Calculus Examples

$f (- 1) = \frac{1}{\sqrt{2 π}} e^{- (\frac{1}{2}) x^{2}}$

Step 1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \frac{1}{\sqrt{2 π}} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2

Simplify the result.

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Step 2.1

Multiply $\frac{1}{\sqrt{2 π}}$ by $\frac{\sqrt{2 π}}{\sqrt{2 π}}$ .

$f (- 1) = \frac{1}{\sqrt{2 π}} \cdot \frac{\sqrt{2 π}}{\sqrt{2 π}} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.2

Combine and simplify the denominator.

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Step 2.2.1

Multiply $\frac{1}{\sqrt{2 π}}$ by $\frac{\sqrt{2 π}}{\sqrt{2 π}}$ .

$f (- 1) = \frac{\sqrt{2 π}}{\sqrt{2 π} \sqrt{2 π}} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.2.2

Raise $\sqrt{2 π}$ to the power of $1$ .

$f (- 1) = \frac{\sqrt{2 π}}{\sqrt{2 π} \sqrt{2 π}} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.2.3

Raise $\sqrt{2 π}$ to the power of $1$ .

$f (- 1) = \frac{\sqrt{2 π}}{\sqrt{2 π} \sqrt{2 π}} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 1) = \frac{\sqrt{2 π}}{{\sqrt{2 π}}^{1 + 1}} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.2.5

Add $1$ and $1$ .

$f (- 1) = \frac{\sqrt{2 π}}{{\sqrt{2 π}}^{2}} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.2.6

Rewrite ${\sqrt{2 π}}^{2}$ as $2 π$ .

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Step 2.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 π}$ as ${(2 π)}^{\frac{1}{2}}$ .

$f (- 1) = \frac{\sqrt{2 π}}{{({(2 π)}^{\frac{1}{2}})}^{2}} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- 1) = \frac{\sqrt{2 π}}{{(2 π)}^{\frac{1}{2} \cdot 2}} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$f (- 1) = \frac{\sqrt{2 π}}{{(2 π)}^{\frac{2}{2}}} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.6.4.1

Cancel the common factor.

$f (- 1) = \frac{\sqrt{2 π}}{{(2 π)}^{\frac{2}{2}}} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.2.6.4.2

Rewrite the expression.

$f (- 1) = \frac{\sqrt{2 π}}{2 π} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.2.6.5

Simplify.

$f (- 1) = \frac{\sqrt{2 π}}{2 π} \cdot e^{- \frac{1}{2} \cdot {(- 1)}^{2}}$

Step 2.3

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.3.1

Move ${(- 1)}^{2}$ .

$f (- 1) = \frac{\sqrt{2 π}}{2 π} \cdot e^{{(- 1)}^{2} \cdot (- 1 (\frac{1}{2}))}$

Step 2.3.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 2.3.2.1

Raise $- 1$ to the power of $1$ .

$f (- 1) = \frac{\sqrt{2 π}}{2 π} \cdot e^{{(- 1)}^{2} \cdot ({(- 1)}^{1} (\frac{1}{2}))}$

Step 2.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 1) = \frac{\sqrt{2 π}}{2 π} \cdot e^{{(- 1)}^{2 + 1} (\frac{1}{2})}$

Step 2.3.3

Add $2$ and $1$ .

$f (- 1) = \frac{\sqrt{2 π}}{2 π} \cdot e^{{(- 1)}^{3} (\frac{1}{2})}$

Step 2.4

Raise $- 1$ to the power of $3$ .

$f (- 1) = \frac{\sqrt{2 π}}{2 π} \cdot e^{- \frac{1}{2}}$

Step 2.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f (- 1) = \frac{\sqrt{2 π}}{2 π} \cdot \frac{1}{e^{\frac{1}{2}}}$

Step 2.6

Combine.

$f (- 1) = \frac{\sqrt{2 π} \cdot 1}{2 π e^{\frac{1}{2}}}$

Step 2.7

Multiply $\sqrt{2 π}$ by $1$ .

$f (- 1) = \frac{\sqrt{2 π}}{2 π e^{\frac{1}{2}}}$

Step 2.8

The final answer is $\frac{\sqrt{2 π}}{2 π e^{\frac{1}{2}}}$ .

$\frac{\sqrt{2 π}}{2 π e^{\frac{1}{2}}}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2 π}}{2 π e^{\frac{1}{2}}}$

Decimal Form:

$0.24197072 \dots$

Step 4