Math 285 Review practice problem solutions Partial fractions A 1 and B 1 so 1 1 1 x 2 x 3 x 3 x 2 Properties of the exponential and the logarithmic functions ex 2 ex e2 False ex 2 ex e2 True ln 3x ln 3 ln x True 4 ln x ln x4 True ex 2 e2x True ln 3x ln 3 ln x False ln x 2 ln x ln 2 True e2 ln x x2 True Complex numbers 4 2i 1 3i 10 10i 5 xi 5 xi 25 x2 Simple algebraic linear system x y 3 2x 3y 4 Solution x 1 y 2 x 2y z 1 x 2z 5 y z 3 Solution x 1 y 1 z 2 Derivatives of common functions and rules 1 3 2i 2 5 12i Calculate the derivative with respect to x of y x for each of the following cases y 0 x 3 x2 4 y x x3 4 x y 0 x 2 e2x y x e2x y x ln 3x y x x2 1 y 0 x cos x y x sin x y 0 x 2 sin 2x y x cos 2x y x x2 cos x y 0 x 2 x cos x x2 sin x y 0 x ln x 1 y x x ln x y x 1 x x y 0 x 2 x 1 y 0 x ln x x y 0 x y x ln sin x2 4 1 ln x x2 x2 y 0 x x cot x2 4 x2 4 Integrals of common functions and rules Z x ex dx x ex ex Z 2 3x x e dx Z 2 x2 2x 3 9 27 e3x Z 1 1 dx ln 1 2x 1 2x 2 Z 1 1 1 dx 1 2x 2 2 1 2x 1 1 3 x dx ln 3 x 2 x 5 2 x 2 Z x cos 2x dx Z x2 sin 3x dx 1 x sin 2x cos 2x 2 4 x2 2x 2 cos 3x sin 3x cos 3x 3 9 27 3
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