Basic Differentiation Rules

Product and Quotient Rule

Chain Rule

Inverse Functions

Polar and Parametric Functions

The AP Calculus BC exam also includes polar and parametric functions and their derivatives.

The derivative of a polar function, r = f(θ):

The derivative of a parametric function, x = f(t) and y = g(t):

Applications of Derivatives — Velocity

It’s important to know the relationship between position, velocity, and acceleration in terms of derivatives.

On the AP Calculus BC test, the position may be a vector function.

Mean Value Theorem and Rolle’s Theorem

There are two related theorems involving differentiable functions, the Mean Value Theorem, and Rolle’s Theorem.

Mean Value Theorem (MVT): Suppose f is a function that is continuous on [a, b] and differentiable on (a, b). Then there is at least one value x = c, where a < c < b, such that

Rolle’s Theorem: Suppose f is a function that is continuous on [a, b], differentiable on (a, b), and f(a) = f(b). Then there is at least one value x = c, where a < c < b, such that f '(c) = 0.

Integrals and Their Applications

On the BC test, you may have to find velocity and speed for a vector position function.

Integration Techniques

The following formulas are useful for working out integrals of more complicated functions. Think of each rule as a potential tool in your toolbox. Sometimes an integral will require multiple tools.

• u-Substitution
  
• Integration By Parts (BC only)
  

The Fundamental Theorem of Calculus (FTC)

The Fundamental Theorem of Calculuscomes in two versions.

If F(x) is any particular antiderivative for f(x), then

Average Value and Mean Value Theorem for Integrals

Mean Value Theorem for Integrals (MVTI): Suppose f is continuous on [a, b]. Then there is at least one value x = c, where a < c < b, such that

Applications of Integrals

On the AP Calculus BC exam, you may also have to find the length of a parametric curve defined by x = f(t) and y = g(t).

Use the washer or shell method to find the volume of a solid of revolution.

Sequences and Series

One of the most important formulas involving series is the Geometric Series Formula:

Convergence Tests

Given a series,

,

the following tests can help to prove that the series converges or diverges.

• p-series test. If the series has general term an = 1/np, then the series converges if p > 1 and diverges if p ≤ 1.
• Alternating series test. If the series is alternating (i.e., the terms alternate in sign forever), then the series converges if and only if an → 0 as n → ∞. And in that case, the error bound for the nth partial sum is |an+1|.
• Ratio test.
  
  However, if the limit is > 1, then the series diverges. No information if the limit equals 1.
• Root test.
  
  Just as in the ratio test, if the limit is > 1, then the series diverges. No information if the limit equals 1.

Taylor and Maclaurin Series

If a function f is differentiable to all orders, then you can build its Taylor seriescentered at c as follows.

A Taylor series centered at c = 0 is called a Maclaurin series. Below are some common Maclaurin series that are worth memorizing.