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Find the linear approximation for f(x)=cos (2 x) at x=\frac{\pi}{6} . Use the linear approximation to approximate the value of cos (\frac{1}{2}) . Please enter your answer

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The linear approximation to estimate the value of
\( \cos\left((1)/(2)\right) \) is approximately
\( 0.541 \) with three significant digits after the decimal point.

To find the linear approximation of
\( f(x) = \cos(2x) \) at
\( x = (\pi)/(6) \), we first compute the derivative
\( f'(x) \), and then use the point-slope form of the linear approximation:

1. Compute the derivative of
\( f(x) \):


\[ f'(x) = (d)/(dx)(\cos(2x)) = -2\sin(2x) \]

2. Evaluate
\( f(x) \) and
\( f'(x) \) at
\( x = (\pi)/(6) \):


\[ f\left((\pi)/(6)\right) = \cos\left(2 \cdot (\pi)/(6)\right) = \cos\left((\pi)/(3)\right) = (1)/(2) \]


\[ f'\left((\pi)/(6)\right) = -2\sin\left(2 \cdot (\pi)/(6)\right) = -2\sin\left((\pi)/(3)\right) = -√(3) \]

3. The linear approximation
\( L(x) \) near
\( x = (\pi)/(6) \) is:


\[ L(x) = f\left((\pi)/(6)\right) + f'\left((\pi)/(6)\right)(x - (\pi)/(6)) \]


\[ L(x) = (1)/(2) - √(3)\left(x - (\pi)/(6)\right) \]

4. Use the linear approximation to estimate
\( f\left((1)/(2)\right) \):


\[ L\left((1)/(2)\right) = (1)/(2) - √(3)\left((1)/(2) - (\pi)/(6)\right) \]

The approximation yields
\( L\left((1)/(2)\right) \approx 0.541 \).

Therefore, The answer is 0.541.

The complete question is here:

Find the linear approximation for
f(x)=cos (2 x) at
x=(\pi)/(6) . Use the linear approximation to approximate the value of cos
((1)/(2)) . Please enter your answer

User Jim Mcnamara
by
7.6k points
3 votes

The linear approximation is


- √(3)x + ( √(3)\pi )/(6) + (1)/(2)

The appropriated value of


\cos( (1)/(2) )

is 0.974.

How to linearize a function.

The formula for linear approximation, also known as linearization, is given by:

L(x) = f(a) + f'(a) * (x - a)

L(x) is the linear approximation function.

f(x) is the original function.

a is the point around which you are linearizing.

f'(a) is the derivative of f(x) evaluated at a.

x is the variable.

Given f(x) = cos2x at π/6

when x = π/6

f(π/6) = cos(2*π/6)

= cos(π/3) or cos60⁰

= 1/2

f'(x) = derivative of f(x)

f'(x) = -2sin2x

When x = π/6

f'(π/6) = -2sin(2*π/6)

= -2sin(π/3) or sin60⁰

= -2*√3/2

= -√3

L(x) = f(a) + f'(a) * (x - a)

= 1/2 + (-√3)(x - π/6)

= 1/2 - √3x + (√3 π)/6

= -√3x + √3π/6 + 1/2

To approximate cos(1/2).

compare the functions

cos(2x) and cos(1/2)

2x = 1/2

4x = 1

x = 1/4

Substitute x = 1/4 into L(x) to find cos


(1)/(2)


cos( (1)/(2)) = - √(3)( (1)/(4)) + ( √(3)\pi )/(6) + (1)/(2)

= 0.974

The approximated value of


\cos( (1)/(2) )

is 0.974

User Phil Hannent
by
9.5k points

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