Find the average value of function

f(t)= 4sec(t)tan(t), interval [0,pi/4]

I assume you have no trouble with the definition of the average value, easily found by using

a) your textbook
b) google

Just integrate over the interval, then divide by the interval length. So, how do we integrate?

Recall that d/dx (sec x) = sec x * tan x

Look familiar?

So, our integral is 4 tan(t) from 0:pi/4

tan(0) = 0
tan(pi/4) = 1

Answer: 1/(pi/4 - 0) * 4(1 - 0)
= 4/pi * 4 = 16/pi

To find the average value of a function over a given interval, we need to calculate the definite integral of the function over that interval and then divide it by the width of the interval.

In this case, we need to find the average value of the function f(t) = 4sec(t)tan(t) over the interval [0, π/4].

Step 1: Find the definite integral of f(t) over the given interval.
∫[0, π/4](4sec(t)tan(t)) dt

To integrate this function, we can use the trigonometric identity: sec(t)tan(t) = sec(t).

∫[0, π/4](4sec(t)) dt = 4∫[0, π/4](sec(t)) dt

Step 2: Evaluate the integral.
To evaluate the integral of sec(t) with respect to t, we can use the natural logarithm function.

Let u = sec(t), then du = sec(t)tan(t) dt.

Using this substitution, the integral becomes:
4∫[0, π/4](sec(t)) dt = 4∫[0, π/4] du = 4[u] from 0 to π/4

Substituting the limits of integration:
4[sec(π/4) - sec(0)]

Step 3: Simplify the expression.
sec(π/4) = 1/ cos(π/4) = 1/√2

sec(0) = 1/ cos(0) = 1

Plugging these values into the expression:
4[sec(π/4) - sec(0)] = 4[(1/√2) - 1]

Step 4: Calculate the average value.
To find the average value, we divide the result of the integral by the width of the interval, which is (π/4 - 0) = π/4.

Average value = (4[(1/√2) - 1]) / (π/4)

Simplifying this expression gives us the average value of the function f(t) = 4sec(t)tan(t) over the interval [0, π/4].