What is the answer to do and how do I solve it?

Question

If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b], then

a). Use this property to estimate the value of the integral. π/8 7tan(2x) dx π/6

Answer 1

The function f(x) = 7tan(2x)is increasing on this interval...you can see this from a graph, and verify it using the First Derivative Test (be careful with this, because tan(2x) is undefined at {..., -pi/4, pi/4, ...}, but the given interval does not contain any of these points).

Therefore, the minimum value (on this interval) of f(x) will occur at the left endpoint, m = 7tan(2*(pi/8)); you can evaluate the exact value. The maximum value will occur at the right endpoint, M = 7tan(2*(pi/6)), and you can evaluate that exactly.

The length of in the interval is L = b - a = pi/6 - pi/8; I leave that for you to compute/simplify.

Putting everything together, you have mL <= (Value of the Integral) <= ML; just substitute the values you've computed above.