Precalculus Examples

$P (θ) = θ^{3} - 4 θ^{2} + 5 θ$ , $[1, 2]$

Step 1

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

$y = θ^{3} - 4 θ^{2} + 5 θ$

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 $θ$ values of the two points.

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

Step 2.2

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

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

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 $3$ .

$\frac{8 - 4 \cdot 2^{2} + 5 (2) - (1^{3} - 4 \cdot 1^{2} + 5 (1))}{2 - (1)}$

Step 3.1.2

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

$\frac{8 - 4 \cdot 4 + 5 (2) - (1^{3} - 4 \cdot 1^{2} + 5 (1))}{2 - (1)}$

Step 3.1.3

Multiply $- 4$ by $4$ .

$\frac{8 - 16 + 5 (2) - (1^{3} - 4 \cdot 1^{2} + 5 (1))}{2 - (1)}$

Step 3.1.4

Multiply $5$ by $2$ .

$\frac{8 - 16 + 10 - (1^{3} - 4 \cdot 1^{2} + 5 (1))}{2 - (1)}$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

One to any power is one.

$\frac{8 - 16 + 10 - (1 - 4 \cdot 1^{2} + 5 (1))}{2 - (1)}$

Step 3.1.5.2

One to any power is one.

$\frac{8 - 16 + 10 - (1 - 4 \cdot 1 + 5 (1))}{2 - (1)}$

Step 3.1.5.3

Multiply $- 4$ by $1$ .

$\frac{8 - 16 + 10 - (1 - 4 + 5 (1))}{2 - (1)}$

Step 3.1.5.4

Multiply $5$ by $1$ .

$\frac{8 - 16 + 10 - (1 - 4 + 5)}{2 - (1)}$

Step 3.1.6

Subtract $4$ from $1$ .

$\frac{8 - 16 + 10 - (- 3 + 5)}{2 - (1)}$

Step 3.1.7

Add $- 3$ and $5$ .

$\frac{8 - 16 + 10 - 1 \cdot 2}{2 - (1)}$

Step 3.1.8

Multiply $- 1$ by $2$ .

$\frac{8 - 16 + 10 - 2}{2 - (1)}$

Step 3.1.9

Subtract $16$ from $8$ .

$\frac{- 8 + 10 - 2}{2 - (1)}$

Step 3.1.10

Add $- 8$ and $10$ .

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

Step 3.1.11

Subtract $2$ from $2$ .

$\frac{0}{2 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{0}{2 - 1}$

Step 3.2.2

Subtract $1$ from $2$ .

$\frac{0}{1}$

Step 3.3

Divide $0$ by $1$ .

$0$