Precalculus Examples

$f (x) = 2 x^{2} - 2 x - 1$ ; $[1, \frac{3}{2}]$

Step 1

Write $f (x) = 2 x^{2} - 2 x - 1$ as an equation.

$y = 2 x^{2} - 2 x - 1$

Step 2

Substitute using the average rate of change formula.

Tap for more steps...

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.

$\frac{f (\frac{3}{2}) - f (1)}{(\frac{3}{2}) - (1)}$

Step 2.2

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

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

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Multiply the numerator and denominator of the fraction by $2$ .

Tap for more steps...

Step 3.1.1

Multiply $\frac{2 {(\frac{3}{2})}^{2} - 2 (\frac{3}{2}) - 1 - (2 \cdot 1^{2} - 2 \cdot 1 - 1)}{\frac{3}{2} - (1)}$ by $\frac{2}{2}$ .

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

Tap for more steps...

Step 3.4.1

Multiply $2$ by $2$ .

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

Step 3.4.2

Apply the product rule to $\frac{3}{2}$ .

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

Step 3.4.3

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

$\frac{4 \frac{9}{2^{2}} + 2 (- 2 (\frac{3}{2})) + 2 \cdot - 1 + 2 (- (2 \cdot 1^{2} - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.4

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

$\frac{4 (\frac{9}{4}) + 2 (- 2 (\frac{3}{2})) + 2 \cdot - 1 + 2 (- (2 \cdot 1^{2} - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.5

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.4.5.1

Cancel the common factor.

$\frac{4 (\frac{9}{4}) + 2 (- 2 (\frac{3}{2})) + 2 \cdot - 1 + 2 (- (2 \cdot 1^{2} - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.5.2

Rewrite the expression.

$\frac{9 + 2 (- 2 (\frac{3}{2})) + 2 \cdot - 1 + 2 (- (2 \cdot 1^{2} - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.6.1

Factor $2$ out of $- 2$ .

$\frac{9 + 2 (2 (- 1) \frac{3}{2}) + 2 \cdot - 1 + 2 (- (2 \cdot 1^{2} - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.6.2

Cancel the common factor.

$\frac{9 + 2 (2 \cdot - 1 \frac{3}{2}) + 2 \cdot - 1 + 2 (- (2 \cdot 1^{2} - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.6.3

Rewrite the expression.

$\frac{9 + 2 (- 1 \cdot 3) + 2 \cdot - 1 + 2 (- (2 \cdot 1^{2} - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.7

Multiply $- 1$ by $3$ .

$\frac{9 + 2 \cdot - 3 + 2 \cdot - 1 + 2 (- (2 \cdot 1^{2} - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.8

Multiply $2$ by $- 3$ .

$\frac{9 - 6 + 2 \cdot - 1 + 2 (- (2 \cdot 1^{2} - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.9

Multiply $2$ by $- 1$ .

$\frac{9 - 6 - 2 + 2 (- (2 \cdot 1^{2} - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.10

Simplify each term.

Tap for more steps...

Step 3.4.10.1

One to any power is one.

$\frac{9 - 6 - 2 + 2 (- (2 \cdot 1 - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.10.2

Multiply $2$ by $1$ .

$\frac{9 - 6 - 2 + 2 (- (2 - 2 \cdot 1 - 1))}{3 + 2 (- (1))}$

Step 3.4.10.3

Multiply $- 2$ by $1$ .

$\frac{9 - 6 - 2 + 2 (- (2 - 2 - 1))}{3 + 2 (- (1))}$

Step 3.4.11

Subtract $2$ from $2$ .

$\frac{9 - 6 - 2 + 2 (- (0 - 1))}{3 + 2 (- (1))}$

Step 3.4.12

Subtract $1$ from $0$ .

$\frac{9 - 6 - 2 + 2 (- - 1)}{3 + 2 (- (1))}$

Step 3.4.13

Multiply $2 (- - 1)$ .

Tap for more steps...

Step 3.4.13.1

Multiply $- 1$ by $- 1$ .

$\frac{9 - 6 - 2 + 2 \cdot 1}{3 + 2 (- (1))}$

Step 3.4.13.2

Multiply $2$ by $1$ .

$\frac{9 - 6 - 2 + 2}{3 + 2 (- (1))}$

Step 3.4.14

Subtract $6$ from $9$ .

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

Step 3.4.15

Subtract $2$ from $3$ .

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

Step 3.4.16

Add $1$ and $2$ .

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

Step 3.5

Simplify the denominator.

Tap for more steps...

Step 3.5.1

Multiply $2 (- (1))$ .

Tap for more steps...

Step 3.5.1.1

Multiply $- 1$ by $1$ .

$\frac{3}{3 + 2 \cdot - 1}$

Step 3.5.1.2

Multiply $2$ by $- 1$ .

$\frac{3}{3 - 2}$

Step 3.5.2

Subtract $2$ from $3$ .

$\frac{3}{1}$

Step 3.6

Divide $3$ by $1$ .

$3$