Calculus Examples

$f (x) = x^{3} - x^{2} + 5$ , $a = - 3$

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 $a = - 3$ into the linearization function.

$L (x) = f (- 3) + f' (- 3) (x - - 3)$

Step 3

Evaluate $f (- 3)$ .

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Step 3.1

Replace the variable $x$ with $- 3$ in the expression.

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

Step 3.2

Simplify ${(- 3)}^{3} - {(- 3)}^{2} + 5$ .

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Step 3.2.1

Remove parentheses.

${(- 3)}^{3} - {(- 3)}^{2} + 5$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Raise $- 3$ to the power of $3$ .

$- 27 - {(- 3)}^{2} + 5$

Step 3.2.2.2

Raise $- 3$ to the power of $2$ .

$- 27 - 1 \cdot 9 + 5$

Step 3.2.2.3

Multiply $- 1$ by $9$ .

$- 27 - 9 + 5$

Step 3.2.3

Simplify by adding and subtracting.

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Step 3.2.3.1

Subtract $9$ from $- 27$ .

$- 36 + 5$

Step 3.2.3.2

Add $- 36$ and $5$ .

$- 31$

Step 4

Find the derivative and evaluate it at $- 3$ .

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Step 4.1

Find the derivative of $f (x) = x^{3} - x^{2} + 5$ .

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Step 4.1.1

Differentiate.

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Step 4.1.1.1

By the Sum Rule, the derivative of $x^{3} - x^{2} + 5$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [5]$

Step 4.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [5]$

Step 4.1.2

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

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Step 4.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5]$

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 $n = 2$ .

$3 x^{2} - (2 x) + \frac{d}{d x} [5]$

Step 4.1.2.3

Multiply $2$ by $- 1$ .

$3 x^{2} - 2 x + \frac{d}{d x} [5]$

Step 4.1.3

Differentiate using the Constant Rule.

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Step 4.1.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$3 x^{2} - 2 x + 0$

Step 4.1.3.2

Add $3 x^{2} - 2 x$ and $0$ .

$3 x^{2} - 2 x$

Step 4.2

Replace the variable $x$ with $- 3$ in the expression.

$3 {(- 3)}^{2} - 2 \cdot - 3$

Step 4.3

Simplify.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Raise $- 3$ to the power of $2$ .

$3 \cdot 9 - 2 \cdot - 3$

Step 4.3.1.2

Multiply $3$ by $9$ .

$27 - 2 \cdot - 3$

Step 4.3.1.3

Multiply $- 2$ by $- 3$ .

$27 + 6$

Step 4.3.2

Add $27$ and $6$ .

$33$

Step 5

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

$L (x) = - 31 + 33 (x - - 3)$

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.

$L (x) = - 31 + 33 x + 33 \cdot 3$

Step 6.1.2

Multiply $33$ by $3$ .

$L (x) = - 31 + 33 x + 99$

Step 6.2

Add $- 31$ and $99$ .

$L (x) = 33 x + 68$

Step 7