Precalculus Examples

$V = \frac{4}{3} π r^{3}$

Step 1

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

$f (r) = \frac{4}{3} π r^{3}$

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.

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Use the Binomial Theorem.

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

Step 3.1.2.2

Apply the distributive property.

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

Step 3.1.2.3

Remove parentheses.

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

Step 3.1.2.4

Simplify each term.

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Step 3.1.2.4.1

Move $3$ to the left of $π$ .

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

Step 3.1.2.4.2

Move $3$ to the left of $π$ .

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

Step 3.1.2.5

Apply the distributive property.

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

Step 3.1.2.6

Simplify.

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Step 3.1.2.6.1

Multiply $\frac{4}{3} (π r^{3})$ .

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Step 3.1.2.6.1.1

Combine $π$ and $\frac{4}{3}$ .

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

Step 3.1.2.6.1.2

Combine $\frac{π \cdot 4}{3}$ and $r^{3}$ .

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

Step 3.1.2.6.2

Cancel the common factor of $3$ .

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Step 3.1.2.6.2.1

Factor $3$ out of $3 π r^{2} h$ .

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

Step 3.1.2.6.2.2

Cancel the common factor.

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

Step 3.1.2.6.2.3

Rewrite the expression.

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

Step 3.1.2.6.3

Cancel the common factor of $3$ .

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Step 3.1.2.6.3.1

Factor $3$ out of $3 π r h^{2}$ .

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

Step 3.1.2.6.3.2

Cancel the common factor.

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

Step 3.1.2.6.3.3

Rewrite the expression.

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

Step 3.1.2.6.4

Multiply $\frac{4}{3} (π h^{3})$ .

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Step 3.1.2.6.4.1

Combine $π$ and $\frac{4}{3}$ .

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

Step 3.1.2.6.4.2

Combine $\frac{π \cdot 4}{3}$ and $h^{3}$ .

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

Step 3.1.2.7

Simplify each term.

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Step 3.1.2.7.1

Move $4$ to the left of $π$ .

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

Step 3.1.2.7.2

Move $4$ to the left of $π$ .

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

Step 3.1.2.8

The final answer is $\frac{4 π r^{3}}{3} + 4 π r^{2} h + 4 π r h^{2} + \frac{4 π h^{3}}{3}$ .

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

Step 3.2

Reorder.

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Step 3.2.1

Move $r^{2}$ .

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

Step 3.2.2

Move $r$ .

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

Step 3.2.3

Move $\frac{4 π r^{3}}{3}$ .

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

Step 3.2.4

Move $4 π h r^{2}$ .

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

Step 3.2.5

Reorder $4 π h^{2} r$ and $\frac{4 π h^{3}}{3}$ .

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

Step 3.3

Find the components of the definition.

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

$f (r) = \frac{4 π r^{3}}{3}$

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

$f (r) = \frac{4 π r^{3}}{3}$

Step 4

Plug in the components.

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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Step 5.1.3.1

Factor $4 π h$ out of $\frac{4 π h^{3}}{3}$ .

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

Step 5.1.3.2

Factor $4 π h$ out of $4 π h^{2} r$ .

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

Step 5.1.3.3

Factor $4 π h$ out of $4 π h r^{2}$ .

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

Step 5.1.3.4

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

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

Step 5.1.3.5

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

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

Step 5.1.4

To write $h r$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.1.5

Combine $h r$ and $\frac{3}{3}$ .

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

Step 5.1.6

Combine the numerators over the common denominator.

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

Step 5.1.7

Simplify the numerator.

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Step 5.1.7.1

Factor $h$ out of $h^{2} + h r \cdot 3$ .

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Step 5.1.7.1.1

Factor $h$ out of $h^{2}$ .

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

Step 5.1.7.1.2

Factor $h$ out of $h r \cdot 3$ .

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

Step 5.1.7.1.3

Factor $h$ out of $h (h) + h (r \cdot 3)$ .

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

Step 5.1.7.2

Move $3$ to the left of $r$ .

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

Step 5.1.8

To write $r^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.1.9

Combine $r^{2}$ and $\frac{3}{3}$ .

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

Step 5.1.10

Combine the numerators over the common denominator.

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

Step 5.1.11

Simplify the numerator.

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Step 5.1.11.1

Apply the distributive property.

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

Step 5.1.11.2

Multiply $h$ by $h$ .

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

Step 5.1.11.3

Rewrite using the commutative property of multiplication.

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

Step 5.1.11.4

Move $3$ to the left of $r^{2}$ .

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

Step 5.1.12

Combine exponents.

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Step 5.1.12.1

Combine $4$ and $\frac{h^{2} + 3 h r + 3 r^{2}}{3}$ .

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

Step 5.1.12.2

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

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

Step 5.1.12.3

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

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

Step 5.1.13

Remove unnecessary parentheses.

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

Step 5.1.14

Move $4$ to the left of $h π$ .

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

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3

Combine.

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

Step 5.4

Cancel the common factor of $h$ .

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Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Rewrite the expression.

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

Step 5.5

Multiply $4$ by $1$ .

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

Step 6