Precalculus Examples

$f (x) = 3 x^{2} + 3$ on $[3, 5]$

Step 1

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

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

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.

$\frac{f (5) - f (3)}{(5) - (3)}$

Step 2.2

Substitute the equation $y = 3 x^{2} + 3$ for $f (5)$ and $f (3)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(3 {(5)}^{2} + 3) - (3 {(3)}^{2} + 3)}{(5) - (3)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $5$ to the power of $2$ .

$\frac{3 \cdot 25 + 3 - (3 \cdot 3^{2} + 3)}{5 - (3)}$

Step 3.1.2

Multiply $3$ by $25$ .

$\frac{75 + 3 - (3 \cdot 3^{2} + 3)}{5 - (3)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

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

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Step 3.1.3.1.1

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

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Step 3.1.3.1.1.1

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

$\frac{75 + 3 - (3^{1} \cdot 3^{2} + 3)}{5 - (3)}$

Step 3.1.3.1.1.2

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

$\frac{75 + 3 - (3^{1 + 2} + 3)}{5 - (3)}$

Step 3.1.3.1.2

Add $1$ and $2$ .

$\frac{75 + 3 - (3^{3} + 3)}{5 - (3)}$

Step 3.1.3.2

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

$\frac{75 + 3 - (27 + 3)}{5 - (3)}$

Step 3.1.4

Add $27$ and $3$ .

$\frac{75 + 3 - 1 \cdot 30}{5 - (3)}$

Step 3.1.5

Multiply $- 1$ by $30$ .

$\frac{75 + 3 - 30}{5 - (3)}$

Step 3.1.6

Add $75$ and $3$ .

$\frac{78 - 30}{5 - (3)}$

Step 3.1.7

Subtract $30$ from $78$ .

$\frac{48}{5 - (3)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $3$ .

$\frac{48}{5 - 3}$

Step 3.2.2

Subtract $3$ from $5$ .

$\frac{48}{2}$

Step 3.3

Divide $48$ by $2$ .

$24$