Calculus Examples

$f (x) = \sin (x)$ , $a = 0$

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 $a = 0$ into the linearization function.

$L (x) = f (0) + f' (0) (x - 0)$

Step 3

Evaluate $f (0)$ .

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Step 3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \sin (0)$

Step 3.2

Simplify $\sin (0)$ .

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Step 3.2.1

Remove parentheses.

$\sin (0)$

Step 3.2.2

The exact value of $\sin (0)$ is $0$ .

$0$

Step 4

Find the derivative and evaluate it at $0$ .

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Step 4.1

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

Step 4.2

Replace the variable $x$ with $0$ in the expression.

$\cos (0)$

Step 4.3

The exact value of $\cos (0)$ is $1$ .

$1$

Step 5

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

$L (x) = 0 + 1 (x - 0)$

Step 6

Simplify.

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Step 6.1

Add $0$ and $1 (x - 0)$ .

$L (x) = 1 (x - 0)$

Step 6.2

Multiply $x - 0$ by $1$ .

$L (x) = x - 0$

Step 6.3

Subtract $0$ from $x$ .

$L (x) = x$

Step 7