   Chapter 10.5, Problem 9E 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 lev h = 0 el of x, where x is as indicated and 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 ) = 10 , 000 + 5 x − x 2 10 , 000 ; x = 1 , 000

To determine

To calculate: The average rate of change for the function C(x)=10,000+5xx210,000 which represents the cost of manufacturing x items over the interval [x,x+h], when x=1,000 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)=10,000+5xx210,000 which represents cost of manufacturing x items and evaluate the average rate of change when x=1,000.

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)=10,000+5xx210,000 and calculate the average rate of change when x=1,000.

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 [1000,1000+h] is C(1000+h)C(1000)h

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

For h=10,

Evaluate the values of C(1000) and C(1000+10) as:

C(1000)=10000+5(1000)(1000)210000=10000+5000100=14900

And,

C(1000+10)=C(1010)=10000+5(1010)(1010)210000=10000+5050102

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