Math 142 quiz 6 Print Name ______________________________________ Section or class time_______________________ Help is legal. 1. The standard normal bell curve is Graph f ( x ) calculator. e x F (x) 1 2 x e 2 2 . 2 2 and find its max point and inflection points by hand and in the a) From the calculator graph, find lim f ( x ) x and lim f ( x ) x and note that f(x) is symmetric about the y-axis since f(x)=f(-x). b) Find f '(x) and make a sign chart. Find the maximum point for f(x). c) Find f "(x) , make a sign chart and find the inflection points of f(x). 2. Another symmetric bell shaped curve is g(x) 3 3 x 2 . Repeat a through c of problem 1 for g(x). d) Graph f(x) and g(x) in the same graph. Use the window Xmin = -5 Xmax = 5 Ymin=0 and Ymax=1 3. A logistic function is of the form L( x) c 1 ae bx where a, b, and c are constants and a is positive. Use c=50, a=2 and b=1 in this whole problem. a) Find L '(x) b) Graph L(x) and L '(x) in the same window Xmin = -10 Xmax = 10 Ymin=0 Ymax = 50 Note: In general the derivative of a logistic is bell shaped but is only symmetrical if a = 1, so this bell is not quite symmetric since a = 2. c) Find the max point of L '(x) using 2nd trace maximum in the calculator. Use the result to find the inflection point of L(x). d) Find the inflection point of L(x) by hand using the following fact. If y 50 1 2e x then x ln 2 ln y ln( 50 y ) . Find the inflection point of x(y) using derivatives. 4. Elasticity. a) Given that E(50)=1.6, find the approximate per-cent change in demand if p increases from $50 to $52. Will demand increase or decrease? Will revenue increase or decrease? b) Given that E(20)=0.7, find the approximate per-cent change in demand if p increases from $20 to $21. Will demand increase or increase? Will revenue increase or decrease? c) The demand quantity at price p dollars is given by f ( p) 1000 1 . 04 p 2 . On what price interval is demand inelastic? On what price interval is demand elastic? At what price is revenue a maximum? d) For the demand function of part c, find and graph the revenue function, R(p). Label the inelastic and elastic ranges.