The document discusses Taylor series and their applications. It introduces Taylor series as a way to approximate functions using their derivatives. Examples are provided for linear, quadratic, and higher order Taylor approximations. Applications discussed include using Taylor series in physics for concepts like special relativity equations.
6. +
Approximating Functions
f(x) = sin(x)
f(0) = 1, f’(0) = 1, f’’(0) = 0, f’’’(0) = -1,…
What is f(1)? i.e . What is sin(1)?
7. +
Famous Dead People
James Gregory (1671)
Brook Taylor (1712)
Colin Maclaurin (1698-1746)
Joseph-Louis Lagrange (1736-1813)
Augustin-Louis Cauchy (1789-1857)
8. +
Approximations
Linear Approximation
f (x) = f (a) + f ʹ′(a)(x − a) + R1 (x)(x − a)
R1 (x)(x − a) = f (x) − f (a) − f ʹ′(x − a)
Quadratic Approximation
f ʹ′ (a)
€ f (x) = f (a) + f ʹ′(a)(x − a) + (x − a) 2 + R2 (x)(x − a) 2
2 f ʹ′ (a)
2
R2 (x)(x − a) = f (x) − f (a) − f ʹ′(a)(x − a) − (x − a) 2
2
€
9. +
Taylor’s Theorem
Letk≥1 be an integer and f : R →R be k
times differentiable at a ∈ R .
Then there exists a function R : R →R such that
k
f ʹ′ (a) f k (a)
f (x) = f (a) − f ʹ′(a)(x − a) + (x − a) 2 + ...+ (x − a) k + Rk (x)(x − a) k
2! k!
€
€
Note: Taylor Polynomial of degree k is:
€
f ʹ′ (a) f k (a)
Pk (x) = f (a) − f ʹ′(a)(x − a) + (x − a) 2 + ...+ (x − a) k
2! k!
10. +
Works for Linear
Approximations
f (x) = c 0 + c1 (x)
f (a) = c 0 + c1 (a)
f ʹ′(a) = c1 f (x)(a)fʹ′= c + c (x − a)
f = f (a) = c (a)
+ 0 11
1
€ f (x) = f (a) + c1€ € € a)
(x −
€ f (x) = c 0 + c1 (a) + c1 (x − a) = c 0 + c1 (x)
€
€
€
11. +
Works for Quadratic
Approximations
f (x) = c 0 + c1 (x) + c 2 (x 2 )
f (a) = c 0 + c1 (a) + c 2 (a 2 )
f ʹ′(a) = c1 + 2c 2 (a)
f ʹ′ (a) = 2c 2
€ = c + c (a) + c (a 2 ) + [c + 2c (a)](x − a) + 2c 2 [x − a]2 =
€
f (x) 0 1 2 1 2
€ 2
c 0€ c1(a) + c 2 (a 2 ) + c1(x − a) + 2c 2 (x − a) + c 2 (x) 2 − c 2 (2ax) + c 2 (a) 2 =
+
f (x) = c 0 + c1 (x) + c 2 (x 2 )
€
17. +
Implications
Any smooth functions with all the same derivatives
at a point MUST be the same function!
18. + Proof: If f and g are smooth functions that agree
over some interval, they MUST be the same function
Let f and g be two smooth functions that agree for some open
interval (a,b), but not over all of R
Define h as the difference, f – g, and note that h is smooth, being the
difference of two smooth functions. Also h=0 on (a,b), but h≠0 at
other points in R
Without loss of generality, we will form S, the set of all x>a, such
that f(x)≠0
Note that a is a lower bound for this set, S, and being a subset of R,
S is complete so S has a real greatest lower bound, call it c.
c, being a greatest lower bound of S, is also an element of S, since S
is closed
Now we see that h=0 on (a,c), but h≠0 at c. So, h is discontinuous at
c, but then h cannot be smooth
Thus we have reached a contradiction, and so f and g must agree
everywhere!
19. +
Suppose f(x) can be rewritten as a
power series…
f (x) = c 0 + c1 (x − a) + c 2 (x − a) 2 + ...+ c n (x − a) n
c 0 = f (a)
f ʹ′(x) = c1 + 2c 2 (x − a) + 3c 3 (x f − a) 2 + ...+ nc n (x − a) n −1
k
(a)
ck =
€ k!
c1 = f ʹ′(a)
€ f ʹ′ (x) = 2c 2 + 3∗2c 3 (x − € + 4 ∗ 3c 4 (x − a) 2 + ...+ n ∗(n −1)c n (x − a) n −2
a)
€
f ʹ′ (a)
c2 =
€ 2!
€ f k (a)
ck =
k!
€
20. +
Entirety (Analytic Functions)
A function f(x) is said to be entire if it is equal to its
Taylor Series everywhere
Entire Not Entire
sin(x) log(1+x)
21. +
Proof: sin(x) is entire
Maclaurin Series
sin(0)=1
sin’(0)=0
∞
sin’(0)=-1 (−1) n 2n +1
sin(x) = ∑ x
sin’(0)=0
n =0 (2n +1)!
sin’(0)=1
sin’(0)=0
sin’(0)=-1
… etc. €
22. +
Proof: sin(x) is entire
∞
(−1) n 2n +1
sin(x) = ∑ x
n =0 (2n +1)!
Lagrange formula for the remainder:
Let f : R →R be k+1 times differentiable on
(a,x) and continuous on [a,x]. Then
f k +1 (z) k +1
Rk (x) = (x − a)
(k +1)!
€ for some z in (x,a)
€
23. +
Proof: sin(x) is entire
First, sin(x)
is continuous and infinitely
differentiable over all of R
If we look at the Taylor Polynomial of degree k
f k +1 (z) k +1
Rk (x) = (x − a)
(k +1)!
Note though f k +1 (z) ≤ 1 for all z in R
k +1
(x − a)
Rk (x) ≤
€ (k +1)!
€
€
24. +
Proof: sin(x) is entire
However, as k goes to infinity, we see Rk (x) ≤ 0
Applyingthe Squeeze Theorem to our original
€
equation, we obtain that as k goes to infinity
f (x) = Tk (x)
and thus sin(x) is complete
€
25. +
Maclaurin Series Examples
∞ ∞
xn xn
log(1 − x) = −∑ log(1+ x) = ∑ (−1) n +1
n =1 n! n =1
n!
∞ ∞
1 (−1) n (2n)!
= ∑ xn 1+ x = ∑ xn
1 − x n =0 n =0
(1 − 2n)(n!) 2 (4) n
∞
€ xn€
ex = ∑
n =0 n!
∞
€ ∞
€
(−1) n
(−1) n 2n
sin(x) = ∑ x 2n +1 cos(x) = ∑ x
n =0
(2n +1)! n =0 (2n)!
€ ix
Note: e = cos(x) + isin(x)
€ €
26. +
Applications
Physics
Special Relativity Equation
Fermat’s Principle (Optics)
Resistivity of Wires
Electric Dipoles
Periods of Pendulums
Surveying (Curvature of the Eart)
27. +
Special Relativity
m0
m= KE = mc 2 − m0c 2
1− v2 c2
m0c 2 ⎡⎛ v 2 ⎞ −1/ 2 ⎤
KE = − m0c 2 = m0c 2 ⎢⎜1 − 2 ⎟ −1⎥
2
1−v c 2 € ⎢⎝ c ⎠
⎣ ⎥
⎦
€
If v ≤ 100 m/s
€
Then according to Taylor’s Inequality
1 3m0c 2 100 4 −10
R1 (x) ≤ 2 2 4 < (4.17 × 10 )m0
2 4(1 −100 /c ) c