Precalculus Examples

$y = e^{- 0.25 t}$

Step 1

Write $y = e^{- 0.25 t}$ as a function.

$f (t) = e^{- 0.25 t}$

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.

$f (t + h) = e^{- 0.25 (t + h)}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

$f (t + h) = e^{- 0.25 t - 0.25 h}$

Step 3.1.2.2

The final answer is $e^{- 0.25 t - 0.25 h}$ .

$e^{- 0.25 t - 0.25 h}$

Step 3.2

Find the components of the definition.

$f (t + h) = e^{- 0.25 t - 0.25 h}$

$f (t) = e^{- 0.25 t}$

$f (t + h) = e^{- 0.25 t - 0.25 h}$

$f (t) = e^{- 0.25 t}$

Step 4

Plug in the components.

$\frac{f (t + h) - f t}{h} = \frac{e^{- 0.25 t - 0.25 h} - (e^{- 0.25 t})}{h}$

Step 5

Multiply $- 1$ by $e^{- 0.25 t}$ .

$\frac{e^{- 0.25 t - 0.25 h} - e^{- 0.25 t}}{h}$

Step 6