The FTC tells us how to build antiderivatives for any continuous function we like. No user manuel required. There are some things we need to know about these antiderivatives.
If we define an antiderivative of f(x) by
then
because we're integrating over an interval of length 0.
1) If we define two antiderivatives of f(x) by
where a < b, then the difference between Fb(x) and Fa(x) is
which is a constant.
2) Define an antiderivative of f(x) by
Since the derivative of a constant is 0, we can add any constant c that we like to F(x) and get a new antiderivative of f(x). That is, the function
has G '(x) = f(x).
Using the information above, we can build antiderivatives with various properties.
Example 1
Let f(x) be a continuous function. Build an antiderivative F of f that satisfies F(2) = 0. |
Example 2
Let f(x) = cos (x2). Build an antiderivative G(x) of f(x) that satisfies G(2) = 7. |
Exercise 1
Build an antiderivative F(x) of f(x) = eex satisfying F(8) = 0.
Exercise 2
Build an antiderivative F(x) of f(x) = eex satisfying F(8) = -1.
Exercise 3
Build an antiderivative of cos x that is 0 when x = 9.
Exercise 4
Build an antiderivative of sin(x2) that is 3 when x = 8.
