Precalculus Examples

$2 t^{2} - t$

Step 1

Write $2 t^{2} - t$ as a function.

$f (t) = 2 t^{2} - t$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

$f (t + h) = 2 {(t + h)}^{2} - (t + h)$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

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

$f (t + h) = 2 ((t + h) (t + h)) - (t + h)$

Step 3.1.2.1.2

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

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Step 3.1.2.1.2.1

Apply the distributive property.

$f (t + h) = 2 (t (t + h) + h (t + h)) - (t + h)$

Step 3.1.2.1.2.2

Apply the distributive property.

$f (t + h) = 2 (t \cdot t + t h + h (t + h)) - (t + h)$

Step 3.1.2.1.2.3

Apply the distributive property.

$f (t + h) = 2 (t \cdot t + t h + h t + h \cdot h) - (t + h)$

Step 3.1.2.1.3

Simplify and combine like terms.

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Step 3.1.2.1.3.1

Simplify each term.

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Step 3.1.2.1.3.1.1

Multiply $t$ by $t$ .

$f (t + h) = 2 (t^{2} + t h + h t + h \cdot h) - (t + h)$

Step 3.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (t + h) = 2 (t^{2} + t h + h t + h^{2}) - (t + h)$

Step 3.1.2.1.3.2

Add $t h$ and $h t$ .

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Step 3.1.2.1.3.2.1

Reorder $t$ and $h$ .

$f (t + h) = 2 (t^{2} + h t + h t + h^{2}) - (t + h)$

Step 3.1.2.1.3.2.2

Add $h t$ and $h t$ .

$f (t + h) = 2 (t^{2} + 2 h t + h^{2}) - (t + h)$

Step 3.1.2.1.4

Apply the distributive property.

$f (t + h) = 2 t^{2} + 2 (2 h t) + 2 h^{2} - (t + h)$

Step 3.1.2.1.5

Multiply $2$ by $2$ .

$f (t + h) = 2 t^{2} + 4 (h t) + 2 h^{2} - (t + h)$

Step 3.1.2.1.6

Apply the distributive property.

$f (t + h) = 2 t^{2} + 4 h t + 2 h^{2} - t - h$

Step 3.1.2.2

The final answer is $2 t^{2} + 4 h t + 2 h^{2} - t - h$ .

$2 t^{2} + 4 h t + 2 h^{2} - t - h$

Step 3.2

Reorder.

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Step 3.2.1

Move $- t$ .

$2 t^{2} + 4 h t + 2 h^{2} - h - t$

Step 3.2.2

Move $2 t^{2}$ .

$4 h t + 2 h^{2} + 2 t^{2} - h - t$

Step 3.2.3

Reorder $4 h t$ and $2 h^{2}$ .

$2 h^{2} + 4 h t + 2 t^{2} - h - t$

Step 3.3

Find the components of the definition.

$f (t + h) = 2 h^{2} + 4 h t + 2 t^{2} - h - t$

$f (t) = 2 t^{2} - t$

$f (t + h) = 2 h^{2} + 4 h t + 2 t^{2} - h - t$

$f (t) = 2 t^{2} - t$

Step 4

Plug in the components.

$\frac{f (t + h) - f t}{h} = \frac{2 h^{2} + 4 h t + 2 t^{2} - h - t - (2 t^{2} - t)}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$\frac{2 h^{2} + 4 h t + 2 t^{2} - h - t - (2 t^{2}) - - t}{h}$

Step 5.1.2

Multiply $2$ by $- 1$ .

$\frac{2 h^{2} + 4 h t + 2 t^{2} - h - t - 2 t^{2} - - t}{h}$

Step 5.1.3

Multiply $- - t$ .

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Step 5.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{2 h^{2} + 4 h t + 2 t^{2} - h - t - 2 t^{2} + 1 t}{h}$

Step 5.1.3.2

Multiply $t$ by $1$ .

$\frac{2 h^{2} + 4 h t + 2 t^{2} - h - t - 2 t^{2} + t}{h}$

Step 5.1.4

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

$\frac{2 h^{2} + 4 h t - h - t + 0 + t}{h}$

Step 5.1.5

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

$\frac{2 h^{2} + 4 h t - h - t + t}{h}$

Step 5.1.6

Add $- t$ and $t$ .

$\frac{2 h^{2} + 4 h t - h + 0}{h}$

Step 5.1.7

Add $2 h^{2} + 4 h t - h$ and $0$ .

$\frac{2 h^{2} + 4 h t - h}{h}$

Step 5.1.8

Factor $h$ out of $2 h^{2} + 4 h t - h$ .

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Step 5.1.8.1

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

$\frac{h (2 h) + 4 h t - h}{h}$

Step 5.1.8.2

Factor $h$ out of $4 h t$ .

$\frac{h (2 h) + h (4 t) - h}{h}$

Step 5.1.8.3

Factor $h$ out of $- h$ .

$\frac{h (2 h) + h (4 t) + h \cdot - 1}{h}$

Step 5.1.8.4

Factor $h$ out of $h (2 h) + h (4 t)$ .

$\frac{h (2 h + 4 t) + h \cdot - 1}{h}$

Step 5.1.8.5

Factor $h$ out of $h (2 h + 4 t) + h \cdot - 1$ .

$\frac{h (2 h + 4 t - 1)}{h}$

Step 5.2

Reduce the expression by cancelling the common factors.

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Step 5.2.1

Cancel the common factor of $h$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{h (2 h + 4 t - 1)}{h}$

Step 5.2.1.2

Divide $2 h + 4 t - 1$ by $1$ .

$2 h + 4 t - 1$

Step 5.2.2

Reorder $2 h$ and $4 t$ .

$4 t + 2 h - 1$

Step 6