Homework 7 1

CalculusHomework 7

Due Date: November 28 (Wednesday)

1. Reading assignments

(a) Chapter 4: Exponential and Logarithm Functions

2. Analyze and sketch the graph of the function. Label any intercepts, relative extrema,points of inflection, and asymptotes.

(a) f(x) = 4x3 − x4

(b) f(x) = x3 − 6x2 + 3x + 10

(c) f(x) = x−13x2+1

(d) f(x) = x+1x−1

.

Solution:

(a)

2 Calculus H415611 Fall 2012

(b)

(c)

Homework 7 3

(d)

3. The demand function for a product is modeled by

p = 960 − x, 0 ≤ x ≤ 960

4 Calculus H415611 Fall 2012

where p is the price (in dollars) and x is the number of units.

(a) Determine when the demand is elastic, inelastic, and of unit elasticity.

(b) Use the result of part (a) to describe the behavior of the revenue function.

Solution:

4. The profit P (in thousands of dollars) for a company in terms of the amount s spent onadvertising (in thousands of dollars) can be modeled by

P = −4s3 + 72s2 − 240 + 500.

Find the amount of advertising that maximizes the profit. Find the point of diminishingreturns.Solution:

Homework 7 5

5. Find the differential dy.

(a) y = (x2 + 3)(2x + 4)2.

(b) y = x+12x−1

.

Solution:

(a)

(b)

6. A retailer has determined that the monthly sales x of a watch are 150 units when theprice is $50, but decrease to 120 units when the price is $60. Assume that the demand isa linear function of the price. Find the revenue R as function of x and approximate the

6 Calculus H415611 Fall 2012

change in revenue for a one-unit increase in sales when x = 141. Make a sketch showingdR and ∆R.Solution: