practice_exam2

practice_exam2 - 12 A company would like to design a...

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Math 2250, Spring 2011, Practice Exam 2 Solve for dy dt in each of the following cases: 1. x 2 + y 2 = 1 2. y 2 + t 2 = 1 3. yt = ln y Find all critical values of the following functions. 4. f ( x ) = 2 x + ln( x + 1) 2 , on the interval ( - 10 , 5] 5. f ( x ) = x 2 + 3 , on the interval (0 , 1) 6. f ( x ) = x 2 / 3 + 4 , on the interval [ - 1 , 1] Find the absolute minimum and absolute maximum values of the following functions. 7. f ( x ) = x 3 - 3 x, on the interval [ - 3 , 2] 8. f ( x ) = x 2 / 3 - 2 x, on the interval [ - 1 , 1] 9. Consider the function f ( x ) = ln x . i. Find the equation for the tangent line to f ( x ) at x = 1 ii. Use linear approximation to estimate the value f (1 + 1 30 ). 10. Suppose we have a function y = f ( x ) such that x 3 + y 2 + e y = 2 and f (1) = 0. i. Find the equation for the tangent line to f ( x ) at x = 1 ii. Use linear approximation to estimate the value f (1 + 1 10 ). 11. A 50 foot ladder is leaning against a wall, and the bottom is sliding away from the wall at a constant rate of 5 feet per second. Find the speed at which the top of the ladder is falling when the bottom of the ladder is 30 feet from the wall.
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Unformatted text preview: 12. A company would like to design a box (bottom, top and four sides), with square base using exactly 500 square inches of material, and having a height somewhere between 3 inches and 20 inches. How tall should the box be made so that its volume is as large as possible? 13. In inverted conical tank is draining water at a rate of 30 cubic centimeters per second. If the tank is 100 centimeters high and has a radius of 20 centimeters at the top, find the rate at which the level of the water is falling when the level is at 30 centimeters. Reminder: the formula for the volume of a circular cone is V = 1 3 πr 2 h . 14. Consider the function f ( x ) = x + e x . Use Rolle’s theorem to show that there is at most one solution to the equation f ( x ) = 5. Hint: you may use the fact that e x > for all values of x ....
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