Precalculus Examples

$V = \frac{4 π r^{3}}{3}$

Step 1

Write $V = \frac{4 π r^{3}}{3}$ as a function.

$f (r) = \frac{4 π r^{3}}{3}$

Step 2

Consider the difference quotient formula.

$\frac{f (r + h) - f r}{h}$

Step 3

Find the components of the definition.

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Step 3.1

Evaluate the function at $x = r + h$ .

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Step 3.1.1

Replace the variable $r$ with $r + h$ in the expression.

$f (r + h) = \frac{4 π {(r + h)}^{3}}{3}$

Step 3.1.2

The final answer is $\frac{4 π {(r + h)}^{3}}{3}$ .

$\frac{4 π {(r + h)}^{3}}{3}$

Step 3.2

Find the components of the definition.

$f (r + h) = \frac{4 π {(r + h)}^{3}}{3}$

$f (r) = \frac{4 π r^{3}}{3}$

$f (r + h) = \frac{4 π {(r + h)}^{3}}{3}$

$f (r) = \frac{4 π r^{3}}{3}$

Step 4

Plug in the components.

$\frac{f (r + h) - f r}{h} = \frac{\frac{4 π {(r + h)}^{3}}{3} - (\frac{4 π r^{3}}{3})}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Combine the numerators over the common denominator.

$\frac{\frac{4 π {(r + h)}^{3} - 4 π r^{3}}{3}}{h}$

Step 5.1.2

Rewrite $\frac{4 π {(r + h)}^{3} - 4 π r^{3}}{3}$ in a factored form.

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Step 5.1.2.1

Use the Binomial Theorem.

$\frac{\frac{4 π (r^{3} + 3 r^{2} h + 3 r h^{2} + h^{3}) - 4 π r^{3}}{3}}{h}$

Step 5.1.2.2

Apply the distributive property.

$\frac{\frac{4 π r^{3} + 4 π (3 r^{2} h) + 4 π (3 r h^{2}) + 4 π h^{3} - 4 π r^{3}}{3}}{h}$

Step 5.1.2.3

Simplify.

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Step 5.1.2.3.1

Multiply $3$ by $4$ .

$\frac{\frac{4 π r^{3} + 12 π (r^{2} h) + 4 π (3 r h^{2}) + 4 π h^{3} - 4 π r^{3}}{3}}{h}$

Step 5.1.2.3.2

Multiply $3$ by $4$ .

$\frac{\frac{4 π r^{3} + 12 π (r^{2} h) + 12 π (r h^{2}) + 4 π h^{3} - 4 π r^{3}}{3}}{h}$

Step 5.1.2.4

Remove parentheses.

$\frac{\frac{4 π r^{3} + 12 π r^{2} h + 12 π r h^{2} + 4 π h^{3} - 4 π r^{3}}{3}}{h}$

Step 5.1.2.5

Subtract $4 π r^{3}$ from $4 π r^{3}$ .

$\frac{\frac{12 π r^{2} h + 12 π r h^{2} + 4 π h^{3} + 0}{3}}{h}$

Step 5.1.2.6

Add $12 π r^{2} h + 12 π r h^{2} + 4 π h^{3}$ and $0$ .

$\frac{\frac{12 π r^{2} h + 12 π r h^{2} + 4 π h^{3}}{3}}{h}$

Step 5.1.2.7

Factor $4 π h$ out of $12 π r^{2} h + 12 π r h^{2} + 4 π h^{3}$ .

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Step 5.1.2.7.1

Factor $4 π h$ out of $12 π r^{2} h$ .

$\frac{\frac{4 π h (3 r^{2}) + 12 π r h^{2} + 4 π h^{3}}{3}}{h}$

Step 5.1.2.7.2

Factor $4 π h$ out of $12 π r h^{2}$ .

$\frac{\frac{4 π h (3 r^{2}) + 4 π h (3 r h) + 4 π h^{3}}{3}}{h}$

Step 5.1.2.7.3

Factor $4 π h$ out of $4 π h^{3}$ .

$\frac{\frac{4 π h (3 r^{2}) + 4 π h (3 r h) + 4 π h \cdot h^{2}}{3}}{h}$

Step 5.1.2.7.4

Factor $4 π h$ out of $4 π h (3 r^{2}) + 4 π h (3 r h)$ .

$\frac{\frac{4 π h (3 r^{2} + 3 r h) + 4 π h \cdot h^{2}}{3}}{h}$

Step 5.1.2.7.5

Factor $4 π h$ out of $4 π h (3 r^{2} + 3 r h) + 4 π h \cdot h^{2}$ .

$\frac{\frac{4 π h (3 r^{2} + 3 r h + h^{2})}{3}}{h}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 π h (3 r^{2} + 3 r h + h^{2})}{3} \cdot \frac{1}{h}$

Step 5.3

Combine.

$\frac{4 π h (3 r^{2} + 3 r h + h^{2}) \cdot 1}{3 h}$

Step 5.4

Cancel the common factor of $h$ .

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Step 5.4.1

Cancel the common factor.

$\frac{4 π h (3 r^{2} + 3 r h + h^{2}) \cdot 1}{3 h}$

Step 5.4.2

Rewrite the expression.

$\frac{4 π (3 r^{2} + 3 r h + h^{2}) \cdot 1}{3}$

Step 5.5

Multiply $4$ by $1$ .

$\frac{4 π (3 r^{2} + 3 r h + h^{2})}{3}$

Step 6