Precalculus Examples

$c (x) = 0.2 x^{3} + 4 x + 4000$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

$c (x + h) = 0.2 {(x + h)}^{3} + 4 (x + h) + 4000$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Use the Binomial Theorem.

$c (x + h) = 0.2 (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) + 4 (x + h) + 4000$

Step 2.1.2.1.2

Apply the distributive property.

$c (x + h) = 0.2 x^{3} + 0.2 (3 x^{2} h) + 0.2 (3 x h^{2}) + 0.2 h^{3} + 4 (x + h) + 4000$

Step 2.1.2.1.3

Simplify.

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Step 2.1.2.1.3.1

Multiply $3$ by $0.2$ .

$c (x + h) = 0.2 x^{3} + 0.6 (x^{2} h) + 0.2 (3 x h^{2}) + 0.2 h^{3} + 4 (x + h) + 4000$

Step 2.1.2.1.3.2

Multiply $3$ by $0.2$ .

$c (x + h) = 0.2 x^{3} + 0.6 (x^{2} h) + 0.6 (x h^{2}) + 0.2 h^{3} + 4 (x + h) + 4000$

Step 2.1.2.1.4

Remove parentheses.

$c (x + h) = 0.2 x^{3} + 0.6 x^{2} h + 0.6 x h^{2} + 0.2 h^{3} + 4 (x + h) + 4000$

Step 2.1.2.1.5

Apply the distributive property.

$c (x + h) = 0.2 x^{3} + 0.6 x^{2} h + 0.6 x h^{2} + 0.2 h^{3} + 4 x + 4 h + 4000$

Step 2.1.2.2

The final answer is $0.2 x^{3} + 0.6 x^{2} h + 0.6 x h^{2} + 0.2 h^{3} + 4 x + 4 h + 4000$ .

$0.2 x^{3} + 0.6 x^{2} h + 0.6 x h^{2} + 0.2 h^{3} + 4 x + 4 h + 4000$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

$0.2 x^{3} + 0.6 h x^{2} + 0.6 x h^{2} + 0.2 h^{3} + 4 x + 4 h + 4000$

Step 2.2.2

Move $x$ .

$0.2 x^{3} + 0.6 h x^{2} + 0.6 h^{2} x + 0.2 h^{3} + 4 x + 4 h + 4000$

Step 2.2.3

Move $4 x$ .

$0.2 x^{3} + 0.6 h x^{2} + 0.6 h^{2} x + 0.2 h^{3} + 4 h + 4 x + 4000$

Step 2.2.4

Move $0.2 x^{3}$ .

$0.6 h x^{2} + 0.6 h^{2} x + 0.2 h^{3} + 0.2 x^{3} + 4 h + 4 x + 4000$

Step 2.2.5

Move $0.6 h x^{2}$ .

$0.6 h^{2} x + 0.2 h^{3} + 0.6 h x^{2} + 0.2 x^{3} + 4 h + 4 x + 4000$

Step 2.2.6

Reorder $0.6 h^{2} x$ and $0.2 h^{3}$ .

$0.2 h^{3} + 0.6 h^{2} x + 0.6 h x^{2} + 0.2 x^{3} + 4 h + 4 x + 4000$

Step 2.3

Find the components of the definition.

$c (x + h) = 0.2 h^{3} + 0.6 h^{2} x + 0.6 h x^{2} + 0.2 x^{3} + 4 h + 4 x + 4000$

$c (x) = 0.2 x^{3} + 4 x + 4000$

$c (x + h) = 0.2 h^{3} + 0.6 h^{2} x + 0.6 h x^{2} + 0.2 x^{3} + 4 h + 4 x + 4000$

$c (x) = 0.2 x^{3} + 4 x + 4000$

Step 3

Plug in the components.

$\frac{c (x + h) - c x}{h} = \frac{0.2 h^{3} + 0.6 h^{2} x + 0.6 h x^{2} + 0.2 x^{3} + 4 h + 4 x + 4000 - (0.2 x^{3} + 4 x + 4000)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $0.2$ out of $0.2 h^{3} + 0.6 h^{2} x + 0.6 h x^{2} + 0.2 x^{3} + 4 h + 4 x + 4000 - (0.2 x^{3} + 4 x + 4000)$ .

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Step 4.1.1.1

Reorder the expression.

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Step 4.1.1.1.1

Move $h^{2}$ .

$\frac{0.2 h^{3} + 0.6 x h^{2} + 0.6 h x^{2} + 0.2 x^{3} + 4 h + 4 x + 4000 - (0.2 x^{3} + 4 x + 4000)}{h}$

Step 4.1.1.1.2

Move $h$ .

$\frac{0.2 h^{3} + 0.6 x h^{2} + 0.6 x^{2} h + 0.2 x^{3} + 4 h + 4 x + 4000 - (0.2 x^{3} + 4 x + 4000)}{h}$

Step 4.1.1.2

Factor $0.2$ out of $0.2 h^{3}$ .

$\frac{0.2 h^{3} + 0.6 x h^{2} + 0.6 x^{2} h + 0.2 x^{3} + 4 h + 4 x + 4000 - (0.2 x^{3} + 4 x + 4000)}{h}$

Step 4.1.1.3

Factor $0.2$ out of $0.6 x h^{2}$ .

$\frac{0.2 h^{3} + 0.2 (3 x h^{2}) + 0.6 x^{2} h + 0.2 x^{3} + 4 h + 4 x + 4000 - (0.2 x^{3} + 4 x + 4000)}{h}$

Step 4.1.1.4

Factor $0.2$ out of $0.6 x^{2} h$ .

$\frac{0.2 h^{3} + 0.2 (3 x h^{2}) + 0.2 (3 x^{2} h) + 0.2 x^{3} + 4 h + 4 x + 4000 - (0.2 x^{3} + 4 x + 4000)}{h}$

Step 4.1.1.5

Factor $0.2$ out of $0.2 x^{3}$ .

$\frac{0.2 h^{3} + 0.2 (3 x h^{2}) + 0.2 (3 x^{2} h) + 0.2 (x^{3}) + 4 h + 4 x + 4000 - (0.2 x^{3} + 4 x + 4000)}{h}$

Step 4.1.1.6

Factor $0.2$ out of $4 h$ .

$\frac{0.2 h^{3} + 0.2 (3 x h^{2}) + 0.2 (3 x^{2} h) + 0.2 (x^{3}) + 0.2 (20 h) + 4 x + 4000 - (0.2 x^{3} + 4 x + 4000)}{h}$

Step 4.1.1.7

Factor $0.2$ out of $4 x$ .

$\frac{0.2 h^{3} + 0.2 (3 x h^{2}) + 0.2 (3 x^{2} h) + 0.2 (x^{3}) + 0.2 (20 h) + 0.2 (20 x) + 4000 - (0.2 x^{3} + 4 x + 4000)}{h}$

Step 4.1.1.8

Factor $0.2$ out of $4000$ .

$\frac{0.2 h^{3} + 0.2 (3 x h^{2}) + 0.2 (3 x^{2} h) + 0.2 (x^{3}) + 0.2 (20 h) + 0.2 (20 x) + 0.2 (20000) - (0.2 x^{3} + 4 x + 4000)}{h}$

Step 4.1.1.9

Factor $0.2$ out of $- (0.2 x^{3} + 4 x + 4000)$ .

$\frac{0.2 h^{3} + 0.2 (3 x h^{2}) + 0.2 (3 x^{2} h) + 0.2 (x^{3}) + 0.2 (20 h) + 0.2 (20 x) + 0.2 (20000) + 0.2 (- 5 (0.2 x^{3} + 4 x + 4000))}{h}$

Step 4.1.1.10

Factor $0.2$ out of $0.2 h^{3} + 0.2 (3 x h^{2})$ .

$\frac{0.2 (h^{3} + 3 x h^{2}) + 0.2 (3 x^{2} h) + 0.2 (x^{3}) + 0.2 (20 h) + 0.2 (20 x) + 0.2 (20000) + 0.2 (- 5 (0.2 x^{3} + 4 x + 4000))}{h}$

Step 4.1.1.11

Factor $0.2$ out of $0.2 (h^{3} + 3 x h^{2}) + 0.2 (3 x^{2} h)$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h) + 0.2 (x^{3}) + 0.2 (20 h) + 0.2 (20 x) + 0.2 (20000) + 0.2 (- 5 (0.2 x^{3} + 4 x + 4000))}{h}$

Step 4.1.1.12

Factor $0.2$ out of $0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h) + 0.2 (x^{3})$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3}) + 0.2 (20 h) + 0.2 (20 x) + 0.2 (20000) + 0.2 (- 5 (0.2 x^{3} + 4 x + 4000))}{h}$

Step 4.1.1.13

Factor $0.2$ out of $0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3}) + 0.2 (20 h)$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h) + 0.2 (20 x) + 0.2 (20000) + 0.2 (- 5 (0.2 x^{3} + 4 x + 4000))}{h}$

Step 4.1.1.14

Factor $0.2$ out of $0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h) + 0.2 (20 x)$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h + 20 x) + 0.2 (20000) + 0.2 (- 5 (0.2 x^{3} + 4 x + 4000))}{h}$

Step 4.1.1.15

Factor $0.2$ out of $0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h + 20 x) + 0.2 (20000)$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h + 20 x + 20000) + 0.2 (- 5 (0.2 x^{3} + 4 x + 4000))}{h}$

Step 4.1.1.16

Factor $0.2$ out of $0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h + 20 x + 20000) + 0.2 (- 5 (0.2 x^{3} + 4 x + 4000))$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h + 20 x + 20000 - 5 (0.2 x^{3} + 4 x + 4000))}{h}$

Step 4.1.2

Apply the distributive property.

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h + 20 x + 20000 - 5 (0.2 x^{3}) - 5 (4 x) - 5 \cdot 4000)}{h}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Multiply $0.2$ by $- 5$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h + 20 x + 20000 - 1 x^{3} - 5 (4 x) - 5 \cdot 4000)}{h}$

Step 4.1.3.2

Multiply $4$ by $- 5$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h + 20 x + 20000 - 1 x^{3} - 20 x - 5 \cdot 4000)}{h}$

Step 4.1.3.3

Multiply $- 5$ by $4000$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h + 20 x + 20000 - 1 x^{3} - 20 x - 20000)}{h}$

Step 4.1.4

Rewrite $- 1 x^{3}$ as $- x^{3}$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + x^{3} + 20 h + 20 x + 20000 - x^{3} - 20 x - 20000)}{h}$

Step 4.1.5

Subtract $x^{3}$ from $x^{3}$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + 20 h + 20 x + 20000 + 0 - 20 x - 20000)}{h}$

Step 4.1.6

Add $h^{3}$ and $0$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + 20 h + 20 x + 20000 - 20 x - 20000)}{h}$

Step 4.1.7

Subtract $20 x$ from $20 x$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + 20 h + 0 + 20000 - 20000)}{h}$

Step 4.1.8

Add $h^{3}$ and $0$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + 20 h + 20000 - 20000)}{h}$

Step 4.1.9

Subtract $20000$ from $20000$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + 20 h + 0)}{h}$

Step 4.1.10

Add $h^{3} + 3 x h^{2} + 3 x^{2} h + 20 h$ and $0$ .

$\frac{0.2 (h^{3} + 3 x h^{2} + 3 x^{2} h + 20 h)}{h}$

Step 4.1.11

Factor $h$ out of $h^{3} + 3 x h^{2} + 3 x^{2} h + 20 h$ .

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Step 4.1.11.1

Factor $h$ out of $h^{3}$ .

$\frac{0.2 (h \cdot h^{2} + 3 x h^{2} + 3 x^{2} h + 20 h)}{h}$

Step 4.1.11.2

Factor $h$ out of $3 x h^{2}$ .

$\frac{0.2 (h (h^{2}) + h (3 x h) + 3 x^{2} h + 20 h)}{h}$

Step 4.1.11.3

Factor $h$ out of $3 x^{2} h$ .

$\frac{0.2 (h (h^{2}) + h (3 x h) + h (3 x^{2}) + 20 h)}{h}$

Step 4.1.11.4

Factor $h$ out of $20 h$ .

$\frac{0.2 (h (h^{2}) + h (3 x h) + h (3 x^{2}) + h \cdot 20)}{h}$

Step 4.1.11.5

Factor $h$ out of $h (h^{2}) + h (3 x h)$ .

$\frac{0.2 (h (h^{2} + 3 x h) + h (3 x^{2}) + h \cdot 20)}{h}$

Step 4.1.11.6

Factor $h$ out of $h (h^{2} + 3 x h) + h (3 x^{2})$ .

$\frac{0.2 (h (h^{2} + 3 x h + 3 x^{2}) + h \cdot 20)}{h}$

Step 4.1.11.7

Factor $h$ out of $h (h^{2} + 3 x h + 3 x^{2}) + h \cdot 20$ .

$\frac{0.2 (h (h^{2} + 3 x h + 3 x^{2} + 20))}{h}$

$\frac{0.2 h (h^{2} + 3 x h + 3 x^{2} + 20)}{h}$

Step 4.2

Simplify terms.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.2 h (h^{2} + 3 x h + 3 x^{2} + 20)}{h}$

Step 4.2.1.2

Divide $0.2 (h^{2} + 3 x h + 3 x^{2} + 20)$ by $1$ .

$0.2 (h^{2} + 3 x h + 3 x^{2} + 20)$

Step 4.2.2

Apply the distributive property.

$0.2 h^{2} + 0.2 (3 x h) + 0.2 (3 x^{2}) + 0.2 \cdot 20$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $3$ by $0.2$ .

$0.2 h^{2} + 0.6 (x h) + 0.2 (3 x^{2}) + 0.2 \cdot 20$

Step 4.3.2

Multiply $3$ by $0.2$ .

$0.2 h^{2} + 0.6 (x h) + 0.6 x^{2} + 0.2 \cdot 20$

Step 4.3.3

Multiply $0.2$ by $20$ .

$0.2 h^{2} + 0.6 (x h) + 0.6 x^{2} + 4$

Step 4.4

Move $0.2 h^{2}$ .

$0.6 x h + 0.6 x^{2} + 0.2 h^{2} + 4$

Step 4.5

Reorder $0.6 x h$ and $0.6 x^{2}$ .

$0.6 x^{2} + 0.6 x h + 0.2 h^{2} + 4$

Step 5