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Precalculus : Rate of Change Problems

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Example Questions

Example Question #1 : Rate Of Change Problems

Find the average rate of change of the function y=2e^x over the interval from x=0 to x=2 .

Possible Answers:

e^2-1

e^2

1-e^2

e(e-1)

Correct answer:

e^2-1

Explanation:

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

Here, f(2)=2e^2 , and f(0)=2e^0=2(1)=2 .

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

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Example Question #2 : Rate Of Change Problems

Let a function be defined by f(x)=-ln(x+1) .

Find the average rate of change of the function over [0,4] .

Possible Answers:

$ln\frac{-5}{4}$

$\frac{ln5}{4}$

$-\frac{ln5}{4}$

$-ln\frac{5}{4}$

Correct answer:

$-\frac{ln5}{4}$

Explanation:

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

Now f(4)=-ln(4+1)=-ln5 , and f(0)=-ln(0+1)=-ln1=0 .

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

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Example Question #2 : Derivatives

Suppose we can model the profit, , in dollars from selling items with the equation P(x)=3x^2-4x+2 .

Find the average rate of change of the profit from x=1 to x=4 .

Possible Answers:

Correct answer:

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

Since P(4)=3(4)^2-4(4)+2=3(16)-16+2=48-16+2=34 , and P(1)=3(1)^2-4(1)+2=3(1)-4+2=3-4+2=1 , we find that the average rate of change is $\frac{34-1}{4-1}=\frac{33}{3}=11$ .

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Example Question #3 : Derivatives

Let the profit, , (in thousands of dollars) earned from producing items be found by P(x)=2x^2-6x+25 .

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

Possible Answers:

$\frac{4}{3}$

Correct answer:

Explanation:

Since P(5)=2(5)^2-6(5)+25 , we see that this equals 2(25)-30+25=50-30+25=20+25=45 . Now let's examine P(4) . P(4)=2(4)^2-6(4)+25=2(16)-24+25 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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Example Question #2 : Rate Of Change Problems

Suppose that a customer purchases dog treats based on the sale price , where D(p)=15-p^2+p , 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.

Possible Answers:

Correct answer:

Explanation:

D(3)=15-9+3=6+3=9

D(2)=15-4+2=13

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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Example Question #4 : 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 $A(t)=100e^{0.05t}$ .

Find the average rate of change of the amount in the account between t=0 and t=3 , the year the student expects to graduate.

Possible Answers:

$-\$5.39$

$\$38.73$

$\$5.39$

$\$38.39$

Correct answer:

$\$5.39$

Explanation:

$A(3)=100e^{0.05*3}=100e^{0.15}\approx \$116.18$ .

$A(0)=100e^{0.05*0}=100e^{0}=\$100.00$ .

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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Example Question #5 : Rate Of Change Problems

Find the average rate of change of $y=\sqrt{x-3}$ between x=4 and x=7 .

Possible Answers:

$\frac{1}{3}$

$-\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

Explanation:

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

Here f(7) 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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Example Question #4 : Rate Of Change Problems

Find the average rate of change of y=2x^2+3x over the interval from x=1 to x=3 .

Possible Answers:

Correct answer:

Explanation:

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

f(3)=2(3)^2+3(3)=2(9)+9=18+9=27 .

f(1)=2(1)^2+3(1)=2(1)+3=2+3=5 .

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

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Example Question #5 : Rate Of Change Problems

Find the average rate of change of y=-x^2+3x-6 over the interval from x=-2 to x=0 .

Possible Answers:

Correct answer:

Explanation:

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

Now f(0)=-(0)^2+3(0)-6=-0+0-6=-6 .

We also know f(-2)=-(-2)^2+3(-2)-6=-4-6-6=-16 .

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

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Example Question #6 : Rate Of Change Problems

Why can we make an educated guess that the average rate of change of f(x)=x^2 , between x=-1 and x=1 would be ?

Possible Answers:

We know f(x) is horizontal on that interval.

We know f(x) is a polynomial.

We know f(x) is vertical on that interval.

We know f(x) is symmetrical on that interval.

We know f(x) is odd on that interval.

Correct answer:

We know f(x) is symmetrical on that interval.

Explanation:

Because f(x) 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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Precalculus : Rate of Change Problems

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Rate Of Change Problems

Example Question #2 : Rate Of Change Problems

Example Question #2 : Derivatives

Example Question #3 : Derivatives

Example Question #2 : Rate Of Change Problems

Example Question #4 : Rate Of Change Problems

Example Question #5 : Rate Of Change Problems

Example Question #4 : Rate Of Change Problems

Example Question #5 : Rate Of Change Problems

Example Question #6 : Rate Of Change Problems

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