System Dynamics
System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Chapter 1, Problem 1.17P
To determine

Two linear approximations of the function f(θ)=sinθ.

Expert Solution & Answer
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Answer to Problem 1.17P

The linear approximation, f(θ) at θ=π4rad  is 0.7071(1+xπ4).

The linear approximation, f(θ) at θ=3π4rad  is 0.7071(1x+3π4).

Explanation of Solution

Given Information:

Function f(θ)=sinθ

Consider the following data.

Function f(θ)=sinθ

Angles,

θ=π4

And

θ=3π4

Differentiate given equation with respect to θ

ddθf(θ)=ddθsinθf1(θ)=cosθ[ddθf(θ)isrespresentedasf1(θ)]

The linear approximation near θ=π4rad is,

f(θ)=f(θ)+f1(θ)(xθ)=f(π4)+f1(π4)(xπ4)=sin(π4)+cos(π4)(xπ4)=0.7071+(0.7071)(xπ4)=0.7071(1+xπ4)

Calculate the linear approximation, near θ=3π4rad.

f(θ)=f(θ)+f1(θ)(xθ)=f(3π4)+f1(3π4)(x3π4)=sin(3π4)+cos(3π4)(x3π4)=0.7071+(0.7071)(x3π4)=0.7071(1x+3π4).

Conclusion:

The linear approximation, f(θ) at θ=π4rad  is 0.7071(1+xπ4).

The linear approximation, f(θ) at θ=3π4rad  is 0.7071(1x+3π4).

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