Unit 8 - Advanced Calculus (HL only)
Required Prior Knowledge
Questions
a) Expand \(\left(1+x-x^{2}\right)^{2} \)
State the Maclaurin series, and the interval of convergence, for:
b) \(e^{x}\)
c) \(\cos x\)
d) \(\sin x\)
e) \(\left(1+x\right)^{n}, n\in\mathbb{R}\)
Solutions
Get Ready
Questions
Compute the 6th order Maclaurin polynomial of $$f\left(x\right)=e^{x^{2}}$$
Solutions
Notes
When we know a Maclaurin Series for a function, we can use this to find the Maclaurin Series for composites of this function.
There are four common ways we might manipulate a known series for \(f\left(x\right)\) to obtain a new series for \(g\left(x\right)\):
Substitution into a composite function
Multiplication of two known functions
Differentiation of a known result
Integration of a known result
Examples and Your Turns
Example
Derive the 6th order Maclaurin polynomial of $$f\left(x\right)=x\cos 2x$$
Your Turn
Derive the Maclaurin series for $$f\left(x\right)=e^{x^{2}}$$
Your Turn
Derive the Maclaurin series for $$f\left(x\right)=xe^{-2x}$$
Your Turn
Find the first three non-zero terms of the Maclaurin series for $$f\left(x\right)=\frac{\sin x}{x}$$
Your Turn
Compute the Maclaurin series for $$f\left(x\right)=\ln \frac{1+x}{1-x}$$
Your Turn
Compute the Maclaurin series for $$f\left(x\right)=\ln \frac{x}{\left(1+x\right)^{2}}$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Why do we often only need a result for the first 3 terms of a Maclaurin Series?
Which method is more efficient for computing Maclaurin Series: the direct calculation or via manipulation? Which is more rigorous?
Will the two methods for computing a Maclaurin Series always give the same final result?
Common Mistakes / Misconceptions
The most common mistake is an algebraic slip due to heavy load on the working memory. Check you work carefully to avoid this.
It is common to forget to substitute the new ‘\(x\)’ in all the parts of the known series.
Connecting This to Other Skills
Obviously this skill builds directly upon the work on Maclaurin Series (8.2), but also involves ideas from Differential Calculus (Unit 4) and Integral Calculus (Unit 5), particularly knowing particular results.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?