Antiderivatives

and

uses of derivatives and antiderivatives

Ann Newsome

Definition of Antiderivative: Let f be a function of x. If F is a function such that F’(x) = f(x), then F is an antiderivative of f.

Ex.

so F(x) is an antiderivative of f.

Antiderivatives are not unique.

If F(x) is an antiderivative of f, and C is any constant,

then F(x) + C is also an antiderivative of f.

Ex.

Power Rule for Antiderivatives:

If

If k were -1, the denominator would be zero.

To verify this formula:

Ex.

Confirm by taking the derivative:

✔

Using your graphing calculator, examine the graph of

If this graph is the derivative of f(x), what do we know about f ?

Q. Is f increasing or decreasing?A. We know f is increasing because its derivative is positive.

Q. What is the concavity of f ?

A. We know f is concave down because f’ is decreasing.

Can you think of a function that is always increasing, always concave down, and has a domain (0,∞)?

Hint: It’s not a polynomial.

has the right characteristics.

Explore this possibility using the graphing calculator.

Fact: The antiderivative of

is

Ex. What function has as its derivative?

Ex. What function has as its derivative?

Are there any other possible antiderivatives?

Yes, is an example.

Confirm the results by taking the derivative.

✔

Last problem: p. 121, #44:

Q. For which values of x does the slope of the line tangent to the curve take on its largest value?

A. To find the slope I will take the derivative of the function.

I now need to find where this function has a maximum value. To find this I will look at the derivative of f’.

Where f’ has a maximum value, f” will have a zero.

+ 1 −

f’ has a local max at x = 1, where f” changes from positive to negative. f’ increases before 1 and decreases afterwards, so the greatest slope is at x = 1.