Section 2.5 Applications of Derivatives
1) The cost function for producing x units of a certain
product is:
C(x) = 100 + 8x + 0.1x2,
1a) Find the cost of producing 100 units of the product.
1b) Create the marginal cost function C’(x) for this
product.
1c) Find the marginal cost when 100 units of the product
are produced.
1d) Interpret your answer to question 1c.
2) The cost function for producing x units of a certain
product is:
C(x) = 0.4x2 +7x + 8,
2a) Find the cost of producing 4 units of the product.
2b) Create the marginal cost function C’(x) for this
product.
2c) Find the marginal cost when 4 units of the product
are produced.
2d) Interpret your answer to question 2c.
3) From past data analysis a manufacturing company
finds that the cost of manufacturing for a certain product
can be modeled by:
𝐶 (𝑥 ) = 3000 + 11𝑥 − 7√𝑥 + 0.03𝑥 3⁄2
where x represents the number of units produced and
C(x) represents the cost in dollars.
a) Create the marginal cost function C’(x) for this
product.
b) Evaluate and interpret C(100).
c) Evaluate and interpret C’(100).
4) From past data analysis a manufacturing company
finds that the cost of manufacturing for a certain product
can be modeled by:
𝐶 (𝑥 ) = 3000 + 10𝑥 2 − 7√𝑥 + 3𝑥 5⁄2
where x represents the number of units produced and
C(x) represents the cost in dollars.
a) Create the marginal cost function C’(x) for this
product.
b) Evaluate and interpret C(36).
c) Evaluate and interpret C’(36).
5) Suppose that the cost in dollars to make x smart
phone cases is given by: 𝐶 (𝑥 ) = 5 ln(𝑥 ) + 10
a) Create the marginal cost function C’(x) for this
product.
b) Evaluate and interpret C(10). (round your answer to 2
decimals)
c) Evaluate and interpret C’(10).
6) Suppose that the cost in dollars to make a x pairs of
socks is given by:
𝐶 (𝑥 ) = 3 ln(𝑥 2 ) + 2
a) Create the marginal cost function C’(x) for this
product.
b) Evaluate and interpret C(10). (round your answer to 2
decimals)
c) Evaluate and interpret C’(10).Section 2.5 Applications
of Derivatives
7) Bob’s Bobble heads company determines the profit
function for producing and selling a certain bobble head
can be modeled by:
𝑃(𝑥 ) = −0.001𝑥 2 + 8𝑥 − 1000 0 ≤ 𝑥 ≤ 7000.
Where x represents the number of bobble heads sold
and P(x) represents the monthly profit in dollars.
a) Determine P’(x)
b) Evaluate and interpret P(1000). (round your answer to
2 decimals)
c) Evaluate and interpret P’(1000).
8) The Radio Corporation determines the weekly profit
(P(x)) from selling x radios can be modeled by:
𝑃(𝑥 ) = −0.01𝑥 2 + 12𝑥 − 2000 0 ≤ 𝑥 ≤ 1000.
a) Determine P’(x)
b) Evaluate and interpret P(500). (round your answer to
2 decimals)
c) Evaluate and interpret P’(500).
9) A newspaper courier determines that the monthly
profit from his current newspaper route can be modeled
by:
𝑃(𝑥 ) = 2𝑥 − √𝑥
0 ≤ 𝑥 ≤ 200;
where x represents the number of papers delivered and
P(x) represents the monthly profit.
a) Determine P’(x)
b) Evaluate and interpret P(100).
c) Evaluate and interpret P’(100).
10) A telemarketing company has determined that the
monthly profit (P(x)) from selling x magazine
subscriptions can be modeled by:
𝑃(𝑥 ) = 5𝑥 + √𝑥 0 ≤ 𝑥 ≤ 100
a) Determine P’(x)
b) Evaluate and interpret P(20). (round your answer to 2
decimals)
c) Evaluate and interpret P’(20). (round your answer to 2
decimals)
11) The number of dollars spent on advertising for a
product influences the number of items of the product
that will be purchased by customers. The number of
Eddie Van Halen limited edition guitars that will be
purchased by customers is a function of the amount
spent on advertising and can be modeled by: Where x
represents the number of thousands of dollars spent on
advertising.
𝑁(𝑥 ) = 150 −
200
𝑥
a) Determine N’(x)
b) Evaluate and interpret N(5).
c) Evaluate and interpret N’(5).
12) The number of dollars spent on advertising for a
product influences the number of items of the product
that will be purchased by customers. The number of
diamond rings that will be purchased by customers is a
function of the amount spent on advertising and can be
modeled by: Where x represents the number of
thousands of dollars spent on advertising.
𝑁(𝑥 ) = 150 −
200
𝑥
a) Determine N’(x)
b) Evaluate and interpret N(5).
c) Evaluate and interpret N’(5).
13) A Sun City couple has a small garden and they grow
blueberries. They have found the price-demand function
is: 𝑝(𝑥 ) = −0.25𝑥 + 5.50
Where x is the number of quarts of blueberries
demanded and p(x) represents the price per quart in
dollars.
a) Find and interpret p(5) round to 2 decimals.
b) Create a revenue function R(x). Hint: R(x) = x*p(x)
(revenue = quantity*price)
c) Find R’(x)
d) Evaluate and interpret R(5). (Round your answer to 2
decimals.)
e) Evaluate and interpret R’(5).
14) A Boy Scout troop builds pinewood derby cars. They
have found the price-demand function is:
𝑝(𝑥 ) = −0.50𝑥 + 25
Where x is the number of pinewood derby cars
demanded and p(x) represents the price of a car in
dollars.
a) Find and interpret p(10) round to 2 decimals.
b) Create a revenue function R(x) hint R(x) = x*p(x)
(revenue = quantity*price)
c) Find R’(x)
d) Evaluate and interpret R(10). (round your answer to 2
decimals)
e) Evaluate and interpret R’(10).