Question

The revenue​ (in dollars) from the sale of x car seats for infants is given by the following function.

R (x)= 64x - 0.020 x^2    0 <= x <= 3200

a. Find the average change in revenue if production is changed from​ 1,000 car seats to​ 1,050 car seats. ​
b. Use the​ four-step process to find R (x).
​c. Find the revenue and the instantaneous rate of change of revenue at a production level of​ 1,000 car​ seats, and interpret the results.

Answer 1

a) Average rate of change: 23 $/seat

b) $R'(x)=64-0.040x$

c) R(1000) = 44,000 $, R'(1000) = 23 $seat

Explanation:

a)

The average rate of change of a function f(x) in a certain interval $x_1\leq x\leq x_2$ is given by

$r=\frac{f(x_2)-f(x_1)}{x_2-x_1}$

The function that gives the revenue in dollars from the sale of x car seats is

$R(x)=64x-0.020x^2$

where

0 <= x <= 3200

We want to calculate the average rate of change in the interval

$x_1 = 1000\\x_2 = 1050$

Calculating the value of the revenue function for these two values,

$f(x_1)=64\cdot 1000 - 0.020\cdot 1000^2=44,000\\f(x_2)=64\cdot 1050 -0.020 \cdot 1050^2=45,150$

Therefore, the average rate of change is

$r=\frac{45,150-44,000}{1050-1000}=23$

b)

Here we want to find R'(x), the derivative of R(x).

Step 1:

Evaluating $R(x+h)$:

$R(x+h)=64(x+h)-0.020(x+h)^2=64x+64h-0.020x^2-0.040hx-0.020h^2$

Step 2:

Evaluating $R(x+h)-R(x)$:

$R(x+h)-R(x)=64x+64h-0.020x^2-0.040hx-0.020h^2-(64x-0.020x^2)=\\=64h-0.040hx-0.020h^2$

Step 3:

Evaluating $\frac{R(x+h)-R(x)}{h}=\frac{64h-0.040hx-0.020h^2}{h}=64-0.040x-0.020h$

Step 4:

Find the limit for $h\rightarrow 0$:

$\lim_{h \to 0} (64-0.040x-0.020h) =64-0.040x$

c)

When the production level is 1000 car seats,

x = 1000

Therefore the revenue is:

$R(1000)=64\cdot 1000-0.020\cdot 1000^2=44,000$

The instantaneous rate of change of revenue instead is given by the derivative of R(x), found in part b):

$R'(x)=64-0.040 x$

Therefore, substituting x = 1000,

$R'(1000)=64-0.040\cdot 1000=24$

The two data can be interpreted as follows:

- When the production level is 1000 car seats, the revenue is 44,000$

- When the production level is 1000 car seats, the revenue is changing by 23$/new seat produced