Calculus Examples

t (x) = x^{3} - 5 x^{2} - 9 x + 45

$t (x) = x^{3} - 5 x^{2} - 9 x + 45$ ,

x - 5

$x - 5$

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

$a = x^{3} - 5 x^{2} - 9 x + 45$ into the linearization function.

$L (x) = f (x^{3} - 5 x^{2} - 9 x + 45) + f' (x^{3} - 5 x^{2} - 9 x + 45) (x - x^{3} - 5 x^{2} - 9 x + 45)$

Step 3

Evaluate

$f (x^{3} - 5 x^{2} - 9 x + 45)$ .

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Step 3.1

Replace the variable

$x$ with

$x^{3} - 5 x^{2} - 9 x + 45$ in the expression.

$f (x^{3} - 5 x^{2} - 9 x + 45) = (x^{3} - 5 x^{2} - 9 x + 45) - 5$

Step 3.2

Simplify

$(x^{3} - 5 x^{2} - 9 x + 45) - 5$ .

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Step 3.2.1

Remove parentheses.

$(x^{3} - 5 x^{2} - 9 x + 45) - 5$

Step 3.2.2

Subtract

$5$ from

$45$ .

$x^{3} - 5 x^{2} - 9 x + 40$

Step 4

Find the derivative of

$f (x) = x - 5$ .

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Step 4.1

By the Sum Rule, the derivative of

$x - 5$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$

Step 4.2

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$1 + \frac{d}{d x} [- 5]$

Step 4.3

Since

$- 5$ is constant with respect to

$x$ , the derivative of

$- 5$ 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

$a$ .

$L (x) = x^{3} - 5 x^{2} - 9 x + 40 + 1 (x - x^{3} - 5 x^{2} - 9 x + 45)$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply

$x - x^{3} - 5 x^{2} - 9 x + 45$ by

$1$ .

$L (x) = x^{3} - 5 x^{2} - 9 x + 40 + x - x^{3} - 5 x^{2} - 9 x + 45$

Step 6.1.2

Subtract

$9 x$ from

$x$ .

$L (x) = x^{3} - 5 x^{2} - 9 x + 40 - x^{3} - 5 x^{2} - 8 x + 45$

Step 6.2

Simplify by adding terms.

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Step 6.2.1

Combine the opposite terms in

$x^{3} - 5 x^{2} - 9 x + 40 - x^{3} - 5 x^{2} - 8 x + 45$ .

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Step 6.2.1.1

Subtract

$x^{3}$ from

$x^{3}$ .

$L (x) = - 5 x^{2} - 9 x + 40 + 0 - 5 x^{2} - 8 x + 45$

Step 6.2.1.2

Add

$- 5 x^{2} - 9 x + 40$ and

$0$ .

$L (x) = - 5 x^{2} - 9 x + 40 - 5 x^{2} - 8 x + 45$

Step 6.2.2

Subtract

$5 x^{2}$ from

$- 5 x^{2}$ .

$L (x) = - 10 x^{2} - 9 x + 40 - 8 x + 45$

Step 6.2.3

Subtract

$8 x$ from

$- 9 x$ .

$L (x) = - 10 x^{2} - 17 x + 40 + 45$

Step 6.2.4

Add

$40$ and

$45$ .

$L (x) = - 10 x^{2} - 17 x + 85$

Step 7