Calculus Examples

$f (x) = \sqrt{4 - x}$ , $a = 0$

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 $a = 0$ into the linearization function.

$L (x) = f (0) + f' (0) (x - 0)$

Step 3

Evaluate $f (0)$ .

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Step 3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \sqrt{4 - (0)}$

Step 3.2

Simplify $\sqrt{4 - (0)}$ .

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Step 3.2.1

Remove parentheses.

$\sqrt{4 - (0)}$

Step 3.2.2

Multiply $- 1$ by $0$ .

$\sqrt{4 + 0}$

Step 3.2.3

Add $4$ and $0$ .

$\sqrt{4}$

Step 3.2.4

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

$\sqrt{2^{2}}$

Step 3.2.5

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

$2$

Step 4

Find the derivative and evaluate it at $0$ .

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Step 4.1

Find the derivative of $f (x) = \sqrt{4 - x}$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x}$ as ${(4 - x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]$

Step 4.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 - x$ .

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Step 4.1.2.1

To apply the Chain Rule, set $u$ as $4 - x$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [4 - x]$

Step 4.1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [4 - x]$

Step 4.1.2.3

Replace all occurrences of $u$ with $4 - x$ .

$\frac{1}{2} {(4 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} [4 - x]$

Step 4.1.3

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

$\frac{1}{2} {(4 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 - x]$

Step 4.1.4

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

$\frac{1}{2} {(4 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 - x]$

Step 4.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(4 - x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [4 - x]$

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(4 - x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [4 - x]$

Step 4.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(4 - x)}^{\frac{- 1}{2}} \frac{d}{d x} [4 - x]$

Step 4.1.7

Combine fractions.

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Step 4.1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(4 - x)}^{- \frac{1}{2}} \frac{d}{d x} [4 - x]$

Step 4.1.7.2

Combine $\frac{1}{2}$ and ${(4 - x)}^{- \frac{1}{2}}$ .

$\frac{{(4 - x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 - x]$

Step 4.1.7.3

Move ${(4 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} \frac{d}{d x} [4 - x]$

Step 4.1.8

By the Sum Rule, the derivative of $4 - x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x]$ .

$\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} (\frac{d}{d x} [4] + \frac{d}{d x} [- x])$

Step 4.1.9

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x])$

Step 4.1.10

Add $0$ and $\frac{d}{d x} [- x]$ .

$\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} \frac{d}{d x} [- x]$

Step 4.1.11

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} (- \frac{d}{d x} [x])$

Step 4.1.12

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

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

Step 4.1.13

Combine fractions.

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Step 4.1.13.1

Multiply $- 1$ by $1$ .

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

Step 4.1.13.2

Combine $\frac{1}{2 {(4 - x)}^{\frac{1}{2}}}$ and $- 1$ .

$\frac{- 1}{2 {(4 - x)}^{\frac{1}{2}}}$

Step 4.1.13.3

Move the negative in front of the fraction.

$- \frac{1}{2 {(4 - x)}^{\frac{1}{2}}}$

Step 4.2

Replace the variable $x$ with $0$ in the expression.

$- \frac{1}{2 {(4 - (0))}^{\frac{1}{2}}}$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the denominator.

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Step 4.3.1.1

Subtract $0$ from $4$ .

$- \frac{1}{2 \cdot 4^{\frac{1}{2}}}$

Step 4.3.1.2

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

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

Step 4.3.1.3

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

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

Step 4.3.1.4

Cancel the common factor of $2$ .

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Step 4.3.1.4.1

Cancel the common factor.

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

Step 4.3.1.4.2

Rewrite the expression.

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

Step 4.3.1.5

Evaluate the exponent.

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

Step 4.3.2

Multiply $2$ by $2$ .

$- \frac{1}{4}$

Step 5

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

$L (x) = 2 - \frac{1}{4} (x - 0)$

Step 6

Simplify each term.

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Step 6.1

Subtract $0$ from $x$ .

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

Step 6.2

Combine $x$ and $\frac{1}{4}$ .

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

Step 7