The University of Sydney
School of Mathematics and Statistics
Board Tutorial 7
MATH1021: Calculus of One Variable
Material covered
(1) Taylor polynomials including the Lagrange remainder
Outcomes
After completing this tutorial, you should be able to
LO1 apply mathematical logic and rigour to solve problems
LO10 find Taylor polynomials and the Taylor series expansion of a function
LO15 express mathematical ideas and arguments coherently in written form.
Summary of essential material
Note (Taylor polynomial). The 𝑛-th order Taylor polynomial 𝑇𝑛 of a 𝑛 times differentiable function 𝑓
at a point 𝑎 is the polynomial of degree at most 𝑛 providing the best possible approximation of 𝑓 near
𝑎. It is given by
𝑇𝑛 (𝑥) = 𝑓 (𝑎) + 𝑓 ′ (𝑎)(𝑥 − 𝑎) +
+···+
𝑓 ′′ (𝑎)
(𝑥 − 𝑎) 2
2
𝑓 (𝑘) (𝑎)
𝑓 (𝑛) (𝑎)
(𝑥 − 𝑎) 𝑘 + · · · +
(𝑥 − 𝑎) 𝑛
𝑘!
𝑛!
Note that 𝑓 (𝑘) (𝑎) = 𝑇𝑛(𝑘) (𝑎) for 𝑘 = 0, . . . , 𝑛. We call 𝑇𝑛 the 𝑛-th order Taylor polynomial of 𝑓 about
𝑥 = 𝑎.
Substitution method: If 𝑇𝑛 is the 𝑛-th order Taylor polynomial of 𝑓 about 𝑎 = 0 and 𝑚 ∈ N, then
𝑇𝑛 (𝑥 𝑚 ) is the Taylor polynomial of order 𝑚𝑛 of 𝑓 (𝑥 𝑚 ) about 0.
Note (Lagrange form of remainder). If 𝑇𝑛 is the 𝑛-th order Taylor polynomial of 𝑓 centred at 𝑎 we call
𝑅𝑛 (𝑥) = 𝑓 (𝑥) − 𝑇𝑛 (𝑥)
the 𝑛-th order remainder. It satsifies the limit
lim
𝑥→𝑎
𝑅𝑛 (𝑥)
= 0.
(𝑥 − 𝑎) 𝑛
If 𝑓 has 𝑛 + 1 derivatives, then 𝑅𝑛 (𝑥) can be represented in the Lagrange form: There exists 𝑐 strictly
between 𝑎 and 𝑥 such that
𝑓 (𝑛+1) (𝑐)
𝑅𝑛 (𝑥) =
(𝑥 − 𝑎) 𝑛+1 .
(𝑛 + 1)!
It is almost like the (𝑛 + 1)-st term, but with 𝑓 (𝑛+1) (𝑎) replaced by 𝑓 (𝑛+1) (𝑐).
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Questions to complete during the tutorial
1.
(a) Find the Taylor polynomial of order five for 𝑓 (𝑥) = cosh(𝑥) about 𝑥 = 0.
(b) Find the Taylor polynomial of order 2𝑛 for 𝑓 (𝑥) = cosh(𝑥) about 𝑥 = 0.
2. Let 𝑓 (𝑥) = ln(1 + 𝑥).
(a) Find an expression for 𝑓 (𝑛) (𝑥).
(b) Find the Taylor polynomial 𝑃𝑛 (𝑥) of order 𝑛 for 𝑓 (𝑥) about 𝑥 = 0.
(c) Use Lagrange’s formula to find a value of 𝑛 such that | ln 2 − 𝑃𝑛 (1)| ≤ 0.01.
3.
(a) Use a 6th order Taylor polynomial for cosh(𝑥) to obtain an approximation for the value
of the integral
∫ 1
𝐼=
cosh(𝑡 2 ) 𝑑𝑡.
0
(Note that this is an integral that cannot be computed in exact terms).
(b) Use Lagrange’s formula to estimate the error in your approximation.
4. Find the Taylor polynomial of order six, about 𝑥 = 𝜋/2, for the function sin 𝑥.
5. Show that if 𝑥 ∈ [−1, 1] then 𝑒 𝑥 can be approximated to within 0.025 by the polynomial
1+𝑥+
𝑥2 𝑥3 𝑥4
+
+ .
2! 3! 4!
You may use the fact that 𝑒 < 3.
6. Find the Taylor polynomial of order 5 for 𝑓 (𝑥) = tan−1 (𝑥) about 𝑥 = 0.
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7. You will have seen in the previous question that finding the Taylor polynomial of tan−1 𝑥 is not
so easy to do directly because the derivatives get rather messy. Remarkably, it turns out that
there is still a neat general formula for the Taylor polynomial of tan−1 𝑥. We will derive this
formula in this question.
Let 𝑓 (𝑥) be a function, and let 𝑃𝑛 (𝑥), 𝑄 𝑛 (𝑥), and 𝑇𝑛 (𝑥) be the Taylor polynomials of order 𝑛
about 𝑥 = 0 for the functions 𝑓 (𝑥), 𝑓 ′ (𝑥), and 𝑓 (𝑥 2 ) (respectively). It is true that
𝑄 𝑛−1 (𝑥) = 𝑃′𝑛 (𝑥)
and 𝑇2𝑛 (𝑥) = 𝑃𝑛 (𝑥 2 )
(you are not asked to prove these facts – however note that the first fact is quite easy, while the
second is a little tricky to prove).
(a) Find the Taylor polynomial of order 𝑛 of 1/(1 + 𝑥) around 𝑥 = 0.
(b) Hence find the Taylor polynomial of order 2𝑛 of 1/(1 + 𝑥 2 ) around 𝑥 = 0.
Hint: Take 𝑓 (𝑥) = 1/(1 + 𝑥) and use 𝑇2𝑛 (𝑥) = 𝑃𝑛 (𝑥 2 ).
(c) Hence show that the Taylor polynomial of order 2𝑛 + 1 of tan−1 (𝑥) around 𝑥 = 0 is
𝑥−
𝑥3 𝑥5 𝑥7
𝑥 2𝑛+1
+
−
+ · · · + (−1) 𝑛
.
3
5
7
2𝑛 + 1
′
Hint: This time take 𝑓 (𝑥) = tan−1 (𝑥) and use 𝑃2𝑛+1
(𝑥) = 𝑄 2𝑛 (𝑥).
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