Precalculus Examples

$f (x) = 3 x^{2}$ , $[1, 3]$

Step 1

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

$y = 3 x^{2}$

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

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 3.1.1.1

Multiply $3$ by $3^{2}$ .

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Step 3.1.1.1.1

Raise $3$ to the power of $1$ .

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

Step 3.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.1.2

Add $1$ and $2$ .

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

Step 3.1.2

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

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

Step 3.1.3

Multiply $- 1$ by $3$ .

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

Step 3.1.4

One to any power is one.

$\frac{27 - 3 \cdot 1}{3 - (1)}$

Step 3.1.5

Multiply $- 3$ by $1$ .

$\frac{27 - 3}{3 - (1)}$

Step 3.1.6

Subtract $3$ from $27$ .

$\frac{24}{3 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{24}{3 - 1}$

Step 3.2.2

Subtract $1$ from $3$ .

$\frac{24}{2}$

Step 3.3

Divide $24$ by $2$ .

$12$