Find the Average Rate of Change

$f (x) = 2 x^{2} - 4$ , $[5, 6]$

Step 1

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

$y = 2 x^{2} - 4$

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.

$\frac{f (6) - f (5)}{(6) - (5)}$

Step 2.2

Substitute the equation $y = 2 x^{2} - 4$ for $f (6)$ and $f (5)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(2 {(6)}^{2} - 4) - (2 {(5)}^{2} - 4)}{(6) - (5)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $6$ to the power of $2$ .

$\frac{2 \cdot 36 - 4 - (2 \cdot 5^{2} - 4)}{6 - (5)}$

Step 3.1.2

Multiply $2$ by $36$ .

$\frac{72 - 4 - (2 \cdot 5^{2} - 4)}{6 - (5)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

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

$\frac{72 - 4 - (2 \cdot 25 - 4)}{6 - (5)}$

Step 3.1.3.2

Multiply $2$ by $25$ .

$\frac{72 - 4 - (50 - 4)}{6 - (5)}$

Step 3.1.4

Subtract $4$ from $50$ .

$\frac{72 - 4 - 1 \cdot 46}{6 - (5)}$

Step 3.1.5

Multiply $- 1$ by $46$ .

$\frac{72 - 4 - 46}{6 - (5)}$

Step 3.1.6

Subtract $4$ from $72$ .

$\frac{68 - 46}{6 - (5)}$

Step 3.1.7

Subtract $46$ from $68$ .

$\frac{22}{6 - (5)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $5$ .

$\frac{22}{6 - 5}$

Step 3.2.2

Subtract $5$ from $6$ .

$\frac{22}{1}$

Step 3.3

Divide $22$ by $1$ .

$22$

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