Calculus Examples

$f (θ) = \sin (θ + \frac{π}{3})$ ,

$θ = 0$

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

$θ$ with

$0$ in the expression.

$f (0) = \sin ((0) + \frac{π}{3})$

Step 3.2

Simplify

$\sin ((0) + \frac{π}{3})$ .

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Step 3.2.1

Remove parentheses.

$\sin ((0) + \frac{π}{3})$

Step 3.2.2

Add

$0$ and

$\frac{π}{3}$ .

$\sin (\frac{π}{3})$

Step 3.2.3

The exact value of

$\sin (\frac{π}{3})$ is

$\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3}}{2}$

Step 4

Find the derivative and evaluate it at

$0$ .

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Step 4.1

Find the derivative of

$f (θ) = \sin (θ + \frac{π}{3})$ .

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Step 4.1.1

Differentiate using the chain rule, which states that

$\frac{d}{d θ} [f (g (θ))]$ is

$f' (g (θ)) g' (θ)$ where

$f (θ) = \sin (θ)$ and

$g (θ) = θ + \frac{π}{3}$ .

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Step 4.1.1.1

To apply the Chain Rule, set

$u$ as

$θ + \frac{π}{3}$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d θ} [θ + \frac{π}{3}]$

Step 4.1.1.2

The derivative of

$\sin (u)$ with respect to

$u$ is

$\cos (u)$ .

$\cos (u) \frac{d}{d θ} [θ + \frac{π}{3}]$

Step 4.1.1.3

Replace all occurrences of

$u$ with

$θ + \frac{π}{3}$ .

$\cos (θ + \frac{π}{3}) \frac{d}{d θ} [θ + \frac{π}{3}]$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of

$θ + \frac{π}{3}$ with respect to

$θ$ is

$\frac{d}{d θ} [θ] + \frac{d}{d θ} [\frac{π}{3}]$ .

$\cos (θ + \frac{π}{3}) (\frac{d}{d θ} [θ] + \frac{d}{d θ} [\frac{π}{3}])$

Step 4.1.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d θ} [θ^{n}]$ is

$n θ^{n - 1}$ where

$n = 1$ .

$\cos (θ + \frac{π}{3}) (1 + \frac{d}{d θ} [\frac{π}{3}])$

Step 4.1.2.3

Since

$\frac{π}{3}$ is constant with respect to

$θ$ , the derivative of

$\frac{π}{3}$ with respect to

$θ$ is

$0$ .

$\cos (θ + \frac{π}{3}) (1 + 0)$

Step 4.1.2.4

Simplify the expression.

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Step 4.1.2.4.1

Add

$1$ and

$0$ .

$\cos (θ + \frac{π}{3}) \cdot 1$

Step 4.1.2.4.2

Multiply

$\cos (θ + \frac{π}{3})$ by

$1$ .

$\cos (θ + \frac{π}{3})$

Step 4.2

Replace the variable

$θ$ with

$0$ in the expression.

$\cos ((0) + \frac{π}{3})$

Step 4.3

Simplify.

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Step 4.3.1

Add

$0$ and

$\frac{π}{3}$ .

$\cos (\frac{π}{3})$

Step 4.3.2

The exact value of

$\cos (\frac{π}{3})$ is

$\frac{1}{2}$ .

$\frac{1}{2}$

Step 5

Substitute the components into the linearization function in order to find the linearization at

$a$ .

$L (x) = \frac{\sqrt{3}}{2} + \frac{1}{2} (x - 0)$

Step 6

Simplify each term.

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Step 6.1

Subtract

$0$ from

$x$ .

$L (x) = \frac{\sqrt{3}}{2} + \frac{1}{2} x$

Step 6.2

Combine

$\frac{1}{2}$ and

$x$ .

$L (x) = \frac{\sqrt{3}}{2} + \frac{x}{2}$

Step 7