Calculus Examples

$g (x) = x^{3} - x^{2} - 20 x$ , $x + 4$

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 $a = x^{3} - x^{2} - 20 x$ into the linearization function.

$L (x) = f (x^{3} - x^{2} - 20 x) + f' (x^{3} - x^{2} - 20 x) (x - x^{3} - x^{2} - 20 x)$

Step 3

Evaluate $f (x^{3} - x^{2} - 20 x)$ .

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Step 3.1

Replace the variable $x$ with $x^{3} - x^{2} - 20 x$ in the expression.

$f (x^{3} - x^{2} - 20 x) = (x^{3} - x^{2} - 20 x) + 4$

Step 3.2

Simplify $(x^{3} - x^{2} - 20 x) + 4$ .

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Step 3.2.1

Remove parentheses.

$(x^{3} - x^{2} - 20 x) + 4$

Step 3.2.2

Remove parentheses.

$x^{3} - x^{2} - 20 x + 4$

Step 4

Find the derivative of $f (x) = x + 4$ .

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Step 4.1

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

$\frac{d}{d x} [x] + \frac{d}{d x} [4]$

Step 4.2

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

$1 + \frac{d}{d x} [4]$

Step 4.3

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

$L (x) = x^{3} - x^{2} - 20 x + 4 + 1 (x - x^{3} - x^{2} - 20 x)$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x - x^{3} - x^{2} - 20 x$ by $1$ .

$L (x) = x^{3} - x^{2} - 20 x + 4 + x - x^{3} - x^{2} - 20 x$

Step 6.1.2

Subtract $20 x$ from $x$ .

$L (x) = x^{3} - x^{2} - 20 x + 4 - x^{3} - x^{2} - 19 x$

Step 6.2

Simplify by adding terms.

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Step 6.2.1

Combine the opposite terms in $x^{3} - x^{2} - 20 x + 4 - x^{3} - x^{2} - 19 x$ .

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Step 6.2.1.1

Subtract $x^{3}$ from $x^{3}$ .

$L (x) = - x^{2} - 20 x + 4 + 0 - x^{2} - 19 x$

Step 6.2.1.2

Add $- x^{2} - 20 x + 4$ and $0$ .

$L (x) = - x^{2} - 20 x + 4 - x^{2} - 19 x$

Step 6.2.2

Subtract $x^{2}$ from $- x^{2}$ .

$L (x) = - 2 x^{2} - 20 x + 4 - 19 x$

Step 6.2.3

Subtract $19 x$ from $- 20 x$ .

$L (x) = - 2 x^{2} - 39 x + 4$

Step 7