MATH 31B Study Guide - Midterm Guide: Exponential Decay, Inverse Function
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Midterm 1: consider the function f (x) = xex. (a) (5 points) find the derivative f(cid:48)(x). Since ln f (x) = ln(xex) = ex ln x, we get f(cid:48)(x) f (x) = (ln f (x))(cid:48) = (ex ln x)(cid:48) = ex x. + ex ln x so f(cid:48)(x) = ( ex x. + ex ln x)xex. (b) (5 points) let g = f 1 denote the inverse function. We have f (1) = 1, so f 1(1) = 1 and g(cid:48)(1) = 1 e: we have a sample consisting of 100 grams of a radioactive isotope. The quantity must satisfy the exponential decay law p (t) = p0e kt. We have so the growth constant k satis es e k = 9/10, i. e. k = ln(9/10). P (t) = 30 implies 100e kt = 30 so kt = ln(3/10). Therefore t = ln(3/10) ln(9/10): a function f (x) satis es the di erential equation f(cid:48)(x) + 4f (x) = 2.
