Calculus Examples

$C (q) = 0.1 q^{2} + 12 q + 400$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

$C (q + h) = 0.1 {(q + h)}^{2} + 12 (q + h) + 400$

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

Rewrite ${(q + h)}^{2}$ as $(q + h) (q + h)$ .

$C (q + h) = 0.1 ((q + h) (q + h)) + 12 (q + h) + 400$

Step 2.1.2.1.2

Expand $(q + h) (q + h)$ using the FOIL Method.

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Step 2.1.2.1.2.1

Apply the distributive property.

$C (q + h) = 0.1 (q (q + h) + h (q + h)) + 12 (q + h) + 400$

Step 2.1.2.1.2.2

Apply the distributive property.

$C (q + h) = 0.1 (q \cdot q + q h + h (q + h)) + 12 (q + h) + 400$

Step 2.1.2.1.2.3

Apply the distributive property.

$C (q + h) = 0.1 (q \cdot q + q h + h q + h \cdot h) + 12 (q + h) + 400$

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply $q$ by $q$ .

$C (q + h) = 0.1 (q^{2} + q h + h q + h \cdot h) + 12 (q + h) + 400$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$C (q + h) = 0.1 (q^{2} + q h + h q + h^{2}) + 12 (q + h) + 400$

Step 2.1.2.1.3.2

Add $q h$ and $h q$ .

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Step 2.1.2.1.3.2.1

Reorder $q$ and $h$ .

$C (q + h) = 0.1 (q^{2} + h q + h q + h^{2}) + 12 (q + h) + 400$

Step 2.1.2.1.3.2.2

Add $h q$ and $h q$ .

$C (q + h) = 0.1 (q^{2} + 2 h q + h^{2}) + 12 (q + h) + 400$

Step 2.1.2.1.4

Apply the distributive property.

$C (q + h) = 0.1 q^{2} + 0.1 (2 h q) + 0.1 h^{2} + 12 (q + h) + 400$

Step 2.1.2.1.5

Multiply $2$ by $0.1$ .

$C (q + h) = 0.1 q^{2} + 0.2 (h q) + 0.1 h^{2} + 12 (q + h) + 400$

Step 2.1.2.1.6

Apply the distributive property.

$C (q + h) = 0.1 q^{2} + 0.2 h q + 0.1 h^{2} + 12 q + 12 h + 400$

Step 2.1.2.2

The final answer is $0.1 q^{2} + 0.2 h q + 0.1 h^{2} + 12 q + 12 h + 400$ .

$0.1 q^{2} + 0.2 h q + 0.1 h^{2} + 12 q + 12 h + 400$

Step 2.2

Reorder.

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Step 2.2.1

Move $12 q$ .

$0.1 q^{2} + 0.2 h q + 0.1 h^{2} + 12 h + 12 q + 400$

Step 2.2.2

Move $0.1 q^{2}$ .

$0.2 h q + 0.1 h^{2} + 0.1 q^{2} + 12 h + 12 q + 400$

Step 2.2.3

Reorder $0.2 h q$ and $0.1 h^{2}$ .

$0.1 h^{2} + 0.2 h q + 0.1 q^{2} + 12 h + 12 q + 400$

Step 2.3

Find the components of the definition.

$C (q + h) = 0.1 h^{2} + 0.2 h q + 0.1 q^{2} + 12 h + 12 q + 400$

$C (q) = 0.1 q^{2} + 12 q + 400$

$C (q + h) = 0.1 h^{2} + 0.2 h q + 0.1 q^{2} + 12 h + 12 q + 400$

$C (q) = 0.1 q^{2} + 12 q + 400$

Step 3

Plug in the components.

$\frac{C (q + h) - C q}{h} = \frac{0.1 h^{2} + 0.2 h q + 0.1 q^{2} + 12 h + 12 q + 400 - (0.1 q^{2} + 12 q + 400)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $0.1$ out of $0.1 h^{2} + 0.2 h q + 0.1 q^{2} + 12 h + 12 q + 400 - (0.1 q^{2} + 12 q + 400)$ .

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Step 4.1.1.1

Move $h$ .

$\frac{0.1 h^{2} + 0.2 q h + 0.1 q^{2} + 12 h + 12 q + 400 - (0.1 q^{2} + 12 q + 400)}{h}$

Step 4.1.1.2

Factor $0.1$ out of $0.1 h^{2}$ .

$\frac{0.1 h^{2} + 0.2 q h + 0.1 q^{2} + 12 h + 12 q + 400 - (0.1 q^{2} + 12 q + 400)}{h}$

Step 4.1.1.3

Factor $0.1$ out of $0.2 q h$ .

$\frac{0.1 h^{2} + 0.1 (2 q h) + 0.1 q^{2} + 12 h + 12 q + 400 - (0.1 q^{2} + 12 q + 400)}{h}$

Step 4.1.1.4

Factor $0.1$ out of $0.1 q^{2}$ .

$\frac{0.1 h^{2} + 0.1 (2 q h) + 0.1 (q^{2}) + 12 h + 12 q + 400 - (0.1 q^{2} + 12 q + 400)}{h}$

Step 4.1.1.5

Factor $0.1$ out of $12 h$ .

$\frac{0.1 h^{2} + 0.1 (2 q h) + 0.1 (q^{2}) + 0.1 (120 h) + 12 q + 400 - (0.1 q^{2} + 12 q + 400)}{h}$

Step 4.1.1.6

Factor $0.1$ out of $12 q$ .

$\frac{0.1 h^{2} + 0.1 (2 q h) + 0.1 (q^{2}) + 0.1 (120 h) + 0.1 (120 q) + 400 - (0.1 q^{2} + 12 q + 400)}{h}$

Step 4.1.1.7

Factor $0.1$ out of $400$ .

$\frac{0.1 h^{2} + 0.1 (2 q h) + 0.1 q^{2} + 0.1 (120 h) + 0.1 (120 q) + 0.1 \cdot 4000 - (0.1 q^{2} + 12 q + 400)}{h}$

Step 4.1.1.8

Factor $0.1$ out of $- (0.1 q^{2} + 12 q + 400)$ .

$\frac{0.1 h^{2} + 0.1 (2 q h) + 0.1 q^{2} + 0.1 (120 h) + 0.1 (120 q) + 0.1 \cdot 4000 + 0.1 (- 10 (0.1 q^{2} + 12 q + 400))}{h}$

Step 4.1.1.9

Factor $0.1$ out of $0.1 h^{2} + 0.1 (2 q h)$ .

$\frac{0.1 (h^{2} + 2 q h) + 0.1 q^{2} + 0.1 (120 h) + 0.1 (120 q) + 0.1 \cdot 4000 + 0.1 (- 10 (0.1 q^{2} + 12 q + 400))}{h}$

Step 4.1.1.10

Factor $0.1$ out of $0.1 (h^{2} + 2 q h) + 0.1 q^{2}$ .

$\frac{0.1 (h^{2} + 2 q h + q^{2}) + 0.1 (120 h) + 0.1 (120 q) + 0.1 \cdot 4000 + 0.1 (- 10 (0.1 q^{2} + 12 q + 400))}{h}$

Step 4.1.1.11

Factor $0.1$ out of $0.1 (h^{2} + 2 q h + q^{2}) + 0.1 (120 h)$ .

$\frac{0.1 (h^{2} + 2 q h + q^{2} + 120 h) + 0.1 (120 q) + 0.1 \cdot 4000 + 0.1 (- 10 (0.1 q^{2} + 12 q + 400))}{h}$

Step 4.1.1.12

Factor $0.1$ out of $0.1 (h^{2} + 2 q h + q^{2} + 120 h) + 0.1 (120 q)$ .

$\frac{0.1 (h^{2} + 2 q h + q^{2} + 120 h + 120 q) + 0.1 \cdot 4000 + 0.1 (- 10 (0.1 q^{2} + 12 q + 400))}{h}$

Step 4.1.1.13

Factor $0.1$ out of $0.1 (h^{2} + 2 q h + q^{2} + 120 h + 120 q) + 0.1 \cdot 4000$ .

$\frac{0.1 (h^{2} + 2 q h + q^{2} + 120 h + 120 q + 4000) + 0.1 (- 10 (0.1 q^{2} + 12 q + 400))}{h}$

Step 4.1.1.14

Factor $0.1$ out of $0.1 (h^{2} + 2 q h + q^{2} + 120 h + 120 q + 4000) + 0.1 (- 10 (0.1 q^{2} + 12 q + 400))$ .

$\frac{0.1 (h^{2} + 2 q h + q^{2} + 120 h + 120 q + 4000 - 10 (0.1 q^{2} + 12 q + 400))}{h}$

Step 4.1.2

Apply the distributive property.

$\frac{0.1 (h^{2} + 2 q h + q^{2} + 120 h + 120 q + 4000 - 10 (0.1 q^{2}) - 10 (12 q) - 10 \cdot 400)}{h}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Multiply $0.1$ by $- 10$ .

$\frac{0.1 (h^{2} + 2 q h + q^{2} + 120 h + 120 q + 4000 - 1 q^{2} - 10 (12 q) - 10 \cdot 400)}{h}$

Step 4.1.3.2

Multiply $12$ by $- 10$ .

$\frac{0.1 (h^{2} + 2 q h + q^{2} + 120 h + 120 q + 4000 - 1 q^{2} - 120 q - 10 \cdot 400)}{h}$

Step 4.1.3.3

Multiply $- 10$ by $400$ .

$\frac{0.1 (h^{2} + 2 q h + q^{2} + 120 h + 120 q + 4000 - 1 q^{2} - 120 q - 4000)}{h}$

Step 4.1.4

Rewrite $- 1 q^{2}$ as $- q^{2}$ .

$\frac{0.1 (h^{2} + 2 q h + q^{2} + 120 h + 120 q + 4000 - q^{2} - 120 q - 4000)}{h}$

Step 4.1.5

Subtract $q^{2}$ from $q^{2}$ .

$\frac{0.1 (h^{2} + 2 q h + 120 h + 120 q + 4000 + 0 - 120 q - 4000)}{h}$

Step 4.1.6

Add $h^{2}$ and $0$ .

$\frac{0.1 (h^{2} + 2 q h + 120 h + 120 q + 4000 - 120 q - 4000)}{h}$

Step 4.1.7

Subtract $120 q$ from $120 q$ .

$\frac{0.1 (h^{2} + 2 q h + 120 h + 0 + 4000 - 4000)}{h}$

Step 4.1.8

Add $h^{2}$ and $0$ .

$\frac{0.1 (h^{2} + 2 q h + 120 h + 4000 - 4000)}{h}$

Step 4.1.9

Subtract $4000$ from $4000$ .

$\frac{0.1 (h^{2} + 2 q h + 120 h + 0)}{h}$

Step 4.1.10

Add $h^{2} + 2 q h + 120 h$ and $0$ .

$\frac{0.1 (h^{2} + 2 q h + 120 h)}{h}$

Step 4.1.11

Factor $h$ out of $h^{2} + 2 q h + 120 h$ .

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Step 4.1.11.1

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

$\frac{0.1 (h \cdot h + 2 q h + 120 h)}{h}$

Step 4.1.11.2

Factor $h$ out of $2 q h$ .

$\frac{0.1 (h (h) + h (2 q) + 120 h)}{h}$

Step 4.1.11.3

Factor $h$ out of $120 h$ .

$\frac{0.1 (h (h) + h (2 q) + h \cdot 120)}{h}$

Step 4.1.11.4

Factor $h$ out of $h (h) + h (2 q)$ .

$\frac{0.1 (h (h + 2 q) + h \cdot 120)}{h}$

Step 4.1.11.5

Factor $h$ out of $h (h + 2 q) + h \cdot 120$ .

$\frac{0.1 (h (h + 2 q + 120))}{h}$

$\frac{0.1 h (h + 2 q + 120)}{h}$

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.

$\frac{0.1 h (h + 2 q + 120)}{h}$

Step 4.2.1.2

Divide $0.1 (h + 2 q + 120)$ by $1$ .

$0.1 (h + 2 q + 120)$

Step 4.2.2

Apply the distributive property.

$0.1 h + 0.1 (2 q) + 0.1 \cdot 120$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $2$ by $0.1$ .

$0.1 h + 0.2 q + 0.1 \cdot 120$

Step 4.3.2

Multiply $0.1$ by $120$ .

$0.1 h + 0.2 q + 12$

Step 4.4

Reorder $0.1 h$ and $0.2 q$ .

$0.2 q + 0.1 h + 12$

Step 5