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Flashcards: Calculate and interpret rate of change over a specified interval

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

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

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\displaystyle 2$

$-\frac{1}{3}$ $\displaystyle -\frac{1}{3}$

$\displaystyle 1$

$\frac{2}{3}$ $\displaystyle \frac{2}{3}$

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Flashcards: Calculate and interpret rate of change over a specified interval

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

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

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\displaystyle 2$

$-\frac{1}{3}$ $\displaystyle -\frac{1}{3}$

$\displaystyle 1$

$\frac{2}{3}$ $\displaystyle \frac{2}{3}$

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Flashcards: Calculate and interpret rate of change over a specified interval

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

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

Your answer:

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

Your answer:

$\displaystyle 2$

Your answer:

$-\frac{1}{3}$ $\displaystyle -\frac{1}{3}$

Your answer:

$\frac{2}{3}$ $\displaystyle \frac{2}{3}$

Correct answer:

$\displaystyle 1$

Explanation:

Given two points

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

the formula for a slope is

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

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) $\displaystyle (0,2)$ and (1,3) $\displaystyle (1,3)$

and use the slope formula to compute the slop.

Thus,

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

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

m=1 $\displaystyle m=1$ .

Flashcards: Calculate and interpret rate of change over a specified interval

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

Give the average rate of change of $\displaystyle f$ over the interval [2,3] $\displaystyle [2,3]$ .

7.2 $\displaystyle 7.2$

$\displaystyle 3$

3.6 $\displaystyle 3.6$

$\displaystyle 9$

$\displaystyle 6$

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Flashcards: Calculate and interpret rate of change over a specified interval

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

Give the average rate of change of $\displaystyle f$ over the interval [2,3] $\displaystyle [2,3]$ .

Your answer:

7.2 $\displaystyle 7.2$

Your answer:

$\displaystyle 3$

Your answer:

3.6 $\displaystyle 3.6$

Your answer:

$\displaystyle 9$

Correct answer:

$\displaystyle 6$

Explanation:

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

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

Setting $\displaystyle a = 2, b= 3$ , this is

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

Evaluate f(3) $\displaystyle f(3)$ and f(2) $\displaystyle f(2)$ by substitution:

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

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

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

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

the correct response.

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