Find the linearization of the function f(x) = sin(x) and use it to estimate sin(5\cdot). What...

Question:

Find the linearization of the function {eq}f(x) = sin(x) {/eq} and use it to estimate {eq}sin(5\cdot). {/eq}

What is the percentage error of your approximation?

Linearization of Function:

The linearization of a function of one real variable {eq}f(x) {/eq} at a point {eq}x=x_0 {/eq} is defined by the followint formula

{eq}L(x) = f(x_0) + f'(x_0)(x-x_0). {/eq}

Answer and Explanation:

The the linearization of the function {eq}f(x) = \sin(x) {/eq} about the point {eq}x=0 {/eq} is found as

{eq}f(0) =0 \\ f'(x) = \cos(x) \Rightarrow f'(0) = 1 \\ L(x) = f(0)+f'(0)x = x. {/eq}

The exact value of the quantity

{eq}\sin(5^\circ ) = \sin(5\cdot\frac{\pi}{180} ) = \sin(0.0873) = 0.0872 {/eq}

can be estimated using linear approximation as

{eq}\sin(5^\circ ) \approx L((0.0873) = 0.0873. {/eq}

As a result, the percentage error made using the linear approximation is equal to

{eq}\displaystyle 100 \times \frac{ | L((0.0873)- \sin(5^\circ ) |} {| L((0.0873) } \\ \displaystyle = 100 \times \frac{ |0.0873-0.0872| }{ 0.0782 } \\ =0.11. {/eq}


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Linearization of Functions

from Math 104: Calculus

Chapter 10 / Lesson 1
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