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**Unformatted text preview: **Math 161-03/06 10–10/11–2007 Review Problems for Test 2 These problems are provided to help you study. The fact that a problem occurs here does not mean that there will be a similar problem on the test. And the absence of a problem from this review sheet does not mean that there won’t be a problem of that kind on the test. 1. Graph y = 2 x 3 / 2 − 6 x 1 / 2 . 2. Graph y = x √ x 2 + 7 . 3. Graph y = 5 x 2 / 5 + 5 7 x 7 / 5 . 4. Graph y = 2 x x 2 − 1 . 5. Graph f ( x ) = ( x 2 − 4 x + 5) e x . 6. Graph f ( x ) = ( x − 2)( x − 3) + 2 ln x . 7. Sketch the graph of y = | x 2 − 6 x + 5 | by first sketching the graph of y = x 2 − 6 x + 5. 8. The function y = f ( x ) is defined for all x . The graph of its derivative y ′ = f ′ ( x ) is shown below: y x-1 1 3 Sketch the graph of y = f ( x ). 9. A function y = f ( x ) is defined for all x . In addition: f ( − 1) = 0 and f ′ (3) is undefined , f ′ ( x ) ≥ for x ≤ 2 and x > 3 , f ′ ( x ) ≤ for 2 ≤ x < 3 , f ′′ ( x ) < for x < 3 , f ′′ ( x ) > for x > 3 . Sketch the graph of f . 1 10. Find the critical points of y = 1 3 x 3 + 3 2 x 2 − 4 x + 5 and classify them as local maxima or local minima using the Second Derivative Test. 11. Suppose y = f ( x ) is a differentiable function, f (3) = 4, and f ′ (3) = − 6. Use differentials to approximate f (3 . 01). 12. The derivative of a function y = f ( x ) is y ′ = 1 x 4 + x 2 + 2 . Approximate the change in the function as x goes from 1 to 0 . 99. 13. Use a linear approximation to approximate √ 1 . 99 3 + 1 to five decimal places. 14. The area of a sphere of radius r is A = 4 πr 2 . Suppose that the radius of a sphere is measured to be 5 meters with an error of ± . 2 meters. Use a linear approximation to approximation the error in the area and the percentage error. 15. x and y are related by the equation x 3 y − 4 y 2 = 6 xy − 8 y. Find the rate at which x is changing when x = 2 and y = 1, if y decreases at 21 units per second. 16. Let x and y be the two legs of a right triangle. Suppose the area is decreasing at 3 square units per second, and x is increasing at 5 units per second. Find the rate at which y is changing when x = 6 and y = 20. 17. A bagel (with lox and cream cheese) moves along the curve y = x 2 +1 in such a way that its x-coordinate increases at 3 units per second. At what rate is its y-coordinate changing when it’s at the point (2 , 5)? 18. Bonzo ties Calvin to a large helium balloon, which floats away at a constant altitude of 600 feet. Bonzo pays out the rope attached to the balloon at 3 feet per second. How rapidly is the balloon moving horizontally at the instant when 1000 feet of rope have been let out?...

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