Calculus Examples

$f (\frac{π}{4}) = \sin (x) \cos (x)$

Step 1

Replace the variable $x$ with $\frac{π}{4}$ in the expression.

$f (\frac{π}{4}) = \sin (\frac{π}{4}) \cos (\frac{π}{4})$

Step 2

Simplify the result.

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Step 2.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (\frac{π}{4}) = \frac{\sqrt{2}}{2} \cdot \cos (\frac{π}{4})$

Step 2.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (\frac{π}{4}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$

Step 2.3

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 2.3.1

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

$f (\frac{π}{4}) = \frac{\sqrt{2} \sqrt{2}}{2 \cdot 2}$

Step 2.3.2

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

$f (\frac{π}{4}) = \frac{\sqrt{2} \sqrt{2}}{2 \cdot 2}$

Step 2.3.3

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

$f (\frac{π}{4}) = \frac{\sqrt{2} \sqrt{2}}{2 \cdot 2}$

Step 2.3.4

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

$f (\frac{π}{4}) = \frac{{\sqrt{2}}^{1 + 1}}{2 \cdot 2}$

Step 2.3.5

Add $1$ and $1$ .

$f (\frac{π}{4}) = \frac{{\sqrt{2}}^{2}}{2 \cdot 2}$

Step 2.3.6

Multiply $2$ by $2$ .

$f (\frac{π}{4}) = \frac{{\sqrt{2}}^{2}}{4}$

Step 2.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.4.1

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

$f (\frac{π}{4}) = \frac{{(2^{\frac{1}{2}})}^{2}}{4}$

Step 2.4.2

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

$f (\frac{π}{4}) = \frac{2^{\frac{1}{2} \cdot 2}}{4}$

Step 2.4.3

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

$f (\frac{π}{4}) = \frac{2^{\frac{2}{2}}}{4}$

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

$f (\frac{π}{4}) = \frac{2^{\frac{2}{2}}}{4}$

Step 2.4.4.2

Rewrite the expression.

$f (\frac{π}{4}) = \frac{2}{4}$

Step 2.4.5

Evaluate the exponent.

$f (\frac{π}{4}) = \frac{2}{4}$

Step 2.5

Cancel the common factor of $2$ and $4$ .

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Step 2.5.1

Factor $2$ out of $2$ .

$f (\frac{π}{4}) = \frac{2 (1)}{4}$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $2$ out of $4$ .

$f (\frac{π}{4}) = \frac{2 \cdot 1}{2 \cdot 2}$

Step 2.5.2.2

Cancel the common factor.

$f (\frac{π}{4}) = \frac{2 \cdot 1}{2 \cdot 2}$

Step 2.5.2.3

Rewrite the expression.

$f (\frac{π}{4}) = \frac{1}{2}$

Step 2.6

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$

Step 4