Precalculus Examples

$f (x) = x^{3} - 4 x^{2} - x + 5$ on $0$ , $4$

Step 1

Write $f (x) = x^{3} - 4 x^{2} - x + 5$ as an equation.

$y = x^{3} - 4 x^{2} - x + 5$

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.

$\frac{f (4) - f (0)}{(4) - (0)}$

Step 2.2

Substitute the equation $y = x^{3} - 4 x^{2} - x + 5$ for $f (4)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({(4)}^{3} - 4 {(4)}^{2} - (4) + 5) - ({(0)}^{3} - 4 {(0)}^{2} - (0) + 5)}{(4) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $4$ to the power of $3$ .

$\frac{64 - 4 \cdot 4^{2} - (4) + 5 - (0^{3} - 4 \cdot 0^{2} - (0) + 5)}{4 - (0)}$

Step 3.1.2

Raise $4$ to the power of $2$ .

$\frac{64 - 4 \cdot 16 - (4) + 5 - (0^{3} - 4 \cdot 0^{2} - (0) + 5)}{4 - (0)}$

Step 3.1.3

Multiply $- 4$ by $16$ .

$\frac{64 - 64 - (4) + 5 - (0^{3} - 4 \cdot 0^{2} - (0) + 5)}{4 - (0)}$

Step 3.1.4

Multiply $- 1$ by $4$ .

$\frac{64 - 64 - 4 + 5 - (0^{3} - 4 \cdot 0^{2} - (0) + 5)}{4 - (0)}$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

Raising $0$ to any positive power yields $0$ .

$\frac{64 - 64 - 4 + 5 - (0 - 4 \cdot 0^{2} - (0) + 5)}{4 - (0)}$

Step 3.1.5.2

Raising $0$ to any positive power yields $0$ .

$\frac{64 - 64 - 4 + 5 - (0 - 4 \cdot 0 - (0) + 5)}{4 - (0)}$

Step 3.1.5.3

Multiply $- 4$ by $0$ .

$\frac{64 - 64 - 4 + 5 - (0 + 0 - (0) + 5)}{4 - (0)}$

Step 3.1.5.4

Multiply $- 1$ by $0$ .

$\frac{64 - 64 - 4 + 5 - (0 + 0 + 0 + 5)}{4 - (0)}$

Step 3.1.6

Add $0$ and $0$ .

$\frac{64 - 64 - 4 + 5 - (0 + 0 + 5)}{4 - (0)}$

Step 3.1.7

Add $0$ and $0$ .

$\frac{64 - 64 - 4 + 5 - (0 + 5)}{4 - (0)}$

Step 3.1.8

Add $0$ and $5$ .

$\frac{64 - 64 - 4 + 5 - 1 \cdot 5}{4 - (0)}$

Step 3.1.9

Multiply $- 1$ by $5$ .

$\frac{64 - 64 - 4 + 5 - 5}{4 - (0)}$

Step 3.1.10

Subtract $64$ from $64$ .

$\frac{0 - 4 + 5 - 5}{4 - (0)}$

Step 3.1.11

Subtract $4$ from $0$ .

$\frac{- 4 + 5 - 5}{4 - (0)}$

Step 3.1.12

Add $- 4$ and $5$ .

$\frac{1 - 5}{4 - (0)}$

Step 3.1.13

Subtract $5$ from $1$ .

$\frac{- 4}{4 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{- 4}{4 + 0}$

Step 3.2.2

Add $4$ and $0$ .

$\frac{- 4}{4}$

Step 3.3

Divide $- 4$ by $4$ .

$- 1$