Given the function defined on the interval below, find the value of c where [tex]f(c)[/tex] = the average value.
[tex]f(x)=\sec ^2 x[/tex]
on the interval [tex]\left[0, \frac{\pi}{3}\right][/tex]
[tex]c=[?][/tex]
Round to the nearest thousandth.
Answer (1)
Calculate the average value of f ( x ) = sec 2 x on the interval [ 0 , 3 π ] using the formula b − a 1 ∫ a b f ( x ) d x , which results in π 3 3 .
Set f ( c ) = sec 2 c equal to the average value: sec 2 c = π 3 3 .
Solve for c by taking the reciprocal and square root: cos c = 3 3 π .
Find c by taking the inverse cosine: c = arccos ( 3 3 π ) ≈ 0.680 .
Explanation
Problem Setup We are given the function f ( x ) = sec 2 x on the interval [ 0 , 3 π ] . We need to find the value c in this interval such that f ( c ) is equal to the average value of f ( x ) on the interval.
Finding the Average Value First, we need to find the average value of f ( x ) on the interval [ 0 , 3 π ] . The average value is given by the formula: Average value = b − a 1 ∫ a b f ( x ) d x In our case, a = 0 , b = 3 π , and f ( x ) = sec 2 x . So, we have: Average value = 3 π − 0 1 ∫ 0 3 π sec 2 x d x = π 3 ∫ 0 3 π sec 2 x d x
Evaluating the Integral Now, we need to evaluate the integral: ∫ 0 3 π sec 2 x d x = [ tan x ] 0 3 π = tan ( 3 π ) − tan ( 0 ) = 3 − 0 = 3
Calculating the Average Value So, the average value is: Average value = π 3 3 = π 3 3 ≈ 1.653986686
Solving for c Next, we need to find c such that f ( c ) = sec 2 c is equal to the average value. Thus, we have: sec 2 c = π 3 3 Taking the reciprocal of both sides, we get: cos 2 c = 3 3 π Taking the square root of both sides, we get: cos c = 3 3 π Therefore, c = arccos ( 3 3 π ) c ≈ 0.680019987
Final Answer Rounding to the nearest thousandth, we get c ≈ 0.680 .
Examples
Imagine you are designing a solar panel system, and you need to optimize the angle of the panels to maximize energy absorption throughout the day. The function f ( x ) = sec 2 x could represent the intensity of sunlight at different angles. Finding the 'average value' allows you to determine an optimal fixed angle that provides energy absorption equivalent to the average intensity over a specific range of angles. This ensures efficient energy collection throughout the day.
