Calculus Examples

f (x) = x + 7

$f (x) = x + 7$ ,

x = 6

$x = 6$

Step 1

Consider the function used to find the linearization at

a

$a$ .

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

Step 2

Substitute the value of

$a = 6$ into the linearization function.

$L (x) = f (6) + f' (6) (x - 6)$

Step 3

Evaluate

$f (6)$ .

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Step 3.1

Replace the variable

$x$ with

$6$ in the expression.

$f (6) = (6) + 7$

Step 3.2

Simplify

$(6) + 7$ .

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Step 3.2.1

Remove parentheses.

$(6) + 7$

Step 3.2.2

Add

$6$ and

$7$ .

$13$

Step 4

Find the derivative of

$f (x) = x + 7$ .

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Step 4.1

By the Sum Rule, the derivative of

$x + 7$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [7]$

Step 4.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$1 + \frac{d}{d x} [7]$

Step 4.3

Since

$7$ is constant with respect to

$x$ , the derivative of

$7$ with respect to

$x$ is

$0$ .

$1 + 0$

Step 4.4

Add

$1$ and

$0$ .

$1$

Step 5

Substitute the components into the linearization function in order to find the linearization at

$a$ .

$L (x) = 13 + 1 (x - 6)$

Step 6

Simplify.

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Step 6.1

Multiply

$x - 6$ by

$1$ .

$L (x) = 13 + x - 6$

Step 6.2

Subtract

$6$ from

$13$ .

$L (x) = x + 7$

Step 7

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