Precalculus Examples

$D (t) = 64 - 16 t^{2}$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

$D (t + h) = 64 - 16 {(t + h)}^{2}$

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 ${(t + h)}^{2}$ as $(t + h) (t + h)$ .

$D (t + h) = 64 - 16 ((t + h) (t + h))$

Step 2.1.2.1.2

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

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Step 2.1.2.1.2.1

Apply the distributive property.

$D (t + h) = 64 - 16 (t (t + h) + h (t + h))$

Step 2.1.2.1.2.2

Apply the distributive property.

$D (t + h) = 64 - 16 (t \cdot t + t h + h (t + h))$

Step 2.1.2.1.2.3

Apply the distributive property.

$D (t + h) = 64 - 16 (t \cdot t + t h + h t + h \cdot h)$

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 $t$ by $t$ .

$D (t + h) = 64 - 16 (t^{2} + t h + h t + h \cdot h)$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$D (t + h) = 64 - 16 (t^{2} + t h + h t + h^{2})$

Step 2.1.2.1.3.2

Add $t h$ and $h t$ .

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Step 2.1.2.1.3.2.1

Reorder $t$ and $h$ .

$D (t + h) = 64 - 16 (t^{2} + h t + h t + h^{2})$

Step 2.1.2.1.3.2.2

Add $h t$ and $h t$ .

$D (t + h) = 64 - 16 (t^{2} + 2 h t + h^{2})$

Step 2.1.2.1.4

Apply the distributive property.

$D (t + h) = 64 - 16 t^{2} - 16 (2 h t) - 16 h^{2}$

Step 2.1.2.1.5

Multiply $2$ by $- 16$ .

$D (t + h) = 64 - 16 t^{2} - 32 h t - 16 h^{2}$

Step 2.1.2.2

The final answer is $64 - 16 t^{2} - 32 h t - 16 h^{2}$ .

$64 - 16 t^{2} - 32 h t - 16 h^{2}$

Step 2.2

Reorder.

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Step 2.2.1

Move $64$ .

$- 16 t^{2} - 32 h t - 16 h^{2} + 64$

Step 2.2.2

Move $- 16 t^{2}$ .

$- 32 h t - 16 h^{2} - 16 t^{2} + 64$

Step 2.2.3

Reorder $- 32 h t$ and $- 16 h^{2}$ .

$- 16 h^{2} - 32 h t - 16 t^{2} + 64$

Step 2.3

Find the components of the definition.

$D (t + h) = - 16 h^{2} - 32 h t - 16 t^{2} + 64$

$D (t) = 64 - 16 t^{2}$

$D (t + h) = - 16 h^{2} - 32 h t - 16 t^{2} + 64$

$D (t) = 64 - 16 t^{2}$

Step 3

Plug in the components.

$\frac{D (t + h) - D t}{h} = \frac{- 16 h^{2} - 32 h t - 16 t^{2} + 64 - (64 - 16 t^{2})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{- 16 h^{2} - 32 h t - 16 t^{2} + 64 - 1 \cdot 64 - (- 16 t^{2})}{h}$

Step 4.1.2

Multiply $- 1$ by $64$ .

$\frac{- 16 h^{2} - 32 h t - 16 t^{2} + 64 - 64 - (- 16 t^{2})}{h}$

Step 4.1.3

Multiply $- 16$ by $- 1$ .

$\frac{- 16 h^{2} - 32 h t - 16 t^{2} + 64 - 64 + 16 t^{2}}{h}$

Step 4.1.4

Add $- 16 t^{2}$ and $16 t^{2}$ .

$\frac{- 16 h^{2} - 32 h t + 0 + 64 - 64}{h}$

Step 4.1.5

Add $- 16 h^{2}$ and $0$ .

$\frac{- 16 h^{2} - 32 h t + 64 - 64}{h}$

Step 4.1.6

Subtract $64$ from $64$ .

$\frac{- 16 h^{2} - 32 h t + 0}{h}$

Step 4.1.7

Add $- 16 h^{2} - 32 h t$ and $0$ .

$\frac{- 16 h^{2} - 32 h t}{h}$

Step 4.1.8

Factor $16 h$ out of $- 16 h^{2} - 32 h t$ .

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Step 4.1.8.1

Factor $16 h$ out of $- 16 h^{2}$ .

$\frac{16 h (- h) - 32 h t}{h}$

Step 4.1.8.2

Factor $16 h$ out of $- 32 h t$ .

$\frac{16 h (- h) + 16 h (- 2 t)}{h}$

Step 4.1.8.3

Factor $16 h$ out of $16 h (- h) + 16 h (- 2 t)$ .

$\frac{16 h (- h - 2 t)}{h}$

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.

$\frac{16 h (- h - 2 t)}{h}$

Step 4.2.1.2

Divide $16 (- h - 2 t)$ by $1$ .

$16 (- h - 2 t)$

Step 4.2.2

Apply the distributive property.

$16 (- h) + 16 (- 2 t)$

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Multiply $- 1$ by $16$ .

$- 16 h + 16 (- 2 t)$

Step 4.2.3.2

Multiply $- 2$ by $16$ .

$- 16 h - 32 t$

Step 4.2.3.3

Reorder $- 16 h$ and $- 32 t$ .

$- 32 t - 16 h$

Step 5