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1024worksheet11

1024worksheet11 - f x increase and the intervals on which f...

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Polytechnic Institute of NYU MA 1024/1324 Worksheet 11 Print Name: Signature: ID #: Instructor/Section: Directions: Show all your work for every problem. Problem Possible Points 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100 1
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2 (1) Find the tangent line approximation to f ( x ) = 4 x at x = 16. Use this to estimate the value of 4 15.
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3 (2) Find the tangent line approximation to f ( x ) = sin x at x = 60 o . Use this to estimate the value of sin 62 . (Remember to convert to radians)
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4 (3) The radius of a ball of ice shrinks from 10 to 9 . 8 inches. Estimate, with linear approximation, the decrease in (a) the volume. (b) the surface area.
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5 (4) (a) Estimate f (2 . 8) given that f (3) = 2 and f 0 ( x ) = ( x 3 + 5) 1 / 5 (b) Estimate f (5 . 4) given that f (5) = 1 and f 0 ( x ) = 3 x 2 + 2
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6 (5) Assume that the earth is a sphere of radius 4000 miles. The volume of ice at the north and south poles is estimated to be 8 million cubic miles. Suppose that this ice melts and the water produced distributes uniformly over the surface of the earth. Estimate, with linear approximation, the depth of the added water at any point on the earth.
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7 (6) Find the intervals on which
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Unformatted text preview: f ( x ) increase and the intervals on which f ( x ) decreases. (a) f ( x ) = r 1 + x 2 2 + x 2 (b) f ( x ) = 2 x 5 / 3 + 5 x 2 / 3 8 (7) Find the critical points of f ( x ) and the local extreme values. (a) f ( x ) = (1-x ) 2 (1 + x ) (b) f ( x ) = | x 2-16 | 9 (8) Describe the concavity of the graph of f ( x ) and find the points of inflection (if any). (a) f ( x ) = (1-x ) 2 (1 + x ) 2 (b) f ( x ) = 2 cos 2 x-x 2 , x ∈ [0 ,π ] 10 (9) Let f ( x ) = x 1 / 3 ( x-6) 2 / 3 . Find (a) the intervals on which f ( x ) increases or decreases (b) the local maxima and minima (c) the intervals on which the graph of f ( x ) is concave up and the intervals on which it is concave down (d) the points of inflection 11 (10) Let f ( x ) = ax 3 + bx 2 + cx + d such that a 6 = 0. Under what conditions on a , b , and c will f ( x ) have: (a) two local extrema? (b) only one local extremum? (c) no local extrema?...
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