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# Taylor series

## by John Weiss, Working at Pomona College on May 07, 2011

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## Taylor seriesPresentation Transcript

• Taylor Series
John Weiss
• Approximating Functions
f(0)= 4
What is f(1)?
f(x) = 4?
f(1) = 4?
• Approximating Functions
f(0)= 4, f’(0)= -1
What is f(1)?
f(x) = 4 - x?
f(1) = 3?
• Approximating Functions
f(0)= 4, f’(0)= -1, f’’(0)= 2
What is f(1)?
f(x) = 4 – x + x2?
(same concavity)
f(1) = 4?
• Approximating Functions
f(x) = sin(x)
What is f(1)?
f(0) = 0, f’(0) = 1
f(x) = 0 + x?
f(1) = 1?
• 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)?
• Famous Mathematicians
James Gregory (1671)
Brook Taylor (1712)
Colin Maclaurin (1698-1746)
Joseph-Louis Lagrange (1736-1813)
Augustin-Louis Cauchy (1789-1857)
• Approximations
Linear Approximation
• Taylor’s Theorem
Let k≥1 be an integer and be k times differentiable at .
Then there exists a function such that
Note: Taylor Polynomial of degree k is:
• Works for Linear Approximations
• f(x) = sin(x)Degree 1
• f(x) = sin(x)Degree 3
• f(x) = sin(x)Degree 5
• f(x) = sin(x)Degree 7
• f(x) = sin(x)Degree 11
• Implications
If fand g have the same value and all of the same derivatives at a point, then they must be the same function!
• 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, so then h cannot be smooth
Thus we have reached a contradiction, and so f and g must agree everywhere!
• Suppose f(x) can be rewritten as a power series…
• Entirety (Analytic Functions)
A function f(x) is said to be entire if it is equal to its Taylor Series everywhere
Entire
sin(x)
Not Entire
log(1+x)
• Proof: sin(x) is entire
Maclaurin Series
sin(0)=0
sin’(0)=1
sin’’(0)=0
sin’’’(0)=-1
sin’’’’(0)=0
sin’’’’’(0)=1
sin’’’’’’(0)=0
… etc.
• Proof: sin(x) is entire
Lagrange formula for the remainder:
Let be k+1 times differentiable on (a,x) and continuous on [a,x]. Then
for some z in (a,x)
• 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
Note though for all z in R
• Proof: sin(x) is entire
However, as k goes to infinity, we see
Applying the Squeeze Theorem to our original equation, we obtain that as k goes to infinity
and thus sin(x) is entire since it is equal to its Taylor series
• Maclaurin Series Examples
Note:
• Applications
Physics
Special Relativity Equation
Fermat’s Principle (Optics)
Resistivity of Wires
Electric Dipoles
Periods of Pendulums
Surveying (Curvature of the Earth)
• Special Relativity
Let . If v ≤ 100 m/s
Then according to Taylor’s Inequality (Lagrange)
• Lagrange Remainder
Lagrange formula for the remainder:
Let be k+1 times differentiable on (a,x) and continuous on [a,x]. Then
for some z in (a,x)
• Special Relativity
Let . If v ≤ 100 m/s
Then according to Taylor’s Inequality (Lagrange)
• The End