   Chapter 10.5, Problem 12E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

In Exercises 9–12 the function gives the cost to manufacture x items. Find the average cost per unit of manufacturing h more items (i.e., the average rate of change of the total cost) at a production level of x, where x is as indicated and h = 0 and 1. Hence, estimate the instantaneous rate of change of the total cost at the given production level x, specifying the units of measurement. (Use smaller values of h to check your estimates.) [HINT: See Example 1.] C ( x ) = 20 , 000 + 50 x + 10 , 000 x ; x = 100

To determine

To calculate: The average rate of change for the function C(x)=20,000+50x+10,000x which represents the cost of manufacturing x items over the interval [x,x+h], when x=100 and h=10 and 1. Also estimate the instantaneous rate of change of the total cost at the production level x.

Explanation

Given information:

The provided function is C(x)=20,000+50x+10,000x which represents cost of manufacturing x items and evaluate the average rate of change when x=100.

Formula used:

The formula for the average rate of change of function f(x) over the interval [a,a+h] is as follows:

Average rate of change of f(x)=f(a+h)f(a)h.

Calculation:

Consider the function, C(x)=20,000+50x+10,000x and calculate the average rate of change when x=100.

The average rate of change of function C(x) over the interval [x,x+h] is:

Average rate of change of C(x) is C(x+h)C(x)h.

Average rate of change of C(x) over [100,100+h] is C(100+h)C(100)h.

Now, calculate the average rate of change for different values of h which is follows:

For h=10,

Evaluate the values of C(100) and C(100+10) as:

C(100)=20000+50(100)+10000100=20000+5000+100=25100

And,

C(100+10)=C(110)=20000+50(110)+10000110=20000+5500+90.9090=25590.909

Substitute the values of C(100) and C(100+10) in the expression C(100+h)C(100)h to calculate the average rate of change of function C(x) as,

as,

C(x)=C(100+10)C(100)h=25590

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