Cogito
AP Calculus BC · Chapter 5 · Lesson 1
Taylor and Maclaurin Series
A function rebuilt from its derivatives at one point.
11 problems · about 26 minutes · AP Calculus BC 10.11, 10.13, 10.14
What this lesson teaches
The student constructs a Taylor polynomial and recognises the common Maclaurin series.
- A Taylor polynomial matches a function’s derivatives at one point.
- A Maclaurin series is simply a Taylor series centred at zero.
- The series represents the function only inside its interval of convergence.
Warm Up
Straightforward practice. Get the method working first.
4 problemsWhat is 6! ?
Answer 720
Why 720.
What is 3! ?
Answer 6
Why 3 × 2 × 1.
What is 5! ?
Answer 120
Why 5 × 24.
In the series for eˣ, what is the coefficient of x²? Give the denominator.
Answer 2
Why 2! = 2.
Build It Up
The same ideas with more to keep track of.
3 problemsThe series for sin x has only which powers? 1 for odd, 2 for even.
Answer 1
Why Sine is an odd function.
The series for cos x has only which powers? 1 for odd, 2 for even.
Answer 2
Why Cosine is even.
A Taylor series centred at 3 is called a Maclaurin series. 1 for yes, 0 for no.
Answer 0
Why Only zero qualifies.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsEvaluate 1 + x + x²/2 at x = 0.
Answer 1
Why Only the constant survives.
Select every true statement about Taylor series.
Answer It matches the derivatives at the centre.; It is only valid inside its interval of convergence.
Why Centring at zero is Maclaurin, a special case.
The Factorial: What is 4! ?
Answer 24
Why 24.
The Centre: A Maclaurin series is centred at which value?
Answer 0
Why 0.