Rate of Change Problems - Pre-Calculus

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Question

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

Answer

The average rate of change will be $\frac{f(3)-f(1)}{3-1}$ .

.

.

This gives us $\frac{27-5}{3-1}=\frac{22}{2}=11$ .

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Question

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

Answer

The average rate of change will be $\frac{f(0)-f(-2)}{0-(-2)}$ .

Now.

We also know .

So we have $\frac{-6-(-16)}{0-(-2)}=\frac{-6+16}{0+2}=\frac{10}{2}=5$ .

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Question

Find the average rate of change of $y=\sqrt{x-3}$ between and .

Answer

The solution will be found by the formula $\frac{f(7)-f(4)}{7-4}$ .

Here gives us $\sqrt{7-3}=\sqrt{4}=2$ , and $f(4)=\sqrt{4-3}=\sqrt{1}=1$ .

Thus, we find that the average rate of change is $\frac{2-1}{7-4}=\frac{1}{3}$ .

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Question

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

Answer

The average rate of change will be found by $\frac{f(2)-f(0)}{2-0}$ .

Here, , and .

Now, we have $\frac{2e^2-2}{2-0}=\frac{2e^2-2}{2}=e^2-1$ .

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Question

Let a function be defined by .

Find the average rate of change of the function over .

Answer

We use the average rate of change formula, which gives us $\frac{f(4)-f(0)}{4-0}$ .

Now , and .

Therefore, the answer becomes $\frac{-ln5-0}{4-0}=-\frac{ln5}{4}$ .

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Question

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 .

Answer

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 $\frac{P(4)-P(1)}{4-1}$ .

Since , and , we find that the average rate of change is $\frac{34-1}{4-1}=\frac{33}{3}=11$ .

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Question

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.

Answer

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

Therefore the average rate of change formula gives us $\frac{f(5)-f(4)}{5-4}=\frac{45-33}{5-4}=\frac{12}{1}=12$ .

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Question

Suppose that a customer purchases dog treats based on the sale price , where , where $1\leq p\leq3$ .

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

Answer

Thus the average rate of change formula yields $\frac{9-13}{3-2}=\frac{-4}{1}=-4$ .

This implies that the demand drops as the price increases.

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Question

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 $A(t)=100e^{0.05t}$ .

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

Answer

$A(3)=100e^{0.053}=100e^{0.15}\approx $116.18$ .

$A(0)=100e^{0.050}=100e^{0}=$100.00$ .

Hence, the average rate of change formula gives us $\frac{A(3)-A(0)}{3-0}=\frac{$116.18-$100.00}{3-0}=\frac{$16.18}{3}\approx $5.39$ .

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Question

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

Answer

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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Question

If the average rate of change of between and , where , is positive, then what can be said about on that interval?

Answer

If the average rate of change is positive, then the formula gives us $\frac{f(b)-f(a)}{b-a}>0$ , so . We know because it is a given in the proble, so . Hence

and . This shows that must increase over the interval from to .

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Question

If the average rate of change of between and , where , is negative, then what can be said about on that interval?

Answer

If the average rate of change is negative, then the function is changing in a negative direction overall. Hence, the graph of the function will be decreasing on that interval.

$\frac{f(b)-f(a)}{b-a}<0$

and since , is decreasing

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