Calculus Examples

$f (x) = \sqrt{x}$ , $a = 9$

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 $a = 9$ into the linearization function.

$L (x) = f (9) + f' (9) (x - 9)$

Step 3

Evaluate $f (9)$ .

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Step 3.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = \sqrt{9}$

Step 3.2

Simplify $\sqrt{9}$ .

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Step 3.2.1

Remove parentheses.

$(\sqrt{9})$

Step 3.2.2

Remove parentheses.

$\sqrt{9}$

Step 3.2.3

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}}$

Step 3.2.4

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

$3$

Step 4

Find the derivative and evaluate it at $9$ .

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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 $9$ in the expression.

$\frac{1}{2 {(9)}^{\frac{1}{2}}}$

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 $9$ as $3^{2}$ .

$\frac{1}{2 \cdot {(3^{2})}^{\frac{1}{2}}}$

Step 4.3.1.2

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

$\frac{1}{2 \cdot 3^{2 (\frac{1}{2})}}$

Step 4.3.1.3

Cancel the common factor of $2$ .

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Step 4.3.1.3.1

Cancel the common factor.

$\frac{1}{2 \cdot 3^{2 (\frac{1}{2})}}$

Step 4.3.1.3.2

Rewrite the expression.

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

Step 4.3.1.4

Evaluate the exponent.

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

Step 4.3.2

Multiply $2$ by $3$ .

$\frac{1}{6}$

Step 5

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

$L (x) = 3 + \frac{1}{6} (x - 9)$

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.

$L (x) = 3 + \frac{1}{6} x + \frac{1}{6} \cdot - 9$

Step 6.1.2

Combine $\frac{1}{6}$ and $x$ .

$L (x) = 3 + \frac{x}{6} + \frac{1}{6} \cdot - 9$

Step 6.1.3

Cancel the common factor of $3$ .

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Step 6.1.3.1

Factor $3$ out of $6$ .

$L (x) = 3 + \frac{x}{6} + \frac{1}{3 (2)} \cdot - 9$

Step 6.1.3.2

Factor $3$ out of $- 9$ .

$L (x) = 3 + \frac{x}{6} + \frac{1}{3 \cdot 2} \cdot (3 \cdot - 3)$

Step 6.1.3.3

Cancel the common factor.

$L (x) = 3 + \frac{x}{6} + \frac{1}{3 \cdot 2} \cdot (3 \cdot - 3)$

Step 6.1.3.4

Rewrite the expression.

$L (x) = 3 + \frac{x}{6} + \frac{1}{2} \cdot - 3$

Step 6.1.4

Combine $\frac{1}{2}$ and $- 3$ .

$L (x) = 3 + \frac{x}{6} + \frac{- 3}{2}$

Step 6.1.5

Move the negative in front of the fraction.

$L (x) = 3 + \frac{x}{6} - \frac{3}{2}$

Step 6.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$L (x) = \frac{x}{6} + 3 \cdot \frac{2}{2} - \frac{3}{2}$

Step 6.3

Combine $3$ and $\frac{2}{2}$ .

$L (x) = \frac{x}{6} + \frac{3 \cdot 2}{2} - \frac{3}{2}$

Step 6.4

Combine the numerators over the common denominator.

$L (x) = \frac{x}{6} + \frac{3 \cdot 2 - 3}{2}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $3$ by $2$ .

$L (x) = \frac{x}{6} + \frac{6 - 3}{2}$

Step 6.5.2

Subtract $3$ from $6$ .

$L (x) = \frac{x}{6} + \frac{3}{2}$

Step 7