Examples

$f (x) = 2 x^{3} + 7 x$ , $(1, 3)$

Step 1

Write $f (x) = 2 x^{3} + 7 x$ as an equation.

$y = 2 x^{3} + 7 x$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $x$ values of the two points.

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

Step 2.2

Substitute the equation $y = 2 x^{3} + 7 x$ for $f (3)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(2 {(3)}^{3} + 7 (3)) - (2 {(1)}^{3} + 7 (1))}{(3) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $3$ to the power of $3$ .

$\frac{2 \cdot 27 + 7 (3) - (2 \cdot 1^{3} + 7 (1))}{3 - (1)}$

Step 3.1.2

Multiply $2$ by $27$ .

$\frac{54 + 7 (3) - (2 \cdot 1^{3} + 7 (1))}{3 - (1)}$

Step 3.1.3

Multiply $7$ by $3$ .

$\frac{54 + 21 - (2 \cdot 1^{3} + 7 (1))}{3 - (1)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

One to any power is one.

$\frac{54 + 21 - (2 \cdot 1 + 7 (1))}{3 - (1)}$

Step 3.1.4.2

Multiply $2$ by $1$ .

$\frac{54 + 21 - (2 + 7 (1))}{3 - (1)}$

Step 3.1.4.3

Multiply $7$ by $1$ .

$\frac{54 + 21 - (2 + 7)}{3 - (1)}$

Step 3.1.5

Add $2$ and $7$ .

$\frac{54 + 21 - 1 \cdot 9}{3 - (1)}$

Step 3.1.6

Multiply $- 1$ by $9$ .

$\frac{54 + 21 - 9}{3 - (1)}$

Step 3.1.7

Add $54$ and $21$ .

$\frac{75 - 9}{3 - (1)}$

Step 3.1.8

Subtract $9$ from $75$ .

$\frac{66}{3 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{66}{3 - 1}$

Step 3.2.2

Subtract $1$ from $3$ .

$\frac{66}{2}$

Step 3.3

Divide $66$ by $2$ .

$33$

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