Calculus Examples

$f (x) = 12 x^{2} + 8 x$ , $x = 9$

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 $a = 9$ into the linearization function.

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

Step 3

Evaluate $f (9)$ .

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Step 3.1

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

$f (9) = 12 {(9)}^{2} + 8 (9)$

Step 3.2

Simplify $12 {(9)}^{2} + 8 (9)$ .

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Step 3.2.1

Remove parentheses.

$12 {(9)}^{2} + 8 (9)$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Raise $9$ to the power of $2$ .

$12 \cdot 81 + 8 (9)$

Step 3.2.2.2

Multiply $12$ by $81$ .

$972 + 8 (9)$

Step 3.2.2.3

Multiply $8$ by $9$ .

$972 + 72$

Step 3.2.3

Add $972$ and $72$ .

$1044$

Step 4

Find the derivative and evaluate it at $9$ .

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Step 4.1

Find the derivative of $f (x) = 12 x^{2} + 8 x$ .

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Step 4.1.1

By the Sum Rule, the derivative of $12 x^{2} + 8 x$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [8 x]$ .

$\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [8 x]$

Step 4.1.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

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Step 4.1.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x]$

Step 4.1.2.2

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

$12 (2 x) + \frac{d}{d x} [8 x]$

Step 4.1.2.3

Multiply $2$ by $12$ .

$24 x + \frac{d}{d x} [8 x]$

Step 4.1.3

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 4.1.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$24 x + 8 \frac{d}{d x} [x]$

Step 4.1.3.2

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

$24 x + 8 \cdot 1$

Step 4.1.3.3

Multiply $8$ by $1$ .

$24 x + 8$

Step 4.2

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

$24 (9) + 8$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $24$ by $9$ .

$216 + 8$

Step 4.3.2

Add $216$ and $8$ .

$224$

Step 5

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

$L (x) = 1044 + 224 (x - 9)$

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.

$L (x) = 1044 + 224 x + 224 \cdot - 9$

Step 6.1.2

Multiply $224$ by $- 9$ .

$L (x) = 1044 + 224 x - 2016$

Step 6.2

Subtract $2016$ from $1044$ .

$L (x) = 224 x - 972$

Step 7