Precalculus Examples

$f (x) = x^{\frac{3}{2}}$ on $[1, 4]$

Step 1

Write $f (x) = x^{\frac{3}{2}}$ as an equation.

$y = x^{\frac{3}{2}}$

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.

$\frac{f (4) - f (1)}{(4) - (1)}$

Step 2.2

Substitute the equation $y = x^{\frac{3}{2}}$ for $f (4)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.3

Cancel the common factor of $2$ .

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Step 3.1.3.1

Cancel the common factor.

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

Step 3.1.3.2

Rewrite the expression.

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

Step 3.1.4

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

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

Step 3.1.5

One to any power is one.

$\frac{8 - 1 \cdot 1}{4 - (1)}$

Step 3.1.6

Multiply $- 1$ by $1$ .

$\frac{8 - 1}{4 - (1)}$

Step 3.1.7

Subtract $1$ from $8$ .

$\frac{7}{4 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{7}{4 - 1}$

Step 3.2.2

Subtract $1$ from $4$ .

$\frac{7}{3}$