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MATH 142 Business Math II, Week In Review Spring, 2015, Problem Set 5 (4.4, 4.5, 5.1) JoungDong Kim 1. Find the derivative of function. (a) f (x) = e−0.1x (b) f (x) = e4x (c) f (x) = e (d) f (x) = 3 −x−1 √ 3x √ 3 xex (e) f (x) = (x3 − 2)e2x 1 (f) f (x) = ex x2 + 1 (g) f (x) = √ x3 e3x (h) f (x) = √ 2 + e−3x (i) f (x) = 33x (j) f (x) = 7x 7 (k) f (x) = 3x · 5x (l) f (x) = ln(x4 + x2 + 3) 2 (m) f (x) = ln(x2 ) x+1 (n) f (x) = ln(x2 ) ex + e−x (o) f (x) = √ ln x, 2 (p) f (x) = ex · ln x, (q) f (x) = ln(x7 ), (x > 0) (x > 0) (x > 0) 3 2 2. Suppose the profit equation is given by p(x) = x · e−0.5x . Find the marginal profit, and find where the marginal profit is zero. 3. Find f ′ (x). (a) f (x) = ee ex (b) f (x) = ln[ln(ln x)] 4 4. According to Stone and Rowe, the price elasticity of furniture is 3.04. If the price of furniture is increased by 10%, what will happen to the demand for furniture? 5. A restaurant owner sells 100 dinner specials for $10 each. After raising the price to $11, she noticed that only 70 specials were sold. What is the elasticity of demand? 5 6. Find the elasticity E at the given points and determine whether demand is inelastic, elastic, or unit elastic. (a) x = 20 − 4p, 1 (b) x = 10 + , p 10 (c) x = √ , p p = 1, p = 2.5, and p = 4 p = 0.1 and p = 1 p = 1 and p = 3 6 7. Find all critical values, the largest open intervals on which f is increasing, the largest open intervals on which f is decreasing, and all relative maxima and minima. (a) f (x) = −2x2 + 8x − 1 (b) f (x) = −4x6 − 6x4 + 5 7 (c) f (x) = 1 − ln x (d) f (x) = e−x 2 8 8. If the demand equation for a function is given by p = −0.2x + 16, find the value of x for which revenue is maximum. 9