Math 1A Calculus Prof Haiman Fall 2006 Practice Final Exam 1 Differentiate the function x 1 x 2 y 3 x 3 2 Evaluate the limit if it exists possibly as an infinite limit a lim x 1 x 1 x b lim x 1 x 1 x x x 1 1 x c lim 3 Find all points P on the curve y x2 1 with the property that the tangent line at P passes through the origin 4 Use a linear approximation to estimate 37 5 If sin y x y x express dy dx in terms of x and y 6 Find the constant a for which f x x3 ax2 has an inflection point at x 1 For this value of a find the intervals of concavity of f x 7 Use Newton s method to find the root of x4 x 4 0 in the interval 1 2 correct to 6 decimal places 8 Find the points on the parabola y x2 closest to 0 1 9 Find the limit 1 1 ln x x 1 2 x x 1 dx lim x 1 10 Evaluate the integral Z 1 11 Find the area enclosed by the lines x 0 y 1 and the curve y 12 Evaluate the integral Z 2 cos x 0 1 1 dx 2 3 x 13 Differentiate the function Z 2x f x x et dt t 14 Find the most general function f x for which f 00 x cos x 15 Find an interval 0 c on which the average value of the function f x x2 2 is equal to 5 16 Set up an integral for the volume of the solid obtained by rotating the region enclosed by the x axis the line x 2 and the curve y ln x about the y axis using a the method of slices b the method of cylindrical shells Evaluate one of these integrals to find the volume 17 Find the volume of a pyramid with a square base of length 2 on each side and height 3 18 Evaluate the limit by expressing it as an integral n 1 X i2 lim n n n2 i 1 2
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