Precalculus Examples

$f (x) = 5 x^{3} - 4 x - 4$ ; between $1$ and $5$

Step 1

Write $f (x) = 5 x^{3} - 4 x - 4$ as an equation.

$y = 5 x^{3} - 4 x - 4$

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.

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

Step 2.2

Substitute the equation $y = 5 x^{3} - 4 x - 4$ for $f (5)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(5 {(5)}^{3} - 4 \cdot 5 - 4) - (5 {(1)}^{3} - 4 \cdot 1 - 4)}{(5) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $5$ by $5^{3}$ by adding the exponents.

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Step 3.1.1.1

Multiply $5$ by $5^{3}$ .

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Step 3.1.1.1.1

Raise $5$ to the power of $1$ .

$\frac{5^{1} \cdot 5^{3} - 4 \cdot 5 - 4 - (5 \cdot 1^{3} - 4 \cdot 1 - 4)}{5 - (1)}$

Step 3.1.1.1.2

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

$\frac{5^{1 + 3} - 4 \cdot 5 - 4 - (5 \cdot 1^{3} - 4 \cdot 1 - 4)}{5 - (1)}$

Step 3.1.1.2

Add $1$ and $3$ .

$\frac{5^{4} - 4 \cdot 5 - 4 - (5 \cdot 1^{3} - 4 \cdot 1 - 4)}{5 - (1)}$

Step 3.1.2

Raise $5$ to the power of $4$ .

$\frac{625 - 4 \cdot 5 - 4 - (5 \cdot 1^{3} - 4 \cdot 1 - 4)}{5 - (1)}$

Step 3.1.3

Multiply $- 4$ by $5$ .

$\frac{625 - 20 - 4 - (5 \cdot 1^{3} - 4 \cdot 1 - 4)}{5 - (1)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

One to any power is one.

$\frac{625 - 20 - 4 - (5 \cdot 1 - 4 \cdot 1 - 4)}{5 - (1)}$

Step 3.1.4.2

Multiply $5$ by $1$ .

$\frac{625 - 20 - 4 - (5 - 4 \cdot 1 - 4)}{5 - (1)}$

Step 3.1.4.3

Multiply $- 4$ by $1$ .

$\frac{625 - 20 - 4 - (5 - 4 - 4)}{5 - (1)}$

Step 3.1.5

Subtract $4$ from $5$ .

$\frac{625 - 20 - 4 - (1 - 4)}{5 - (1)}$

Step 3.1.6

Subtract $4$ from $1$ .

$\frac{625 - 20 - 4 - - 3}{5 - (1)}$

Step 3.1.7

Multiply $- 1$ by $- 3$ .

$\frac{625 - 20 - 4 + 3}{5 - (1)}$

Step 3.1.8

Subtract $20$ from $625$ .

$\frac{605 - 4 + 3}{5 - (1)}$

Step 3.1.9

Subtract $4$ from $605$ .

$\frac{601 + 3}{5 - (1)}$

Step 3.1.10

Add $601$ and $3$ .

$\frac{604}{5 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{604}{5 - 1}$

Step 3.2.2

Subtract $1$ from $5$ .

$\frac{604}{4}$

Step 3.3

Divide $604$ by $4$ .

$151$