwani (amj566) – Homework08 – Fouli – (58395) 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 10.0 points A circle of radius r has area A and circum- ference C are given respectively by A = πr 2 , C = 2 πr . If r varies with time t , for what value of r is the rate of change of A with respect to t half the rate of change of C with respect to t ? 1. r = 1 2. r = π 2 3. r = 1 2 correct 4. r = 2 5. r = π 6. r = 2 π Explanation: Differentiating A = πr 2 , C = 2 πr implicitly with respect to t we see that dA dt = 2 πr dr dt , dC dt = 2 π dr dt . Thus the rate of change, dA/dt , of area is half the rate of change, dC/dt , of circumference when dA dt = 1 2 dC dt , i.e. , when 2 πr dr dt = 1 2 parenleftBig 2 π dr dt parenrightBig . This happens when r = 1 2 . 002 10.0 points Determine the value of dy/dt at x = 2 when y = x 2 − 2 x and dx/dt = 3. 1. dy dt vextendsingle vextendsingle vextendsingle x =2 = 6 correct 2. dy dt vextendsingle vextendsingle vextendsingle x =2 = 14 3. dy dt vextendsingle vextendsingle vextendsingle x =2 = 8 4. dy dt vextendsingle vextendsingle vextendsingle x =2 = 12 5. dy dt vextendsingle vextendsingle vextendsingle x =2 = 10 Explanation: Differentiating implicitly with respect to t we see that dy dt = (2 x − 2) dx dt = 3 (2 x − 2) . At x = 2, therefore, dy dt = 3(2) = 6 . 003 10.0 points A point is moving on the graph of xy = 5. When the point is at (3 , 5 3 ), its x-coordinate is increasing at a rate of 6 units per second. What is the speed of the y-coordinate at that moment and in which direction is it mov- ing? 1. speed = − 10 3 units/sec, decreasing y 2. speed = − 13 3 units/sec, decreasing y 3. speed = 16 3 units/sec, increasing y jiwani (amj566) – Homework08 – Fouli – (58395) 2 4. speed = − 16 3 units/sec, increasing y 5. speed = 10 3 units/sec, decreasing y correct 6. speed = 13 3 units/sec, increasing y Explanation: Provided x, y negationslash = 0, the equation xy = 5 can be written as y = 5 /x . Differentiating implicitly with respect to t we thus see that dy dt = − 5 x 2 dx dt . whenever x negationslash = 0. When x = 3 , dx dt = 6 , therefore, the corresponding rate of change of the y-coordinate is given by dy dt vextendsingle vextendsingle vextendsingle x =3 = − 6 parenleftBig 5 x 2 parenrightBigvextendsingle vextendsingle vextendsingle x =3 = − 10 3 . Consequently, the speed of the y-coordintate is 10 3 units per second and the negative sign indicates that the point is moving in the di- rection of decreasing y . 004 10.0 points The dimensions of a cylinder are changing, but the height is always equal to the diameter of the base of the cylinder. If the height is increasing at a speed of 5 inches per second, determine the speed at which the volume, V , is increasing (in cubic inches per second) when the height is 4 inches....
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This note was uploaded on 07/10/2011 for the course KIN 321M taught by Professor Jensen during the Spring '11 term at University of Texas.

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Homework #8 - jiwani(amj566 – Homework08 – Fouli...

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