Section 10.4
Taylor Series – Part 1
Taylor’s brilliant idea improved on the idea of approximating
function values with points on the tangent line.
Instead, he created polynomials to approximate functions.
Whereas the tangent line only shares the function value and slope
with the function at the point of tangency, Taylor polynomials
share many more of the function’s traits.
A second degree Taylor polynomial has the same slope and second
derivative value as the function (at the point of tangency) and thus
the same concavity or shape as the function.
A third degree Taylor polynomial matches the first three
derivatives, a fourth degree the first four derivatives, and so on.
Each iteration produces a polynomial that better matches the
points on the original function…often on a much wider interval.
For certain functions (categorized as transcendental functions)
if the Taylor Polynomial is infinite, becoming a
Taylor Series, then the polynomial is actually
equivalent to the original function.
WOW!!!
In mathematics, a Taylor series is a representation of a
function as an infinite sum of polynomial terms
calculated from the values of the function's derivatives at
a single point.
The concept of a Taylor series was formally introduced by the English
mathematician Brook Taylor in 1715. If the Taylor series is centered at
zero, then that series is also called a Maclaurin series, named after
the Scottish mathematician Colin Maclaurin, who made extensive use
of this special case of Taylor series in the 18th century.
Colin Maclaurin (February 1698 – June 1746) was
a Scottish mathematician who made important
contributions to geometry and algebra. The
Maclaurin series, a special case of the Taylor
series centered about x = 0, are named after him.
Maclaurin was born in Kilmodan, Argyll. His father, Reverend
and Minister of Glendaruel John Maclaurin, died when Maclaurin was
in infancy, and his mother died before he reached nine years of age.
He was then educated under the care of his uncle, the Reverend
Daniel Maclaurin, minister of Kilfinnan.
At eleven, Maclaurin entered the University of Glasgow. He
graduated with an MA three years later by defending a thesis on the
Power of Gravity, and remain at Glasgow to study divinity until he was
19 in 1717. He was soon elected professor of mathematics in a tenday competition at the Marshall College in the University of
Aberdeen, where Dr. Archibald James Macintyre, father of Susan
Cantey (your teacher) taught later during the twentieth century.
Maclaurin held the record as the world's youngest professor until
March 2008, when the record was officially given to Alia Sabur.
In the vacations of 1719 and 1721, Maclaurin went to London,
where he became acquainted with Sir Isaac Newton, Dr. Hoadley, Dr.
Samuel Clarke, Martin Folkes, and other eminent philosophers. He
was soon admitted as a member of the Royal Society.
In 1722, having provided a substitute for his class at Aberdeen
(birth place and childhood home of Susan Cantey), he traveled on the
Continent as tutor to George Hume, the son of Alexander Hume, 2nd
Earl of Marchmont. During their time in Lorraine, he wrote his essay
on the Percussion of Bodies, which would gain the prize of the Royal
Academy of Sciences in 1724. Upon the death of his pupil at
Montpellier, Maclaurin returned to Aberdeen.
In 1725 Maclaurin was appointed deputy to the mathematical
professor at Edinburgh, James Gregory upon the recommendation of
Isaac Newton. On November 3 of that year Maclaurin took Gregory’s
place. Newton was so impressed with Maclaurin that he had offered
to pay his salary himself.
Taylor series were actually known before Newton and in special
cases by Madhava of Sangamagrama in fourteenth century India.
However, Maclaurin and Taylor still receive credit because of their
use of them. In particular, Maclaurin used these series to characterize
maxima, minima, and points of inflection for infinitely differentiable
functions.
Taylor series were actually known before Newton and in special
cases by Madhava of Sangamagrama in fourteenth century India.
However, Maclaurin and Taylor still receive credit because of their use
of them. In particular, Maclaurin used these series to characterize
maxima, minima, and points of inflection for infinitely differentiable
functions.
Maclaurin also made significant contributions to the gravitation
attraction of ellipsoids, a subject that furthermore attracted the
attention of d'Alembert, Clairaut, Euler, Laplace, Legendre, Poisson and
Gauss. The subject continues to be of scientific interest, and in 1983,
Nobel Laureate Subramanyan Chandrasekhar dedicated a chapter of
his book Ellipsoidal Figures of Equilibrium to Maclaurin spheroids.
In 1733, Maclaurin married Anne Stewart, the daughter of
Walter Stewart, the Solicitor General for Scotland, by whom he had
seven children.
Maclaurin actively opposed the Jacobite Rebellion of 1745
(the attempt by Charles Edward Stuart to regain the British throne for
the exiled House of Stuart) and superintended the operations
necessary for the defence of Edinburgh against the Highland army.
Upon entry into the city, however, he fled to York, where he was
invited to stay by the Archbishop of York. Later, during a journey
south of Edinburgh in 1746, Maclaurin fell from his horse, and the
fatigue, anxiety, and cold to which he was exposed on that occasion
laid the foundations of dropsy. He returned to Edinburgh after the
Jacobite army marched south, but died soon after his return that
same year.
Brook Taylor was born in Edmonton (at that time in Middlesex)
on August 18th 1685 to John Taylor of Kent, and Olivia Tempest,
daughter of Sir Nicholas Tempest, of Durham. Brook entered St John's
College, Cambridge as a fellow-commoner in 1701, and earned 2
degrees in 1709 and 1714, respectively. Having studied mathematics
under John Machin and John Keill, in 1708 he obtained a remarkable
solution to the problem of the "centre of oscillation," which, however,
remained unpublished until May 1714,
when his claim to priority was disputed
by Johann Bernoulli.
Taylor's Methodus Incrementorum Directa
et Inversa (1715) added a new branch to
higher mathematics, now designated the
"calculus of finite differences.”
Among other ingenious applications, he used it to determine
the form of movement of a vibrating string. The same work
contained the celebrated formula known as Taylor's theorem, the
importance of which remained unrecognized until 1772, when J. L.
Lagrange realized its powers and termed it "le principal fondement
du calcul différentiel" ("the main foundation of differential calculus").
In his 1715 essay Linear Perspective, Taylor set forth the
true principles of the art in an original and more general form than
any of his predecessors; but the work suffered from brevity and
obscurity, traits which affected most of his writings.
Taylor was elected a fellow of the Royal Society early in
1712, and in the same year sat on the committee for adjudicating
the claims of Sir Isaac Newton and Gottfried Leibniz. From 1715 on,
his studies took a philosophical and religious bent. He
corresponded, in that year, with the Comte de Montmort on the
subject of Nicolas Malebranche's tenets; and unfinished treatises,
On the Jewish Sacrifices and On the Lawfulness of Eating Blood,
written in 1719, were afterwards found among his papers.
His marriage in 1721 with Miss Brydges of Wallington, Surrey,
led to an estrangement from his father, which ended in 1723 after her
death in giving birth to a son, who also died. The next two years were
spent by him with his family at Bifrons, and in 1725 he married, this
time with his father's approval, Sabetta Sawbridge, who also died in
childbirth in 1730. In this case, however, his daughter, Elizabeth,
survived.
By the date of his father's death in 1729, he had inherited the
Bifrons estate. As a mathematician, he was the only Englishman after
Sir Isaac Newton and Roger Cotes capable of holding his own with the
Bernoulli's, but a great part of the effect of his work was lost due to
his failure to express his ideas fully and clearly.
Taylor's fragile health soon gave way and he fell into a decline
from which he died when he was 46 years old. He was buried in
London on 2 December 1731, near his first wife, in the churchyard of
St Anne's in Soho.
Let’s revisit 𝑓 𝑥 = 𝑒 𝑥 , exploring its Taylor polynomials.
We will use 0, 1 as the point of tangency.
This is known as “building” the polynomials about 𝑥 = 0.
𝑝0 𝑥 = 𝑓 0 = 𝑒 0 = 1
𝑝1 𝑥 = tangent line = 𝑥 + 1
Note: 𝑝1 0 = 𝑓 0 = 1 𝑎𝑛𝑑 𝑝1′ (0) = 𝑓 ′ 0 = 1
We need 𝑝2 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 such that
𝑝2 0 = 𝑓 0 = 1, 𝑝2 ′ 0 = 𝑓′ 0 = 1, 𝑝2 ′′ 0 = 𝑓′′ 0 = 1
𝑝2 0 = 1 = 𝑐
𝑝2′ 𝑥 = 2𝑎𝑥 + 𝑏 → 𝑝2′ 0 = 1 = 𝑏
1
′′
′′
𝑝2 𝑥 = 2𝑎 → 𝑝2 0 = 1 → 𝑎 =
2
Hence: 𝑝2 𝑥 =
1 2
𝑥
2
+𝑥+1
We need 𝑝3 𝑥 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑 such that
𝑝2 0 = 𝑝2 ′ 0 = 𝑝2 ′′ 0 = 𝑝2 ′′′ 0 = 1
𝑝3 0 = 𝑑 = 1
𝑝3′ (𝑥) = 3𝑎𝑥 2 + 2𝑏𝑥 + 𝑐 → 𝑝3′ 0 = 𝑐 = 1
1
1
′′
′′
𝑝3 𝑥 = 3 ∙ 2𝑎𝑥 + 2𝑏 → 𝑝3 0 = 2𝑏 → 𝑏 = =
2
2!
1
′′′
′′′
𝑝3 𝑥 = 3 ∙ 2𝑎 → 𝑝3 0 = 3 ∙ 2𝑎 = 1 → 𝑎 =
3!
1 3 1 2
𝑝3 𝑥 =
𝑥 + 𝑥 +𝑥+1
3!
2!
𝑝3 𝑥 can be rewritten as follows:
𝑥 3 𝑥 2 𝑥1 𝑥 0
𝑝3 𝑥 =
+ + +
3! 2! 1! 0!
…which makes the pattern clear.
0
1
2
3
𝑛
𝑥
𝑥
𝑥
𝑥
𝑥
𝑓 𝑥 = 𝑒 𝑥 ≈ 𝑝𝑛 𝑥 =
+ + + + ⋯+
0! 1! 2! 3!
𝑛!
Let 𝑓 𝑥 be any function with n defined derivatives.
𝑝𝑛 𝑥 = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 + 𝑎4 𝑥 4 + ⋯ + 𝑎𝑛 𝑥 𝑛
𝑝𝑛 0 = 𝑎0 = 𝑓(0)
𝑝𝑛′ (𝑥) = 𝑎1 + 2𝑎2 𝑥 + 3𝑎3 𝑥 2 + 4𝑎4 𝑥 3 + ⋯ + 𝑛𝑎𝑛 𝑥 𝑛−1
𝑝𝑛′ (0) = 𝑎1 = 𝑓′(0)
𝑝𝑛′′ (𝑥) = 2𝑎2 + 3 ∙ 2𝑎3 𝑥 + 4 ∙ 3𝑎4 𝑥 2 + ⋯ + 𝑛(𝑛 − 1)𝑎𝑛 𝑥 𝑛−2
′′ 0
𝑓
𝑝𝑛′′ (0) = 2𝑎2 = 𝑓 ′′ 0 → 𝑎2 =
2
𝑝𝑛′′′ (𝑥) = 3 ∙ 2 ∙ 1𝑎3 + 4 ∙ 3 ∙ 2𝑎4 𝑥 + ⋯ + 𝑛(𝑛 − 1)(𝑛 − 2)𝑎𝑛 𝑥 𝑛−2
𝑓′′′(0)
′′′
′′′
𝑝𝑛 0 = 3 ∙ 2 ∙ 1𝑎3 = 𝑓 0 → 𝑎3 =
3!
′′ 0
′′′ 0
𝑛 0
𝑓
𝑓
𝑓
𝑝𝑛 𝑥 = 𝑓 0 + 𝑓 ′ 0 𝑥 +
𝑥2 +
𝑥3 + ⋯ +
𝑥𝑛
2!
3!
𝑛!
nth Taylor (Maclaurin) Polynomial
(for approximating 𝑓 𝑥 about 𝑥 = 0)
𝑛
𝑝𝑛 𝑥 =
𝑘=0
𝑓
𝑘
𝑘!
0
𝑥𝑘
′′
′′′
𝑛
𝑓
0
𝑓
0
𝑓
0 𝑛
′
2
3
=𝑓 0 +𝑓 0 𝑥+
𝑥 +
𝑥 + ⋯+
𝑥
2!
3!
𝑛!
I guess I’m
expected to know
this formula?
Ex: #3 (read directions)
𝑓 𝑥 = 𝑠𝑖𝑛𝑥, 𝑛 = 4
We will need four derivatives all evaluated at 𝑥 = 0.
𝑓 𝑥 = 𝑠𝑖𝑛𝑥 → 𝑓 0 = 0
𝑓 ′ 𝑥 = 𝑐𝑜𝑠𝑥 → 𝑓 ′ 0 = 1
𝑓 ′′ 𝑥 = −𝑠𝑖𝑛𝑥 → 𝑓 ′′ 0 = 0
𝑓 ′′′ 𝑥 = −𝑐𝑜𝑠𝑥 → 𝑓 ′′′ 0 = −1
𝑓 4 𝑥 = 𝑠𝑖𝑛𝑥 → 𝑓 4 𝑥 = 0
′′
′′′
4
𝑓
0
𝑓
0
𝑓
0 4
′
2
3
𝑝4 𝑥 = 𝑓 0 + 𝑓 0 𝑥 +
𝑥 +
𝑥 +
𝑥
2!
3!
4!
0 2 −1 3 0 4
= 0 + 1𝑥 + 𝑥 +
𝑥 + 𝑥
2!
3!
4!
1 3
𝑓 𝑥 = 𝑠𝑖𝑛𝑥 ≈ 𝑝4 𝑥 = 𝑥 − 𝑥
3!
In this case 𝑝4 𝑥 = 𝑝3 𝑥 , since all the
even numbered coefficients will be zero.
LaGrange Remainder Theorem
The difference between 𝑓 𝑥 and 𝑝𝑛 (𝑥) is
𝑓
𝑛+1
(𝑧)
𝑅𝑛 𝑥 =
𝑥−𝑎
𝑛+1 !
𝑛+1
where 𝑧 is some unknowable
number between 𝑥 𝑎𝑛𝑑 𝑎.
𝑎 is the 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 at the point of tangency.
In the remainder for our Taylor Series for 𝑓 𝑥 = 𝑠𝑖𝑛𝑥,
𝑎 = 0 and since our 𝑛 = 4, 𝑓 5 𝑧 = 𝑐𝑜𝑠𝑧.
𝑅4 𝑥 =
𝑓
5
(𝑧)
𝑥−0
5 !
5
cos 𝑧 5
=
𝑥
5 !
Here’s the graph of
𝜋 𝜋
2 2
Notice that on − ,
𝑓 𝑥 = 𝑠𝑖𝑛𝑥 & 𝑝4 𝑥 = 𝑥 −
1 3
𝑥
3!
the calculator shows almost identical graphs.
1
𝑓 𝑥 =
,
𝑛=4
EX: #4
1−𝑥
1
𝑓 𝑥 =
→𝑓 0 =1
1−𝑥
𝑓 ′ 𝑥 = −(1 − 𝑥)−2 −1 = (1 − 𝑥)−2 → 𝑓 ′ 0 = 1
𝑓 ′′ 𝑥 = −2(1 − 𝑥)−3 −1 = 2(1 − 𝑥)−3 → 𝑓 ′ ′ 0 = 2
𝑓 ′ ′′ 𝑥 = −6(1 − 𝑥)−4 −1 = 6(1 − 𝑥)−4 → 𝑓 ′ ′′ 0 = 6
𝑓 (4) 𝑥 = −24(1 − 𝑥)−5 −1 = 24(1 − 𝑥)−5 → 𝑓 4 0 = 24
′′ 0
′′′ 0
4 0
𝑓
𝑓
𝑓
𝑝4 𝑥 = 𝑓 0 + 𝑓 ′ 0 𝑥 +
𝑥2 +
𝑥3 +
𝑥4
2!
3!
4!
2 2 6 3 24 4
𝑝4 𝑥 = 1 + 1𝑥 + 𝑥 + 𝑥 + 𝑥
2!
3!
4!
𝑝4 𝑥 = 1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4
Look familiar?
1+r
+ 𝑟2
+𝑟 3
+⋯+
𝑟𝑛
1
+⋯=
, 𝑖𝑓 𝑟 < 1
1−𝑟
𝑓
𝑛+1
(𝑧)
𝑅𝑛 𝑥 =
𝑥−𝑎
𝑛+1 !
𝑎 = 0, 𝑛 = 4, 𝑓
𝑅4 𝑥 =
𝑓
5
5
(𝑧)
𝑥−0
5 !
5
𝑛+1
𝑧 = 5! (1 − 𝑧)−6
5! (1 − 𝑧)−6 5
𝑥 = (1 − 𝑧)−6 𝑥 5
=
5 !
Here’s the graph of 𝑓 𝑥 =
1
1−𝑥
& 𝑝4 𝑥 = 1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4
Notice that the graphs are only converging between -1 and 1.
EX: #8
𝑓 𝑥 = 𝑎𝑟𝑐𝑡𝑎𝑛𝑥,
𝑛=2
𝑓 𝑥 = 𝑎𝑟𝑐𝑡𝑎𝑛𝑥 → 𝑓 0 = 0
1
′
′ 0 =1
𝑓 𝑥 =
→
𝑓
1 + 𝑥2
−2𝑥
′′
′′
𝑓 𝑥 =
→
𝑓
0 =0
2
2
1+𝑥
hmm…how boring…
′′ 0
𝑓
𝑝2 𝑥 = 𝑓 0 + 𝑓 ′ 0 𝑥 +
𝑥2
2!
0 2
𝑝2 𝑥 = 0 + 𝑥 + 𝑥 = 𝑥
2!
𝑝2 𝑥 = 𝑡ℎ𝑒 𝑝𝑙𝑎𝑖𝑛 𝑜𝑙𝑑 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒!
𝑓
𝑛+1
(𝑧)
𝑅𝑛 𝑥 =
𝑥−𝑎
𝑛+1 !
𝑛+1
Unfortunately we need the 3rd derivative.
𝑓
′′
−2𝑥
𝑥 =
1 + 𝑥2
𝑓 (3) 𝑥 =
2
1 + 𝑥2
𝑅2 𝑥 =
𝑓
2
−2 − −2𝑥 2 1 + 𝑥 2 2𝑥
=
2
4
1+𝑥
3
(𝑧)
𝑥−0
3 !
3
6𝑧 2 − 2
=
3! 1 + 𝑧 2
6𝑥 2 − 2
1 + 𝑥2 3
3
𝑥
3
This remainder can be estimated for any value of x.
The further x is from zero, the larger the remainder will become.