Calculus Examples

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

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 $a = 6$ 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