Find the Average Rate of Change f(x)=2x^3-5x^2-10x+5 , [-1,2]

$f (x) = 2 x^{3} - 5 x^{2} - 10 x + 5$ , $[- 1, 2]$

Step 1

Write $f (x) = 2 x^{3} - 5 x^{2} - 10 x + 5$ as an equation.

$y = 2 x^{3} - 5 x^{2} - 10 x + 5$

Step 2

Substitute using the average rate of change formula.

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.

$\frac{f (2) - f (- 1)}{(2) - (- 1)}$

Substitute the equation $y = 2 x^{3} - 5 x^{2} - 10 x + 5$ for $f (2)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(2 {(2)}^{3} - 5 {(2)}^{2} - 10 \cdot 2 + 5) - (2 {(- 1)}^{3} - 5 {(- 1)}^{2} - 10 \cdot - 1 + 5)}{(2) - (- 1)}$

Step 3

Simplify the expression.

Simplify the numerator.

Multiply $2$ by $2^{3}$ by adding the exponents.

Multiply $2$ by $2^{3}$ .

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

$\frac{2^{1} \cdot 2^{3} - 5 \cdot 2^{2} - 10 \cdot 2 + 5 - (2 {(- 1)}^{3} - 5 {(- 1)}^{2} - 10 \cdot - 1 + 5)}{2 - (- 1)}$

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

$\frac{2^{1 + 3} - 5 \cdot 2^{2} - 10 \cdot 2 + 5 - (2 {(- 1)}^{3} - 5 {(- 1)}^{2} - 10 \cdot - 1 + 5)}{2 - (- 1)}$

Add $1$ and $3$ .

$\frac{2^{4} - 5 \cdot 2^{2} - 10 \cdot 2 + 5 - (2 {(- 1)}^{3} - 5 {(- 1)}^{2} - 10 \cdot - 1 + 5)}{2 - (- 1)}$

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

$\frac{16 - 5 \cdot 2^{2} - 10 \cdot 2 + 5 - (2 {(- 1)}^{3} - 5 {(- 1)}^{2} - 10 \cdot - 1 + 5)}{2 - (- 1)}$

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

$\frac{16 - 5 \cdot 4 - 10 \cdot 2 + 5 - (2 {(- 1)}^{3} - 5 {(- 1)}^{2} - 10 \cdot - 1 + 5)}{2 - (- 1)}$

Multiply $- 5$ by $4$ .

$\frac{16 - 20 - 10 \cdot 2 + 5 - (2 {(- 1)}^{3} - 5 {(- 1)}^{2} - 10 \cdot - 1 + 5)}{2 - (- 1)}$

Multiply $- 10$ by $2$ .

$\frac{16 - 20 - 20 + 5 - (2 {(- 1)}^{3} - 5 {(- 1)}^{2} - 10 \cdot - 1 + 5)}{2 - (- 1)}$

Simplify each term.

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

$\frac{16 - 20 - 20 + 5 - (2 \cdot - 1 - 5 {(- 1)}^{2} - 10 \cdot - 1 + 5)}{2 - (- 1)}$

Multiply $2$ by $- 1$ .

$\frac{16 - 20 - 20 + 5 - (- 2 - 5 {(- 1)}^{2} - 10 \cdot - 1 + 5)}{2 - (- 1)}$

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

$\frac{16 - 20 - 20 + 5 - (- 2 - 5 \cdot 1 - 10 \cdot - 1 + 5)}{2 - (- 1)}$

Multiply $- 5$ by $1$ .

$\frac{16 - 20 - 20 + 5 - (- 2 - 5 - 10 \cdot - 1 + 5)}{2 - (- 1)}$

Multiply $- 10$ by $- 1$ .

$\frac{16 - 20 - 20 + 5 - (- 2 - 5 + 10 + 5)}{2 - (- 1)}$

Subtract $5$ from $- 2$ .

$\frac{16 - 20 - 20 + 5 - (- 7 + 10 + 5)}{2 - (- 1)}$

Add $- 7$ and $10$ .

$\frac{16 - 20 - 20 + 5 - (3 + 5)}{2 - (- 1)}$

Add $3$ and $5$ .

$\frac{16 - 20 - 20 + 5 - 1 \cdot 8}{2 - (- 1)}$

Multiply $- 1$ by $8$ .

$\frac{16 - 20 - 20 + 5 - 8}{2 - (- 1)}$

Subtract $20$ from $16$ .

$\frac{- 4 - 20 + 5 - 8}{2 - (- 1)}$

Subtract $20$ from $- 4$ .

$\frac{- 24 + 5 - 8}{2 - (- 1)}$

Add $- 24$ and $5$ .

$\frac{- 19 - 8}{2 - (- 1)}$

Subtract $8$ from $- 19$ .

$\frac{- 27}{2 - (- 1)}$

Simplify the denominator.

Multiply $- 1$ by $- 1$ .

$\frac{- 27}{2 + 1}$

Add $2$ and $1$ .

$\frac{- 27}{3}$

Divide $- 27$ by $3$ .

$- 9$

Find the Average Rate of Change f(x)=2x^3-5x^2-10x+5 , [-1,2]

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