Calculus Examples

$f (x) = \sqrt{x}$ , $a = 16$

Step 1

Consider the function used to find the linearization at $a$ .

$L (x) = f (a) + f' (a) (x - a)$

Step 2

Substitute the value of $a = 16$ into the linearization function.

$L (x) = f (16) + f' (16) (x - 16)$

Step 3

Evaluate $f (16)$ .

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Step 3.1

Replace the variable $x$ with $16$ in the expression.

$f (16) = \sqrt{16}$

Step 3.2

Simplify $\sqrt{16}$ .

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Step 3.2.1

Remove parentheses.

$(\sqrt{16})$

Step 3.2.2

Remove parentheses.

$\sqrt{16}$

Step 3.2.3

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

Step 3.2.4

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

$4$

Step 4

Find the derivative and evaluate it at $16$ .

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Step 4.1

Find the derivative of $f (x) = \sqrt{x}$ .

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Step 4.1.1

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

$\frac{d}{d x} [x^{\frac{1}{2}}]$

Step 4.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{1}{2} x^{\frac{1}{2} - 1}$

Step 4.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 4.1.4

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

$\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 4.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}$

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} x^{\frac{1 - 2}{2}}$

Step 4.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} x^{\frac{- 1}{2}}$

Step 4.1.7

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}}$

Step 4.1.8

Simplify.

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Step 4.1.8.1

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

$\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}}$

Step 4.1.8.2

Multiply $\frac{1}{2}$ by $\frac{1}{x^{\frac{1}{2}}}$ .

$\frac{1}{2 x^{\frac{1}{2}}}$

Step 4.2

Replace the variable $x$ with $16$ in the expression.

$\frac{1}{2 {(16)}^{\frac{1}{2}}}$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the denominator.

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Step 4.3.1.1

Rewrite $16$ as $2^{4}$ .

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

Step 4.3.1.2

Multiply the exponents in ${(2^{4})}^{\frac{1}{2}}$ .

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Step 4.3.1.2.1

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

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

Step 4.3.1.2.2

Cancel the common factor of $2$ .

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Step 4.3.1.2.2.1

Factor $2$ out of $4$ .

$\frac{1}{2 \cdot 2^{2 (2) \frac{1}{2}}}$

Step 4.3.1.2.2.2

Cancel the common factor.

$\frac{1}{2 \cdot 2^{2 \cdot 2 \frac{1}{2}}}$

Step 4.3.1.2.2.3

Rewrite the expression.

$\frac{1}{2 \cdot 2^{2}}$

Step 4.3.1.3

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

$\frac{1}{2^{1 + 2}}$

Step 4.3.1.4

Add $1$ and $2$ .

$\frac{1}{2^{3}}$

Step 4.3.2

Raise $2$ to the power of $3$ .

$\frac{1}{8}$

Step 5

Substitute the components into the linearization function in order to find the linearization at $a$ .

$L (x) = 4 + \frac{1}{8} (x - 16)$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the distributive property.

$L (x) = 4 + \frac{1}{8} x + \frac{1}{8} \cdot - 16$

Step 6.1.2

Combine $\frac{1}{8}$ and $x$ .

$L (x) = 4 + \frac{x}{8} + \frac{1}{8} \cdot - 16$

Step 6.1.3

Cancel the common factor of $8$ .

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Step 6.1.3.1

Factor $8$ out of $- 16$ .

$L (x) = 4 + \frac{x}{8} + \frac{1}{8} \cdot (8 (- 2))$

Step 6.1.3.2

Cancel the common factor.

$L (x) = 4 + \frac{x}{8} + \frac{1}{8} \cdot (8 \cdot - 2)$

Step 6.1.3.3

Rewrite the expression.

$L (x) = 4 + \frac{x}{8} - 2$

Step 6.2

Subtract $2$ from $4$ .

$L (x) = \frac{x}{8} + 2$

Step 7