Precalculus Examples

$f (x) = 3 x^{3} - 2 x^{2} + 6$ ; $[2, 6]$

Step 1

Write $f (x) = 3 x^{3} - 2 x^{2} + 6$ as an equation.

$y = 3 x^{3} - 2 x^{2} + 6$

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.

$\frac{f (6) - f (2)}{(6) - (2)}$

Step 2.2

Substitute the equation $y = 3 x^{3} - 2 x^{2} + 6$ for $f (6)$ and $f (2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(3 {(6)}^{3} - 2 {(6)}^{2} + 6) - (3 {(2)}^{3} - 2 {(2)}^{2} + 6)}{(6) - (2)}$

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 $3$ .

$\frac{3 \cdot 216 - 2 \cdot 6^{2} + 6 - (3 \cdot 2^{3} - 2 \cdot 2^{2} + 6)}{6 - (2)}$

Step 3.1.2

Multiply $3$ by $216$ .

$\frac{648 - 2 \cdot 6^{2} + 6 - (3 \cdot 2^{3} - 2 \cdot 2^{2} + 6)}{6 - (2)}$

Step 3.1.3

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

$\frac{648 - 2 \cdot 36 + 6 - (3 \cdot 2^{3} - 2 \cdot 2^{2} + 6)}{6 - (2)}$

Step 3.1.4

Multiply $- 2$ by $36$ .

$\frac{648 - 72 + 6 - (3 \cdot 2^{3} - 2 \cdot 2^{2} + 6)}{6 - (2)}$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

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

$\frac{648 - 72 + 6 - (3 \cdot 8 - 2 \cdot 2^{2} + 6)}{6 - (2)}$

Step 3.1.5.2

Multiply $3$ by $8$ .

$\frac{648 - 72 + 6 - (24 - 2 \cdot 2^{2} + 6)}{6 - (2)}$

Step 3.1.5.3

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

$\frac{648 - 72 + 6 - (24 - 2 \cdot 4 + 6)}{6 - (2)}$

Step 3.1.5.4

Multiply $- 2$ by $4$ .

$\frac{648 - 72 + 6 - (24 - 8 + 6)}{6 - (2)}$

Step 3.1.6

Subtract $8$ from $24$ .

$\frac{648 - 72 + 6 - (16 + 6)}{6 - (2)}$

Step 3.1.7

Add $16$ and $6$ .

$\frac{648 - 72 + 6 - 1 \cdot 22}{6 - (2)}$

Step 3.1.8

Multiply $- 1$ by $22$ .

$\frac{648 - 72 + 6 - 22}{6 - (2)}$

Step 3.1.9

Subtract $72$ from $648$ .

$\frac{576 + 6 - 22}{6 - (2)}$

Step 3.1.10

Add $576$ and $6$ .

$\frac{582 - 22}{6 - (2)}$

Step 3.1.11

Subtract $22$ from $582$ .

$\frac{560}{6 - (2)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $2$ .

$\frac{560}{6 - 2}$

Step 3.2.2

Subtract $2$ from $6$ .

$\frac{560}{4}$

Step 3.3

Divide $560$ by $4$ .

$140$