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Can you help me do these 16 math questions?

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Problem Set 3

1. What is the general rule to find a relative extrema, or in other words, to find a relative

maximum or minimum?

2. Based on data from 1980 to 2000, the median sales price of a new one-family home in the

southern United States may be modeled by

𝑃(𝑡) = 0.0506𝑡 2 + 3.515𝑡 + 58.49

where P(t) is in thousands of dollars and t is the number of years since 1980.

a. Find and interpret (in your own words) the meaning of P(5) .

b. According to the model, was the median sales price increasing more quickly at the end

of 1990 or the end of 2000? Explain.

3. The distance in feet of an object from a starting point is given by

𝑠(𝑡) = 3𝑡 2 − 4𝑡 + 40,

where t is time in seconds.

a.

Find the average velocity of the object from 3 seconds to 5 seconds.

b. Find the instantaneous velocity of the object at 5 seconds.

4.

A company has determined that the cost in dollars to manufacture x cases of its product is

given by 𝑇𝐶(𝑥) = 100 + 20𝑥 − 𝑥 2 (0 ≤ 𝑥 ≤ 6)

a.

b.

c.

5.

Find the average rate of change of manufacturing cost per case between 1 and 5 cases.

Find the instantaneous rate of change of manufacturing cost when the 3rd case is

produced.

Find marginal cost (MC) at x = 3.

Based on data from 2000 to 2016, the average daily volume of the Nasdaq market may be

modeled by

𝑉(𝑡) = 0.06754𝑡 4 − 1.936𝑡 3 + 17.97𝑡 2 − 41.27𝑡 + 39.03

million shares, where t is the number of years since 2000.

a.

b

c.

d.

Find and interpret the meaning of V(20).

Find and interpret the meaning of V’(20).

Use V(20) and V’(20) to estimate market volume in time period 21?.

Is your estimate for time period 21 an overestimate or underestimate for what it will

be? Explain.

6.

Given the following function

𝑇𝑅(𝑄) = 600𝑄 − 4𝑄 2

𝑇𝐶(𝑄) = 100𝑄 + 10,200

a. Find the revenue maximizing level of output (check 2nd order conditions also),

b. Find the maximum revenue

c. Find the corresponding profit function

d. Find the profit maximizing level of output (check 2nd order conditions also)

e. Find the maximum profit

7. Given the following function

𝑇𝑅(𝑥) = −3𝑥 2 + 260𝑥

𝑇𝐶(𝑥) = 5,500 + 50𝑥

a. Find the revenue maximizing level of output (check 2nd order conditions also),

b. Find the maximum revenue

c. Find the corresponding profit function

d. Find the profit maximizing level of output (check 2nd order conditions also)

e. Find the maximum profit

8.

Indicate whether the following functions are (1) increasing or decreasing and (2) concave

or convex at the given points:

(a)

(b)

(c)

9.

Optimize the following functions and test the second-order conditions at the critical points

to distinguish between a relative maximum and a relative minimum.

a.

b.

10.

𝑇𝑅(𝑄) = −2𝑄 2 + 96𝑄

𝑇𝐶(𝑄) = .5𝑄 2 + 3𝑄 + 67

Given that total cost can be expressed as 𝑇𝐶(𝑋) = 𝑥 3 − 140𝑥 2 + 4200, find:

a.

b.

c.

12.

𝑦 = 8𝑥 2 − 208𝑥 + 73

𝑦 = 3𝑥 3 − 45𝑥 2 − 675𝑥 + 13

Find the marginal and average functions for each of the following total functions:

a.

b.

11.

𝑓(𝑥) = 5𝑥 2 + 6𝑥 − 89, at x = 3

𝑓(𝑥) = 7𝑥 3 + 7𝑥 2 − 15𝑥, at x = -4

𝑓(𝑥) = −7𝑥 2 + 14𝑥 − 24, at x = 1

the average cost function

the level of output that minimizes AC (check 2nd order conditions also)

the value of average cost at its minimum level

Given the demand function is 𝑄 = 180 − .5𝑃, find the following:

a.

b.

c.

d.

The revenue function

The revenue maximizing output and price

The own-price elasticity of demand at P = $80

The level of Q and P where the own-price elasticity of demand (𝜀𝐷 ) is equal to one, in

absolute value. What is the nature of total revenue when |𝜀𝐷 | = 1?

13.

Assume that the demand function is 𝑄 = 100 − 3𝑃. Calculate the own-price elasticity of

demand at each of the following prices:

a.

$5

b.

$7

c.

$20

14.

Assume the following demand function: 𝑄 = 400 − .4𝑃.

a. Find the price range where |𝜀𝐷 | > 1 and where |𝜀𝐷 | < 1.
b. At P = $700,
(i) What is demand?
(ii) What is 𝜀𝐷 ?
(iii) Using 𝜀𝐷 , what would you expect to happen to demand if this firm raised price by
7%? (fall by what %)
(iv) What would you expect to happen to total revenue if this firm raised price by 7%?
(go up or down and why)
15.
Assume 𝑇𝑅(𝑄) = 150 + 50𝑄 − 5𝑄 2 and 𝑇𝐶(𝑄) = 100 + 6𝑄
a.
b.
c.
d.
e.
Create a table in Excel showing TR(Q), MR(Q) , TC(Q), 𝑀𝐶(𝑄), 𝜋(𝑄), and marginal
profit, as Q goes from 0 to 10 in increments of 1.
Identify the level of Q that maximizes profit.
At the level of Q that maximizes profit, what is the relation between MR and MC?
At the level of Q that maximizes profit, what is marginal profit?
Check your answer in part b using calculus.
16. Given the following profit function: 𝜋(𝑥) = 𝑥 3 − 6𝑥 2 − 3𝑥 + 25 where x is the number of
days of producing the good.
a. interpret 𝜋(5)
b. interpret 𝜋′(5)
c. based on b, what would you predict sales to be on day 6.
d. Using information from a-c, would predicted sales on day 6 be an overestimate or
under estimate? Explain?
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Tags:
Marginal Profit
average velocity
Relative extrema
median sales price
revenue maximizing output
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