160 STEP 4. Review the Knowledge You Need to Score High
Example 2
Given the cost functionC(x)= 500 + 3 x+ 0. 01 x^2 and the demand function (the price
function) p(x)=10, find the number of units produced in order to have maximum
profit.Solution:
Step 1: Write an equation.
Profit=Revenue−Cost
P=R−C
Revenue=(Units Sold)(Price Per Unit)
R=xp(x)=x(10)= 10 x
P= 10 x−(500+ 3 x+ 0. 01 x^2 )
Step 2: Differentiate.
Enterd(10x−(500+ 3 x+ 0. 01 x∧2,x)) and obtain 7−^0.^02 x.
Step 3: Find critical numbers.
Set 7− 0. 02 x= 0 ⇒x=350.
Critical number isx=350.
Step 4: Apply Second Derivative Test.
Enterd(10x−(500+ 3 x+ 0. 01 x∧2),x,2)|x=350 and obtain− 0 .02.
Sincex=350 is the only relative maximum, it is the absolute maximum.
Step 5: Write a Solution.
Thus, producing 350 units will lead to maximum profit.8.3 Rapid Review
- Find the instantaneous rate of change atx=5 of the functionf(x)=
√
2 x−1.
Answer: f(x)=√
2 x− 1 =(2x−1)^1 /^2
f′(x)=1
2
(2x−1)−^1 /^2 (2)=(2x−1)−^1 /^2f′(5)=1
3
- Ifhis the diameter of a circle andhis increasing at a constant rate of 0.1 cm/sec, find
the rate of change of the area of the circle when the diameter is 4 cm.
Answer: A=πr^2 =π(
h
2) 2
=1
4
πh^2dA
dt=
1
2
πh
dh
dt=
1
2
π(4)(0.1)= 0. 2 πcm^2 /sec.- The radius of a sphere is increasing at a constant rate of 2 inches per minute. In terms
of the surface area, what is the rate of change of the volume of the sphere?
Answer: V=4
3
πr^3 ;
dV
dt
= 4 πr^2
dr
dt
, sinceS=πr^2 ,
dV
dt
=28 in.^3 /min.