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Test Prep

AP® Calculus BC

You + Shmoop = (Calcul)us


If you're feeling a little blue about derivatives, infinite series, and things that rhyme with shmalculus, check out the Shmoop guide to AP Calculus BC 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 polar curves are about more than a sassy polar bear reclining on an ice shelf.

You're in the big leagues now with Calculus BC, but don't worry—we've got your back.

Looking for AP Calculus AB? We got you for that too.

What's Inside Shmoop's Online AP Calculus BC 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 BC exam, you'll find

  • a diagnostic exam to measure where you are.
  • two full-length practice exams that are just like the real thing.
  • answer explanations to figure out where you went wrong…or right.
  • test-taking tips to get you through.
  • practice drills on practice drills.
  • 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.
  • extra help with the stuff that only shows up on the Calculus BC exam, like parametric and polar equations, logistic growth, and two dudes named Taylor and Maclaurin.

Sample Content

Exponential growth is a nice model to work with mathematically (well, nice-ish), but its big flaw is that it's not completely realistic. If a population of bacteria kept growing exponentially, the world would be overrun with bacteria pretty quickly. Nobody wants that.

There's a model that accounts for the fact that resources and space are not infinite and that a population has a "carrying capacity" that it can't exceed in the long run. The model is called logistic growth, and it takes the form of a differential equation. In words, the logistic growth formula says, "The rate of change of a population is jointly proportional to the size of the population and the difference between the population and the carrying capacity."

Uh, let's translate that into math. The differential equation for logistic growth is this guy right here.

y '(t) = ky(ay)

In this equation, k and a are constants, and a is the carrying capacity. The equation might not always appear in this form, so to find the carrying capacity, set the equation equal to 0 and solve for y. One solution is y = 0, and the other solution is y = carrying capacity.

The carrying capacity is a big deal for a couple of reasons. If the initial value y(0) > a, the population will decrease over time until it levels out to y = a, and if y(0) < a, then it will increase until it reaches y = a. The greatest rate of change of the population occurs when the population is at exactly half the value of the carrying capacity.

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