University:
California State University, Northridge
Course:
MATH 103 | Mathematical Methods for Business
Academic year:

2024
Views:

176

Pages:

4
Author:

Courtney Glass

Math 103. Homework 10. Solutions 1. Determine the intervals where the function f(x)=x2 e−x is increasing and the intervals where it is decreasing. Solution. The derivative f′(x) = 2x e−x + (−1) x2 e−x = (2x−x2)e−x Setting f′(x) = 0 gives 2x−x2=0 (because e−x > 0 for all x). Therefore f′(x) = 0 for x=0,2. The sign diagram for the derivative is so that f(x) is increasing on (0,2) and decreasing on (−∞,0)∪(2,∞). 2. Determine the intervals of concavity and inflection points for the function f(x) = x2 lnx Solution. The domain of f(x) is (0,∞). We compute f′(x) = 2x lnx + x2 (1/x) and the second derivative f′′(x) = 2lnx + 2 +1 = 2lnx +3 Setting f′(x)=0 gives lnx = = or 3 2 x = e−3/2 The sign diagram for the second derivative is Therefore f(x) is concave down on (0,e−3/2) and concave up on (e−3/2,∞) 3. Use logarithmic differentiation to find an equation for the tangent line to the graph of f(x) = xlnx at the point (e,e). Solution. We write lnf(x) = ln(xlnx) = ( lnx) (lnx) and taking derivatives f′(x) f(x) = 2lnx x so that f′(x) = 2lnx x f(x) = 2lnx x xlnx Thus f′(e) = 2lne e elne = 2(1) e e1 = 2 An equation for the line through (e,e) with slope 2 is y = 2 x −e 4. Find the present value of $119,346 due in 4 yr at an interest rate of 10%/year compounded continuously. Solution. The formula for continuous compounding of interest is A=Pert We know that A=119, 346, r=0.1 and t=4. Thus the present value is P = (119,346)e(−0.1)(4) ≈ 80,000 5. A hotel was purchased by a conglomerate for $4.5 million and sold 5 years later for $8.2 million. Find the annual rate of return, compounded continuously. Solution. We use the formula A=P ert Here A = 8.2, P=4.5 and t=5. We have to find r. Plugging in these values into the formula we obtain 8.2 = 4.5 er 5 Hence e5r = 8.2 4.5 and taking natural logarithms 5r = ln 8.2 4.5 or r= or 12%/year. 1 5 ln 8.2 4.5 = 0.120011