Respuesta :
Answer:
c₁ = 1/2 cos⁻¹ (2/π) = 0.44
c₂ = -1/2 cos⁻¹ (2/π) = -0.44
Step-by-step explanation:
the average value of f(x)=2 cos(2x) on ( − π/ 4 , π/ 4 ) is
av f(x) =∫[2*cos(2x)] dx /(∫dx) between limits of integration − π/ 4 and π/ 4
thus
av f(x) =∫[cos(2x)] dx /(∫dx) = [sin(2 * π/ 4 ) - sin(2 *(- π/ 4 )] /[ π/ 4 - (-π/ 4)]
= 2*sin (π/2) /(π/2) = 4/π
then the average value of f(x) is 4/π . Thus the values of c such that f(c)= av f(x) are
4/π = 2 cos(2c)
2/π = cos(2c)
c = 1/2 cos⁻¹ (2/π) = 0.44
c= 0.44
since the cosine function is symmetrical with respect to the y axis then also c= -0.44 satisfy the equation
thus
c₁ = 1/2 cos⁻¹ (2/π) = 0.44
c₂ = -1/2 cos⁻¹ (2/π) = -0.44
The two values are,
[tex]c=-\frac{1}{2} cos^{-1}(\frac{2}{\pi}) or\\ c=\frac{1}{2} cos^{-1}(\frac{2}{\pi}) [/tex]
Given that,
[tex]f(x)=2cos(2x)[/tex]
[tex]f_{avg}=\frac{1}{\frac{\pi}{2} }\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}2cos(2x)dx\\ =\frac{8}{2\pi} sin\frac{\pi}{2} \\ =\frac{4}{\pi} [/tex]
[tex]f(c)=2cos(2c)=\frac{4}{\pi} \\ cos2c=\frac{2}{\pi} \\ c=-\frac{1}{2} cos^{-1}(\frac{2}{\pi})or\\ c=\frac{1}{2} cos^{-1}(\frac{2}{\pi})or\\[/tex]
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