Skills Checklist | IB Math AAHL

Skill Checklist

Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.

14 Skills Available

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📖 = included in formula booklet • 🚫 = not in formula booklet

Track your progress:

Don't know

Working on it

Confident

📖 = included in formula booklet • 🚫 = not in formula booklet

Operations on Maclaurin Series

7 skills

Maclaurin Series of Composite Functions

AHL 5.19

The Maclaurin series for a composite function f(g(x)) is

f(g(x))=f(0)+g(x)f′(0)+2![g(x)]2f′′(0)🚫

Basically, we replace xn with [g(x)]n.

Adding and Subtracting Series

AHL 5.19

We can add or subtract Maclaurin polynomials by adding or subtracting the coefficients of each power of x from the original polynomials.

Integration of Maclaurin Series

AHL 5.19

We can integrate a function by integrating its Maclaurin polynomial term by term.

Differentiation of Maclaurin Series

AHL 5.19

We can differentiate a Maclaurin series using the power rule on each term.

Multiplying Maclaurin Series

AHL 5.19

We can find the first few terms in the Maclaurin Series for a product f(x)×g(x) by multiplying out the first few terms in the Maclaurin Series for f and g.

Division of Maclaurin Series

AHL 5.19

We can perform "division" of Maclaurin series by assuming that the result of the division is a series with

p(x)=g(x)f(x)=a0+a1x+a2x2+⋯,

where an are the coefficients of the nth power term. Then, we multiply g(x)f(x) and a0+a1x+a2x2+... by g(x). Finally, we solve for the unknown coefficients using the Maclaurin series of f(x).

Limits using Maclaurin Series

AHL 5.19

Certain limits may be evaluated with L’Hôpital's rule but require difficult or repeated differentiation. Sometimes, expanding a function within the limit using its Maclaurin series and simplifying is an easier method for evaluating these limits.

This kind of simplification often works because we know that after a certain point, every term will tend to 0, allowing us to focus on the few terms that do not.

Maclaurin Series Basics

7 skills

General formula

AHL 5.19

Maclaurin series allow us to approximate arbitrary functions as polynomials.

The Maclaurin series for a function f is given by

f(x)=f(0)+xf′(0)+2!x2f′′(0)+⋯📖

In summation form:

f(x)=n=0∑∞n!f(n)(0)xn🚫

e^x

AHL 5.19

The Maclaurin series for ex is

ex=1+x+2!x2+⋯📖

sin(x)

AHL 5.19

The Maclaurin series for sinx is

sinx=x−3!x3+5!x5−⋯📖

cos(x)

AHL 5.19

The Maclaurin series for cosx is

cosx=1−2!x2+4!x4−⋯📖

ln(x+1)

AHL 5.19

The Maclaurin series for ln(x+1) is

ln(x+1)=x−2x2+3x3−⋯📖

arctan(x)

AHL 5.19

The Maclaurin series for arctanx is

arctanx=x−3x3+5x5−⋯📖

Binomial extension for rational exponents

AHL AA 1.10

The binomial theorem can be extended to expansions with rational exponents (n∈Q):

(a+b)n=an(1+n(ab)+2!n(n−1)(ab)2+⋯)📖