Precalculus : Rate of Change Problems

Example Questions

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Example Question #1 : Rate Of Change Problems

Find the average rate of change of the function  over the interval from  to .

Explanation:

The average rate of change will be found by .

Here, , and .

Now, we have .

Example Question #1 : Rate Of Change Problems

Let a function be defined by .

Find the average rate of change of the function over .

Explanation:

We use the average rate of change formula, which gives us .

Now , and .

Example Question #3 : Rate Of Change Problems

Suppose we can model the profit, , in dollars from selling  items with the equation .

Find the average rate of change of the profit from  to .

Explanation:

We need to apply the formula for the average rate of change to our profit equation. Thus we find the average rate of change is .

Since , and , we find that the average rate of change is .

Example Question #4 : Rate Of Change Problems

Let the profit, , (in thousands of dollars) earned from producing  items be found by .

Find the average rate of change in profit when production increases from 4 items to 5 items.

Explanation:

Since , we see that this equals. Now let's examine  which simplifies to .

Therefore the average rate of change formula gives us .

Example Question #5 : Rate Of Change Problems

Suppose that a customer purchases  dog treats based on the sale price , where , where .

Find the average rate of change in demand when the price increases from $2 per treat to$3 per treat.

Explanation:

Thus the average rate of change formula yields .

This implies that the demand drops as the price increases.

Example Question #6 : Rate Of Change Problems

A college freshman invests \$100 in a savings account that pays 5% interest compounded continuously. Thus, the amount  saved after  years can be calculated by .

Find the average rate of change of the amount in the account between  and , the year the student expects to graduate.

Explanation:

.

.

Hence, the average rate of change formula gives us .

Example Question #7 : Rate Of Change Problems

Find the average rate of change of  between  and .

Explanation:

The solution will be found by the formula .

Here  gives us , and .

Thus, we find that the average rate of change is .

Example Question #8 : Rate Of Change Problems

Find the average rate of change of  over the interval from  to .

Explanation:

The average rate of change will be .

.

.

This gives us .

Example Question #9 : Rate Of Change Problems

Find the average rate of change of  over the interval from  to .

Explanation:

The average rate of change will be .

Now.

We also know .

So we have .

Example Question #10 : Rate Of Change Problems

Why can we make an educated guess that the average rate of change of , between  and  would be ?

We know  is vertical on that interval.

We know is horizontal on that interval.

We know  is odd on that interval.

We know  is a polynomial.

We know  is symmetrical on that interval.

We know  is symmetrical on that interval.

Explanation:

Because  is symmetrical over the y axis, it increases exactly as much as it decreases on the interval from  to . Thus the average rate of change on that interval will be .

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