There are actually two AP Calculus tests offered to students, a Calculus AB exam and a Calculus BC exam. The BC exam covers some more advanced topics; it’s not necessarily a harder test, but its scope is broader. Most students opt to take the AB exam because they find it somewhat easier, but the BC exam is worth more credit at many universities. Grade 11 Pre-Calculus Mathematics. The period, P, measured in seconds, of a pendulum is the time it takes to complete one full swing. The College Board has released two complete exams from prior administrations of the AP Calculus AB exam. The tests are from 1988 and 1998. The 1988 test has an answer key included; however, for some reason, the 1998 exam does not. At the heart of calculus is the concept of functions and their graphs. All of the main areas of calculus, including limits, derivatives, and integrals, require a firm understanding of functions. In this review we will explore what you need to know about functions and their graphs in order to succeed on the AP Calculus exams. AP Biology Exam. The AP Biology test is divided into two sections. Section I is multiple choice and Section II is free response. You have 90 minutes to complete each section, and each section will account for half of your grade. Part A of Section I consists of 63 multiple choice questions.
The AP Calculus exams include a substantial amount of integration. So it’s very important to be familiar with integrals, numerous integration methods, and the interpretations and applications of integration. In this short article, we’ll take a look at some of the most common integrals on the test.
For a quick review of integration (or, antidifferentiation), you might want to check out the following articles first.
And now, without further ado, here are some of the most common integrals found on the AP Calculus exams!
The following seven integrals (or their close cousins) seem to pop up all the time on the AP Calculus AB and BC exams.
1. Remember your Trig Integrals!
Trigonometric functions are popular on the exam!
2. Simple Substitutions
You need to recognize when to use the substitution u = kx, for constant k. This substitution generates a factor of 1/k because du = kdx.
3. Common Integration By Parts
Integrands of the form xf(x) often lend themselves to integration by parts (IBP).
In the following integral, let u = x and dv = sin xdx, and use IBP.
4. Linear Denominators
Integrands of the form a/(bx + c) pop up as a result of partial fractions decomposition. (See AP Calculus BC Review: Partial Fractions). While partial fractions is a BC test topic, it’s not rare to see an integral with linear denominator showing up in the AB test as well.
The key is that substituting u = bx + c (and du = bdx) turns the integrand into a constant times 1/u. Let’s see how this works in general. Keep in mind that a, b, and c must be constants in order to use this rule.
5. Integral of Ln x
The antiderivative of f(x) = ln x is interesting. You have to use a tricky integration by parts.
Let u = ln x, and dv = dx.
By the way, this trick works for other inverse functions too, such as the inverse trig functions, arcsin x, arccos x, and arctan x. For example,
6. Using Trig Identities
For some trigonometric integrals, you have to rewrite the integrand in an equivalent way. In other words, use a trig identity before integrating. One of the most popular (and useful) techniques is the half-angle identity.
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7. Trigonometric Substitution
It’s no secret that the AP Calculus exams consist of challenging problems. Perhaps the most challenging integrals are those that require a trigonometric substitution.
Gaps. The table below summarizes the trigonometric substitutions.
For example, find the integral:
Here, the best substitution would be x = (3/2) sin θ.
Now we’re not out of the woods yet. Use the half-angle identity (see point 6 above). We also get to use the double-angle identity for sine in the second line.
Note, the third line may seem like it comes out of nowhere. But it’s based on the substitution and a right triangle.
If x = (3/2) sin θ, then sin θ = (2x) / 3. Draw a right triangle with angle θ, opposite side 2x, and hypotenuse 3.
By the Pythagorean Theorem, we find the adjacent side is equal to:
That allows us to identify cos θ in the expression (adjacent over hypotenuse).
Finally, θ by itself is equal to arcsin(2x/3).
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About Shaun Ault
Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!
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The revised AP Biology course is focused on enduring, conceptual understandings and content that supports these understandings. The emphasis of the course is on learning essential concepts and reasoning skills. It is the equivalent of a two-semester college introductory class. Our free AP Biology practice test is a great place to start your exam prep.
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AP Biology Practice Tests Organized by Big Ideas
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AP Biology Exam
The AP Biology test is divided into two sections. Section I is multiple choice and Section II is free response. You have 90 minutes to complete each section, and each section will account for half of your grade.
Part A of Section I consists of 63 multiple choice questions. Part B of Section I consists of 6 grid-in questions, in which you will need to integrate your math and science skills in calculating the correct answers. After solving the problem you will write in (or “grid-in”) the exact answer that you have calculated.
Section II begins with a 10 minute reading period. You will then have 80 minutes to answer 2 long free response questions and 6 short free response questions. Students are allowed to use simple four-function calculators on the entire exam.
Get started on your exam prep right now with our free AP Bio practice tests. They feature challenging questions with instant scoring, along with thorough explanations for the correct answers.