Calculus Examples

$f (x) = x^{2}$ , $a = 4$

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 $a = 4$ into the linearization function.

$L (x) = f (4) + f' (4) (x - 4)$

Step 3

Evaluate $f (4)$ .

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Step 3.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = {(4)}^{2}$

Step 3.2

Simplify ${(4)}^{2}$ .

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Step 3.2.1

Remove parentheses.

${(4)}^{2}$

Step 3.2.2

Raise $4$ to the power of $2$ .

$16$

Step 4

Find the derivative and evaluate it at $4$ .

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Step 4.1

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

$2 x$

Step 4.2

Replace the variable $x$ with $4$ in the expression.

$2 (4)$

Step 4.3

Multiply $2$ by $4$ .

$8$

Step 5

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

$L (x) = 16 + 8 (x - 4)$

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.

$L (x) = 16 + 8 x + 8 \cdot - 4$

Step 6.1.2

Multiply $8$ by $- 4$ .

$L (x) = 16 + 8 x - 32$

Step 6.2

Subtract $32$ from $16$ .

$L (x) = 8 x - 16$

Step 7