Find the Linearization at a=9 f(x) = square root of x , a=9

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

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

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

Substitute the value of $a = 9$ into the linearization function.

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

Evaluate $f (9)$ .

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

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

Simplify $\sqrt{9}$ .

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Remove parentheses.

$(\sqrt{9})$

Remove parentheses.

$\sqrt{9}$

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

$\sqrt{3^{2}}$

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

$3$

Find the derivative and evaluate it at $a = 9$ .

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

Simplify.

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Simplify the denominator.

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $2$ by $3$ .

$\frac{1}{6}$

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

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

Simplify.

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Simplify each term.

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Apply the distributive property.

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

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

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

Cancel the common factor of $3$ .

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Factor $3$ out of $6$ .

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

Factor $3$ out of $- 9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Move the negative in front of the fraction.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $3$ by $2$ .

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

Subtract $3$ from $6$ .

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

Write in $y = m x + b$ form.

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

Find the Linearization at a=9 f(x) = square root of x , a=9

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