Finite Math Examples

$f (x) = - 4 x^{3} + 7 x^{2} + 2 x$ , $(- 5, 2)$

Step 1

Write $f (x) = - 4 x^{3} + 7 x^{2} + 2 x$ as an equation.

$y = - 4 x^{3} + 7 x^{2} + 2 x$

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.

$\frac{f (2) - f (- 5)}{(2) - (- 5)}$

Step 2.2

Substitute the equation $y = - 4 x^{3} + 7 x^{2} + 2 x$ for $f (2)$ and $f (- 5)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(- 4 {(2)}^{3} + 7 {(2)}^{2} + 2 (2)) - (- 4 {(- 5)}^{3} + 7 {(- 5)}^{2} + 2 (- 5))}{(2) - (- 5)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $2$ to the power of $3$ .

$\frac{- 4 \cdot 8 + 7 \cdot 2^{2} + 2 (2) - (- 4 {(- 5)}^{3} + 7 {(- 5)}^{2} + 2 (- 5))}{2 - (- 5)}$

Step 3.1.2

Multiply $- 4$ by $8$ .

$\frac{- 32 + 7 \cdot 2^{2} + 2 (2) - (- 4 {(- 5)}^{3} + 7 {(- 5)}^{2} + 2 (- 5))}{2 - (- 5)}$

Step 3.1.3

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

$\frac{- 32 + 7 \cdot 4 + 2 (2) - (- 4 {(- 5)}^{3} + 7 {(- 5)}^{2} + 2 (- 5))}{2 - (- 5)}$

Step 3.1.4

Multiply $7$ by $4$ .

$\frac{- 32 + 28 + 2 (2) - (- 4 {(- 5)}^{3} + 7 {(- 5)}^{2} + 2 (- 5))}{2 - (- 5)}$

Step 3.1.5

Multiply $2$ by $2$ .

$\frac{- 32 + 28 + 4 - (- 4 {(- 5)}^{3} + 7 {(- 5)}^{2} + 2 (- 5))}{2 - (- 5)}$

Step 3.1.6

Simplify each term.

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Step 3.1.6.1

Raise $- 5$ to the power of $3$ .

$\frac{- 32 + 28 + 4 - (- 4 \cdot - 125 + 7 {(- 5)}^{2} + 2 (- 5))}{2 - (- 5)}$

Step 3.1.6.2

Multiply $- 4$ by $- 125$ .

$\frac{- 32 + 28 + 4 - (500 + 7 {(- 5)}^{2} + 2 (- 5))}{2 - (- 5)}$

Step 3.1.6.3

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

$\frac{- 32 + 28 + 4 - (500 + 7 \cdot 25 + 2 (- 5))}{2 - (- 5)}$

Step 3.1.6.4

Multiply $7$ by $25$ .

$\frac{- 32 + 28 + 4 - (500 + 175 + 2 (- 5))}{2 - (- 5)}$

Step 3.1.6.5

Multiply $2$ by $- 5$ .

$\frac{- 32 + 28 + 4 - (500 + 175 - 10)}{2 - (- 5)}$

Step 3.1.7

Add $500$ and $175$ .

$\frac{- 32 + 28 + 4 - (675 - 10)}{2 - (- 5)}$

Step 3.1.8

Subtract $10$ from $675$ .

$\frac{- 32 + 28 + 4 - 1 \cdot 665}{2 - (- 5)}$

Step 3.1.9

Multiply $- 1$ by $665$ .

$\frac{- 32 + 28 + 4 - 665}{2 - (- 5)}$

Step 3.1.10

Add $- 32$ and $28$ .

$\frac{- 4 + 4 - 665}{2 - (- 5)}$

Step 3.1.11

Add $- 4$ and $4$ .

$\frac{0 - 665}{2 - (- 5)}$

Step 3.1.12

Subtract $665$ from $0$ .

$\frac{- 665}{2 - (- 5)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 5$ .

$\frac{- 665}{2 + 5}$

Step 3.2.2

Add $2$ and $5$ .

$\frac{- 665}{7}$

Step 3.3

Divide $- 665$ by $7$ .

$- 95$