Calculus Examples

$f (\sqrt{2}) = x \sqrt{4 - x^{2}}$

Step 1

Replace the variable $x$ with $\sqrt{2}$ in the expression.

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

Step 2

Simplify the result.

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Step 2.1

Combine using the product rule for radicals.

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

$f (\sqrt{2}) = \sqrt{2 (4 - 2)}$

Step 2.2.5

Evaluate the exponent.

$f (\sqrt{2}) = \sqrt{2 (4 - 1 \cdot 2)}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $- 1$ by $2$ .

$f (\sqrt{2}) = \sqrt{2 (4 - 2)}$

Step 2.3.2

Subtract $2$ from $4$ .

$f (\sqrt{2}) = \sqrt{2 \cdot 2}$

Step 2.3.3

Multiply $2$ by $2$ .

$f (\sqrt{2}) = \sqrt{4}$

Step 2.3.4

Rewrite $4$ as $2^{2}$ .

$f (\sqrt{2}) = \sqrt{2^{2}}$

Step 2.4

Pull terms out from under the radical, assuming positive real numbers.

$f (\sqrt{2}) = 2$

Step 2.5

The final answer is $2$ .

$2$

Step 3