Derivatives
On a graph, an integral is the area under the graph; a derivative is the slope of a graph at a given point.
Consider a problem in which you’re asked to find the work done by a non-constant force. If you’re given
a graph of that force vs. position, then all you’ve got to do is find the area under the graph—no
integration necessary .
You should have an idea of the meaning of a derivative or integral, even without evaluating it, or
without graphing the function in question. This isn’t as hard as it looks! Consider the following multiple-
choice problem:
A box is pushed across a frictionless table a distance of 9 m. The horizontal force pushing the box
obeys the function F (x ) = 50(5 – ), where F is in newtons and x is in meters. How much work is
done by the pushing force?(A) 2500 J
(B) 1700 J
(C) 900 J
(D) 250 J
(E) 90 J“Whoa,” you say. “This is a nasty calculus problem, especially without a calculator.” Your first instinct is
to take the integral . That becomes nasty toot sweet. No
chance you can get that done in the minute or so you have on a multiple-choice problem.
So, what to do?
You know in your bones that if this force were constant, then all you’d have to do is multiply the force
by 9 m. This force is not constant. But, we can approximate an average force from the function, can’t we?
Sure ... the initial force is 50(5 – 0) = 250 N. The force at the end of the push is 50(5 – ) = 100 N. So,
the average force is somewhere in between 100 N and 250 N.^4 Guess that this average force is, say, 200
N ... then, the work would be (200 N)(9 m) = 1800 J. So the answer is B .
Note that ANY kind of estimate of the average force would still get you close to the correct answer.
This is a classic calculus concepts question ... it’s not about evaluating the integral, it’s about
understanding the meaning of work.
What Specific Calculus Methods Do I Have to Know?
You will be expected to evaluate straightforward integrals and derivatives. Remember, this is not a math
test—the exam is not trying to test your math skills but rather your ability to apply calculus to physical
situations. This means the actual integrals and derivatives will not be from the most difficult questions on
your AP Calculus BC test!
You should know:
• Derivatives and integrals of polynomial functions
• Derivatives and integrals of sin x and cos x —but we’ve never seen questions that require
trigonometric identities on the exam
• Derivatives and integrals with ln x or ex