Precalculus Examples

$f (x) = \frac{1}{5} x^{5} - 3 x^{3} + 2$ , $[0, 1]$

Step 1

Write $f (x) = \frac{1}{5} x^{5} - 3 x^{3} + 2$ as an equation.

$y = \frac{1}{5} x^{5} - 3 x^{3} + 2$

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.

$\frac{f (1) - f (0)}{(1) - (0)}$

Step 2.2

Substitute the equation $y = \frac{1}{5} x^{5} - 3 x^{3} + 2$ for $f (1)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{1}{5} \cdot {(1)}^{5} - 3 {(1)}^{3} + 2) - (\frac{1}{5} \cdot {(0)}^{5} - 3 {(0)}^{3} + 2)}{(1) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

$\frac{\frac{1}{5} \cdot 1 - 3 \cdot 1^{3} + 2 - (\frac{1}{5} \cdot 0^{5} - 3 \cdot 0^{3} + 2)}{1 - (0)}$

Step 3.1.2

Multiply $\frac{1}{5}$ by $1$ .

$\frac{\frac{1}{5} - 3 \cdot 1^{3} + 2 - (\frac{1}{5} \cdot 0^{5} - 3 \cdot 0^{3} + 2)}{1 - (0)}$

Step 3.1.3

One to any power is one.

$\frac{\frac{1}{5} - 3 \cdot 1 + 2 - (\frac{1}{5} \cdot 0^{5} - 3 \cdot 0^{3} + 2)}{1 - (0)}$

Step 3.1.4

Multiply $- 3$ by $1$ .

$\frac{\frac{1}{5} - 3 + 2 - (\frac{1}{5} \cdot 0^{5} - 3 \cdot 0^{3} + 2)}{1 - (0)}$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

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

$\frac{\frac{1}{5} - 3 + 2 - (\frac{1}{5} \cdot 0 - 3 \cdot 0^{3} + 2)}{1 - (0)}$

Step 3.1.5.2

Multiply $\frac{1}{5}$ by $0$ .

$\frac{\frac{1}{5} - 3 + 2 - (0 - 3 \cdot 0^{3} + 2)}{1 - (0)}$

Step 3.1.5.3

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

$\frac{\frac{1}{5} - 3 + 2 - (0 - 3 \cdot 0 + 2)}{1 - (0)}$

Step 3.1.5.4

Multiply $- 3$ by $0$ .

$\frac{\frac{1}{5} - 3 + 2 - (0 + 0 + 2)}{1 - (0)}$

Step 3.1.6

Add $0$ and $0$ .

$\frac{\frac{1}{5} - 3 + 2 - (0 + 2)}{1 - (0)}$

Step 3.1.7

Add $0$ and $2$ .

$\frac{\frac{1}{5} - 3 + 2 - 1 \cdot 2}{1 - (0)}$

Step 3.1.8

Multiply $- 1$ by $2$ .

$\frac{\frac{1}{5} - 3 + 2 - 2}{1 - (0)}$

Step 3.1.9

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{\frac{1}{5} - 3 \cdot \frac{5}{5} + 2 - 2}{1 - (0)}$

Step 3.1.10

Combine $- 3$ and $\frac{5}{5}$ .

$\frac{\frac{1}{5} + \frac{- 3 \cdot 5}{5} + 2 - 2}{1 - (0)}$

Step 3.1.11

Combine the numerators over the common denominator.

$\frac{\frac{1 - 3 \cdot 5}{5} + 2 - 2}{1 - (0)}$

Step 3.1.12

Simplify the numerator.

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Step 3.1.12.1

Multiply $- 3$ by $5$ .

$\frac{\frac{1 - 15}{5} + 2 - 2}{1 - (0)}$

Step 3.1.12.2

Subtract $15$ from $1$ .

$\frac{\frac{- 14}{5} + 2 - 2}{1 - (0)}$

Step 3.1.13

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{\frac{- 14}{5} + 2 \cdot \frac{5}{5} - 2}{1 - (0)}$

Step 3.1.14

Combine $2$ and $\frac{5}{5}$ .

$\frac{\frac{- 14}{5} + \frac{2 \cdot 5}{5} - 2}{1 - (0)}$

Step 3.1.15

Combine the numerators over the common denominator.

$\frac{\frac{- 14 + 2 \cdot 5}{5} - 2}{1 - (0)}$

Step 3.1.16

Simplify the numerator.

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Step 3.1.16.1

Multiply $2$ by $5$ .

$\frac{\frac{- 14 + 10}{5} - 2}{1 - (0)}$

Step 3.1.16.2

Add $- 14$ and $10$ .

$\frac{\frac{- 4}{5} - 2}{1 - (0)}$

Step 3.1.17

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{\frac{- 4}{5} - 2 \cdot \frac{5}{5}}{1 - (0)}$

Step 3.1.18

Combine $- 2$ and $\frac{5}{5}$ .

$\frac{\frac{- 4}{5} + \frac{- 2 \cdot 5}{5}}{1 - (0)}$

Step 3.1.19

Combine the numerators over the common denominator.

$\frac{\frac{- 4 - 2 \cdot 5}{5}}{1 - (0)}$

Step 3.1.20

Simplify the numerator.

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Step 3.1.20.1

Multiply $- 2$ by $5$ .

$\frac{\frac{- 4 - 10}{5}}{1 - (0)}$

Step 3.1.20.2

Subtract $10$ from $- 4$ .

$\frac{\frac{- 14}{5}}{1 - (0)}$

Step 3.1.21

Move the negative in front of the fraction.

$\frac{- \frac{14}{5}}{1 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{- \frac{14}{5}}{1 + 0}$

Step 3.2.2

Add $1$ and $0$ .

$\frac{- \frac{14}{5}}{1}$

Step 3.3

Divide $- \frac{14}{5}$ by $1$ .

$- \frac{14}{5}$