Finite Math Examples

$f (x) = 3 x^{3} - 2 x^{2} + 5 x + 4$ , $[- 1, 0]$

Step 1

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

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

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.

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

Step 2.2

Substitute the equation $y = 3 x^{3} - 2 x^{2} + 5 x + 4$ for $f (0)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(3 {(0)}^{3} - 2 {(0)}^{2} + 5 (0) + 4) - (3 {(- 1)}^{3} - 2 {(- 1)}^{2} + 5 (- 1) + 4)}{(0) - (- 1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{3 \cdot 0 - 2 \cdot 0^{2} + 5 (0) + 4 - (3 {(- 1)}^{3} - 2 {(- 1)}^{2} + 5 (- 1) + 4)}{0 - (- 1)}$

Step 3.1.2

Multiply $3$ by $0$ .

$\frac{0 - 2 \cdot 0^{2} + 5 (0) + 4 - (3 {(- 1)}^{3} - 2 {(- 1)}^{2} + 5 (- 1) + 4)}{0 - (- 1)}$

Step 3.1.3

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

$\frac{0 - 2 \cdot 0 + 5 (0) + 4 - (3 {(- 1)}^{3} - 2 {(- 1)}^{2} + 5 (- 1) + 4)}{0 - (- 1)}$

Step 3.1.4

Multiply $- 2$ by $0$ .

$\frac{0 + 0 + 5 (0) + 4 - (3 {(- 1)}^{3} - 2 {(- 1)}^{2} + 5 (- 1) + 4)}{0 - (- 1)}$

Step 3.1.5

Multiply $5$ by $0$ .

$\frac{0 + 0 + 0 + 4 - (3 {(- 1)}^{3} - 2 {(- 1)}^{2} + 5 (- 1) + 4)}{0 - (- 1)}$

Step 3.1.6

Simplify each term.

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Step 3.1.6.1

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

$\frac{0 + 0 + 0 + 4 - (3 \cdot - 1 - 2 {(- 1)}^{2} + 5 (- 1) + 4)}{0 - (- 1)}$

Step 3.1.6.2

Multiply $3$ by $- 1$ .

$\frac{0 + 0 + 0 + 4 - (- 3 - 2 {(- 1)}^{2} + 5 (- 1) + 4)}{0 - (- 1)}$

Step 3.1.6.3

Raise $- 1$ to the power of $2$ .

$\frac{0 + 0 + 0 + 4 - (- 3 - 2 \cdot 1 + 5 (- 1) + 4)}{0 - (- 1)}$

Step 3.1.6.4

Multiply $- 2$ by $1$ .

$\frac{0 + 0 + 0 + 4 - (- 3 - 2 + 5 (- 1) + 4)}{0 - (- 1)}$

Step 3.1.6.5

Multiply $5$ by $- 1$ .

$\frac{0 + 0 + 0 + 4 - (- 3 - 2 - 5 + 4)}{0 - (- 1)}$

Step 3.1.7

Subtract $2$ from $- 3$ .

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

Step 3.1.8

Subtract $5$ from $- 5$ .

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

Step 3.1.9

Add $- 10$ and $4$ .

$\frac{0 + 0 + 0 + 4 - - 6}{0 - (- 1)}$

Step 3.1.10

Multiply $- 1$ by $- 6$ .

$\frac{0 + 0 + 0 + 4 + 6}{0 - (- 1)}$

Step 3.1.11

Add $0$ and $0$ .

$\frac{0 + 0 + 4 + 6}{0 - (- 1)}$

Step 3.1.12

Add $0$ and $0$ .

$\frac{0 + 4 + 6}{0 - (- 1)}$

Step 3.1.13

Add $0$ and $4$ .

$\frac{4 + 6}{0 - (- 1)}$

Step 3.1.14

Add $4$ and $6$ .

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{10}{0 + 1}$

Step 3.2.2

Add $0$ and $1$ .

$\frac{10}{1}$

Step 3.3

Divide $10$ by $1$ .

$10$