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HiSET: Math : Rate of change of a function

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Example Questions

Example Question #1 : Calculate And Interpret Rate Of Change Over A Specified Interval

What is the slope of the line given by the following table?

$\begin{tabular}{ | c | c |} \hline x & y \\ \hline 0 & 2 \\ \hline 1 & 3 \\ \hline 2 & 4 \\ \hline \end{tabular}$

Possible Answers:

$\frac{1}{2}$

$-\frac{1}{3}$

$\frac{2}{3}$

Correct answer:

Explanation:

Given two points

$(x_{0},y_{0})$ and $(x_{1},y_{1})$

the formula for a slope is

$m=\frac{y_{1}-y_{0}}{x_{1}-x_{0}}$ .

Thus, since our given table is

$\begin{tabular}{ | c | c |} \hline x & y \\ \hline 0 & 2 \\ \hline 1 & 3 \\ \hline 2 & 4 \\ \hline \end{tabular}$

we select two points, say

(0,2) and (1,3)

and use the slope formula to compute the slop.

Thus,

$m = \frac{3-2}{1-0}=\frac{1}{1} = 1$ .

Hence, the slope of the line generated by the table is

m=1 .

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Example Question #1 : Rate Of Change Of A Function

Define $f(x) = x^{2}+ x$ .

Give the average rate of change of over the interval [2,3] .

Possible Answers:

3.6

7.2

Correct answer:

Explanation:

The average rate of change of a function over an interval [a, b] is equal to

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

Setting a = 2, b= 3 , this is

$\frac{f(3)-f(2)}{3-2} = \frac{f(3)-f(2)}{1} = f(3)-f(2)$

Evaluate f(3) and f(2) by substitution:

$f(x) = x^{2}+ x$

$f(3) = 3^{2}+ 3= 9+ 3 = 12$

$f(2) = 2^{2}+ 2 = 4+ 2 = 6$

$\frac{f(3)-f(2)}{3-2} = 12 - 6 = 6$ ,

the correct response.

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Example Question #2 : Rate Of Change Of A Function

Define $f(x) = \left\{\begin{matrix} x^{2}+x ,&x \le 0 \\ x^{2}-x, & x> 0\end{matrix}\right.$ .

Give the average rate of change of over the interval [-1, 1] .

Possible Answers:

$\frac{1}{2}$

$-\frac{1}{2}$

Correct answer:

Explanation:

The average rate of change of a function over an interval [a, b] is equal to

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

Setting a = -1, b= 1 , this is

$\frac{f(1)-f(-1)}{1-(-1)} = \frac{f(1)-f(-1)}{2}$

Evaluate f(1) using the definition of for x>0 :

$f(x) = x^{2}-x$

$f(1) = 1^{2}-1 = 1 - 1 = 0$

Evaluate f(-1) using the definition of for $x\le 0$ :

$f(x) = x^{2}+x$

$f(-1) = (-)1^{2}+ (-1) = 1 - 1 = 0$

The average rate of change is therefore

$\frac{f(1)-f(-1)}{2} = \frac{0-0}{2} = 0$ .

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Example Question #1 : Estimate Rate Of Change From A Graph

Estimate slope

The graph of a function is given above, with the coordinates of two points on the curve shown. Use the coordinates given to approximate the rate of change of the function between the two points.

Possible Answers:

$\frac{7}{11}$

$\frac{5}{3}$

$-\frac{11}{7}$

$\frac{3}{5}$

$-\frac{3}{5}$

Correct answer:

$\frac{3}{5}$

Explanation:

The rate of change between two points on a curve can be approximated by calculating the change between two points.

Let (x_1,y_1) be the coordinates of the first point and (x_2,y_2) be the coordinates of the second point. Then the formula giving approximate rate of change is:

$\frac{y_2 - y_1} {x_2 - x_1}$

Notice that the numerator is the overall change in y, and the denominator is the overall change in x.

The calculation for the problem proceeds as follows:

Let (3,2) be the first point and (8,5) be the second point. Substitute in the values from these coordinates:

$\frac{y_2 - y_1} {x_2 - x_1} \rightarrow \frac {5-2}{8-3}$

Subtract to get the final answer:

$\frac{3}{5}$

Note that it does not matter which you assign to be the first point and which you assign to be the second, as it will lead to the same value due to negatives canceling out.

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Example Question #1 : Estimate Rate Of Change From A Graph

Graph 1

Above is the graph of a function . Estimate the rate of change of on the interval [-2, -1.5]

Possible Answers:

Correct answer:

Explanation:

The rate of change of a function on the interval [a, b] is equal to

$r = \frac{f(b)-f(a)}{b - a}$ .

Set a = -2, b = -1.5 . Refer to the graph of the function below:

Graph 1

The graph passes through (-2, 2) and (-1.5, 5) .

f(-2) = 2 , f(-1.5)= 5 . Thus,

$r = \frac{f(-1.5)-f(-2)}{-1.5 - (-2)} = \frac{5-2}{0.5} = \frac{3}{0.5} = 6$ ,

the correct response.

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Example Question #1 : Rate Of Change Of A Function

Graph 1

Above is the graph of a function , which is defined and continuous on $(-\infty, \infty)$ . The average rate of change of on the interval [1, 5] is 4. Estimate f(5) .

Possible Answers:

Correct answer:

Explanation:

The rate of change of a function on the interval [a, b] is equal to

$r = \frac{f(b)-f(a)}{b - a}$ .

Set a = 1, b = 5, r =4 . Examine the figure below:

Graph 1

The graph passes through the point (1, 2) , so f(1) = 2 . Therefore,

$4 = \frac{f(5)-f(1)}{5-1}$

and, substituting,

$4 = \frac{f(5)-2}{4}$

Solve for f(5) using algebra:

$\frac{f(5)-2}{4} \cdot 4 = 4 \cdot 4$

f(5)-2 =16

f(5)-2+ 2 =16 + 2

f(5)= 18 ,

the correct response.

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Example Question #1 : Estimate Rate Of Change From A Graph

Graph 1

Above is the graph of a function. The average rate of change of over the interval (0, c) is $-\frac{4}{5}$ . Which of these values comes closest to being a possible value of ?

Possible Answers:

Correct answer:

Explanation:

The average rate of change of a function on the interval (a, b) is equal to

$r = \frac{f(b)-f(a)}{b - a}$ .

Restated, it is the slope of the line that passes through (a, f(a)) and (b, f(b)) .

To find the correct value of that answers this question, it suffices to examine the line with slope $-\frac{4}{5}$ through (0, f(0)) and find the point among those given that is closest to the line. This line falls 4 units for every 5 horizontal units, so the line looks like this:

Graph 1

The -coordinate of the point of intersection is closer to 2 than to any other of the values in the other four choices. This makes 2 the correct choice.

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HiSET: Math : Rate of change of a function

Study concepts, example questions & explanations for HiSET: Math

All HiSET: Math Resources

Example Questions

Example Question #1 : Calculate And Interpret Rate Of Change Over A Specified Interval

Example Question #1 : Rate Of Change Of A Function

Example Question #2 : Rate Of Change Of A Function

Example Question #1 : Estimate Rate Of Change From A Graph

Example Question #1 : Estimate Rate Of Change From A Graph

Example Question #1 : Rate Of Change Of A Function

Example Question #1 : Estimate Rate Of Change From A Graph

All HiSET: Math Resources

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