UNIVERSIDAD IBEROAMERICANA DIFFERENTIAL CALCULUS (CGM-160)

4. FUNCTION ANALYSIS AND OPTIMIZATION

UNIBE Prof. Jorge Ledesma

1. EXERCISES 2. APPLICATION PROBLEMS

Limits. Evaluate each limit. 16. Revenue, Cost and Profit. A company is planning to

𝑥+1 manufacture and market a four-slice electric toaster.

1. lim
𝑥→−1 𝑥 2 − 1 For this toaster, the research department’s estimates
𝑥+1 are a weekly demand of 300 toasters at a price of $25
2. lim
𝑥→1 𝑥 2 − 1
per toaster, and a weekly demand of 400 toasters at a
𝑥+1
3. lim price of $20 per toaster. The financial department’s
𝑥→∞ 𝑥 2 − 1

𝑥 4 − 3𝑥 3 + 𝑥 − 3 estimates are fixed weekly costs of $5,000 and

4. lim
𝑥→3 𝑥2 − 9 variable costs of $5 per toaster.
sin 3𝑥
5. lim
𝑥→0 𝑥 2 a) Assume that the relationship between price 𝑝 and

6. lim 𝑥 ln 𝑥 demand 𝑥 is linear. Use the research

𝑥→0
department’s estimates to express 𝑝 as a function
Function Analysis. For each of the following functions, of 𝑥 and find the domain of this function.
find (a) its critical points and local extrema, (b) describe
b) Find the revenue function 𝑅(𝑥) and its domain.
its monotony, (c) find its inflection points, (d) describe
the concavity. c) Assume that the cost function is linear. Use the
financial department’s estimates to express the
7. 𝑓(𝑥) = 𝑥 3 − 18𝑥 2 + 81𝑥
cost function in terms of 𝑥.

8. 𝑓(𝑥) = 𝑥 3 − 3𝑥 d) Find the break-even points and indicate regions

of loss and profit.
2
9. 𝑓(𝑥) = (𝑥 + 4)(𝑥 − 2) e) Find the profit function in terms of 𝑥.
f) Evaluate the marginal profit at 𝑥 = 325 and 𝑥 =
True or False? Determine whether the statement is true
425 and interpret the results.
or false. Explain why or give an example that supports
your answer.
17. Gold Production Costs. The cost of producing 𝑥
10. If a cost function is linear, then the marginal cost is a
ounces of gold from a new gold mine is 𝐶 = 𝑓(𝑥)
constant.
dollars.
11. If a price-demand equation is linear, then the
marginal revenue function is linear. a) What is the meaning of the derivative 𝑓 ′ (𝑥)?
12. Marginal profit is equal to marginal cost minus What are its units?
marginal revenue. b) What does the statement 𝑓 ′ (800) = 17 mean?
13. Marginal average cost is equal to average marginal c) Do you think the values of 𝑓 ′ (𝑥) will increase or
cost. decrease in the short term? What about the long
14. If the elasticity of demand for a specific price equals term? Explain.
2, it means that a change in price will produce a small
change in demand.
15. When demand is elastic, a price decrease will
increase revenue.

Last Update: April 2020

18. Revenue and elasticity. The price-demand equation 21. Rental income. A 200-room hotel in Reno is filled
for home-delivered large pizzas is every night at a rate of $40 per room. For each $1
increase in the nightly rate, 4 fewer rooms are rented.
𝑝 = 38.2 − 0.002𝑥
If each rented room costs $8 a day to service, how
where 𝑥 is the number of pizzas delivered weekly. much should the management charge per room in
The current price of the pizza is $21. In order to order to maximize gross profit? What is the maximum
generate additional revenue from the sale of large gross profit?
pizzas, would you recommend a price increase or a
price decrease? Explain. 22. Construction. A fence is to be built to enclose a
rectangular area. The fence along three sides is to be
19. Advertising. A retail store estimates that weekly sales made of material that costs $5 per foot. The material
𝑠 and weekly advertising costs 𝑥 (both in dollars) are for the fourth side costs $15 per foot.
related by
a) If the area is 5,000 square feet, fond the
−0.0005𝑥
𝑠 = 60,000 − 40,000𝑒 dimensions of the rectangle that will allow for the
most economical fence.
The current weekly advertising costs are $2,000, and
b) If $3,000 are available for the fencing, find the
these costs are increasing at the rate of $300 per
dimensions of the rectangle that will enclose the
week. Find the current rate of change of sales.
most area.

20. Maximum revenue and profit. A company

manufactures and sells 𝑥 e-book readers per month.
The monthly cost and price-demand equations are,
respectively,
𝐶(𝑥) = 350𝑥 + 50,000

𝑝 = 500 − 0.025𝑥

a) Find the maximum revenue.

b) How many readers should the company
manufacture each month to maximize its profit?
What is the maximum monthly profit? How much
should the company charge for each reader?
c) If the government decides to tax the company $20
for each reader it produces, how many readers
should the company manufacture each month to
maximize its profit? What is the maximum
monthly profit? How much should the company
charge for each reader?

Last Update: December 2017