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AP® Calculus AB

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If you're feeling a little blue about integrals, derivatives, and things that rhyme with shmalculus, check out the Shmoop guide to AP Calculus AB for all your calculating needs.

In this guide, you'll learn

  • what the AP exam is all about, how it's scored, and why guessing can be a good thing.
  • how to keep from bursting into tears during the free response section.
  • why derivatives aren't nearly as terrifying as they sound.

This is an "A" and "B" conversation, so why don't you "C" that you rock this exam?

Looking for AP Calculus BC? We got you covered.

What's Inside Shmoop's Online AP® Calculus AB Test Prep

Shmoop is a labor of love from folks who love to teach. Our Test Prep resources will help you prepare for exams with fun, engaging, and relatable materials that bring the test to life. 

Inside Shmoop's guide to the AP Calculus AB exam, you'll find

  • a diagnostic exam to highlight your strengths and weaknesses.
  • two full-length practice exams that are just as good as the real thing.
  • answer explanations to figure out where you went wrong…or right.
  • test-taking tips to help you weather a lengthy exam.
  • tons of practice problems.
  • a complete walkthrough of every single subject that's likely to show up on the exam, broken down into bite-sized chunks by topic—we're talking limits, derivatives, integrals, and more.

Sample Content

An antiderivative is not married to an unclederivative, and it's not someone who campaigns against derivatives. It's the opposite of a derivative.

What's the opposite of a derivative? Imagine two functions, f and g. If g is the derivative of f, then f is an antiderivative of g. Notice that we say an antiderivative, not the antiderivative. This choice of words is deliberate; a function has an infinite number of antiderivatives, which are often called a family of antiderivatives.

It may seem strange to make finding antiderivatives a family affair, but it's a consequence of the fact that the derivative of a constant term is zero. The derivatives of f(x) = sin x + 3 and g(x) = sin x – 10.4 are identical. Take a look.

f(x) = sin x + 3
f '(x) = cos x + 0 = cos x

g(x) = sin x – 10.4
g'(x) = cos x – 0 = cos x

The derivative of both functions is cos x, so what's the antiderivative of cos x? Part of it is sin x, but it's impossible to tell which constant to add on without some more information. If there's no additional information to be found, the antiderivative of cos x is said to be sin x + C, and C can be any real number constant. Upper-case C is most common for this constant, but lower-case c is also used. However it looks, it's usually referred to by the amazingly creative name of the constant of integration.

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AP® is a trademark registered and/or owned by the College Board, which was not involved in the production of, and does not endorse, this product.