Cracking The Ap Calculus ab Exam 2018

Example 1: Find the average value of f(x) = x^2 from x = 2 to x = 4.

Evaluate the integral x^2 dx.

Example 2: Find the average value of f(x) = sin x on [0, π].

Evaluate sin x dx.

    sin x   dx  =    (−cos  x)  (−cos   π   +   cos 0)  =   

The Second Fundamental Theorem of Calculus

As you saw in the last chapter, we’ve only half-learned the theorem. It has two parts, often referred to as
the First and Second Fundamental Theorems of Calculus.

The First Fundamental Theorem of Calculus (which you’ve already seen):

If  f(x)    is  continuous  at  every   point   of  [a, b], and F(x)    is  an  antiderivative  of  f(x)    on  [a, b], then    

f(x)    dx  =   F(b)    −   F(a).

The Second Fundamental Theorem of Calculus:

If  f(x)    is  continuous  on  [a, b], then    the derivative  of  the function    F(x)    =    f(t)   dt  is

f(t)    dt  =   f(x)

We’ve already made use of the first theorem in evaluating definite integrals. In fact, we use the first
Fundamental Theorem every time we evaluate a definite integral, so we’re not going to give you any