from math 2220 class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfsep. 26, 2014 from...
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![Page 1: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/1.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
From Math 2220 Class 13
Dr. Allen Back
Sep. 26, 2014
![Page 2: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/2.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Prelim and HW
Your next homework on 3.2 and 3.3 will not be due soonerthan Mon 10 days from now.
![Page 3: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/3.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Prelim and HW
No homework quiz today.
![Page 4: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/4.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Prelim and HW
Prelim 1 next Tues at 7:30 pm in Malott 251 will include 3.1and the local part of 3.3 but not 3.2. (Chapter 2 as well.)We can spend as much of Monday on your review questions asyou like.
![Page 5: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/5.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Prelim and HW
The exam has 6 questions and is not very intricatecomputationally. Question 6 is a True/False question (not toohard) with the following directions:
![Page 6: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/6.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Prelim and HW
Problem 6) (15 Points) For the questions below, please justwrite the complete word TRUE or FALSE as your answer. Forthis question ONLY, no explanations are expected.(If a statement is sometimes true but not always, your answerwould be FALSE.)
You will receive 3 points for a correct answer, 1 point for ablank and −1 point for a wrong answer.
(Thus all 5 parts blank would give you 5 out of 15 while all fivewrong will lower your total on the rest of the exam by 5.)
(The exam is 11 pages with space below the questions to writeyour answers.)
![Page 7: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/7.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Prelim and HW
In addition to my usual Monday office hours (Evan has sometoo), I will have office hours
Fri 9/26 2:15-3:15
Tues 9/30 1-2:30
Also I’m willing to talk with anyone who likes rightafter class today though I do have an appt withEvan for 2-2:15.
![Page 8: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/8.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Absolute Extrema
Closed Set contains all its boundary points.
![Page 9: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/9.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Absolute Extrema
Bounded Set lies inside some single ball of some radius.
![Page 10: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/10.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Absolute Extrema
Compact Set both closed and bounded.
![Page 11: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/11.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Absolute Extrema
Theorem A continuous functions whose domain is a compactset always attains (i.e. “has”) an absolute maximum and anabsolute minimum.
![Page 12: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/12.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Absolute Extrema
Counterexamples when not compact:
![Page 13: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/13.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Absolute Extrema
Procedure for finding an absolute extremum for a cont. fcn. ona compact domain:
1 Above theorem guarantees existence since the domain iscompact.
2 Look for critical points (in the interior of the domain.)
3 Investigate the boundary either using lower dimensionalcalculus or Lagrange multipliers.
4 Compare values at all candidates to find the absolute maxand min.
![Page 14: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/14.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Absolute Extrema
Find the absolute max and absolute min of
f (x , y) = xy(2− x − y)
on the square |x | ≤ 1, |y | ≤ 1.
![Page 15: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/15.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Absolute Extrema
Find the absolute max and absolute min of
f (x , y) = xy
on (and inside) the triangle with vertices (2, 0), (10, 0), and(4, 1).
![Page 16: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/16.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Multivariable Taylor approximation follows (using the chainrule) easily from the 1-variable case, so we’ll start by reviewinghow the results there work out.
![Page 17: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/17.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Taylor Series of f (x) about x = a:
Σ∞n=0
f n(a)
n!(x − a)n.
![Page 18: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/18.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Partial sums (the k + 1’st) give the k ’th Taylor Polynomial.
Pk(x) = Σkn=0
f n(a)
n!(x − a)n.
(Pk is of degree k but has k + 1 terms because we start with0.)
![Page 19: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/19.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
So P1(x) is the linear approximation to f (x) at x = a.
![Page 20: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/20.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Examples:
(a)ex about x = 0 (b)ex about x = 2
![Page 21: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/21.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Where does the formula for Pk come from heuristically?
![Page 22: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/22.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Where does the formula for Pk come from heuristically?The k’th Taylor polynomial Pk(x) is the unique polynomial ofdegree k whose value, derivative, 2nd derivative, . . . k’thderivative all agree with those of f (x) at x = a.
![Page 23: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/23.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Where does the formula for Pk come from heuristically?Or integration by parts: u = f ′(x), v ′ = 1, v = (x − b)
f (b) = f (a) +
∫ b
af ′(x) dx
f (b) = f (a) +(
(x − b)f ′(x)∣∣ba−
∫ b
a(x − b)f ′′(x) dx
)f (b) = f (a) + (b − a)f ′(a)+
∫ b
a(b − x)f ′′(x) dx
The last term is one form of the “remainder” in approximatingf (b) by the first Taylor polynomial (aka linear approximant)P1(b).
![Page 24: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/24.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Where does the formula for Pk come from heuristically?
Or integration by parts: u = f ′′(x), v ′ = (b − x), v = −(b−x)2
2
f (b) = f (a) + (b − a)f ′(a)+
∫ b
a(b − x)f ′′(x) dx
)f (b) = f (a) + (b − a)f ′(a)+
(−(b − x)2
2f ′′(x)
∣∣∣∣ba
−∫ b
a
−(b − x)2
2f ′′′(x) dx
)
![Page 25: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/25.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Where does the formula for Pk come from heuristically?Or integration by parts:
f (b) = f (a) + (b − a)f ′(a)+
(−(b − x)2
2f ′′(x)
∣∣∣∣ba
−∫ b
a
−(b − x)2
2f ′′′(x) dx
)f (b) = f (a) + (b − a)f ′(a)+
(b − a)2
2f ′′(a)
+
∫ b
a
(b − x)2
2f ′′′(x) dx
![Page 26: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/26.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
The above “integral formula” for the remainder can be turnedinto something simpler to remember but sufficient for mostapplications:The next Taylor term except with the higher derivativeevaluated at some unknown point c between a and b.
![Page 27: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/27.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Taylor’s Theorem with RemainderIf f (x), f ”(x), . . . f (k)(x) all exist and are continuous on [a, b]and f (k+1)(x) exists on (a, b), then there is a c ∈ (a, b) so that
f (b) = Pk(b) + Rk(b)
where Pk is the k’th Taylor polynomial of f about x = a and
Rk(b) =f k+1(c)
(k + 1)!(b − a)k+1.
Please remember this!
![Page 28: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/28.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Example: Accuracy of sin x byP1(x) (or P3(x)) (about x = 0)for |x | < .1? Or x < .01?
![Page 29: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/29.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
f (x) = 11+x+x2 about x = 1.
Approximation properties of different Taylor polynomials nearx = 1.
![Page 30: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/30.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Taylor polynomials of f (x) = 11+x+x2 about x = 1.
Typically higher order approximate well for a greater distance.
(Bad behavior beyond the radius of convergence.)
![Page 31: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/31.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Show that the Maclaurin series of sin x converges to sin x forall x .
![Page 32: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/32.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Show that the Maclaurin series of sin x converges to sin x forall x .This is different than saying the radius of convergence is ∞.Some functions, e.g.
f (x) =
{0 if x ≤ 0
e−1x if x > 0
have a Taylor series which always converges, but not to f (x)even near 0. In this case, every derivative is 0 at x = 0 and theMaclaurin series is identically 0!
![Page 33: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/33.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Show that the Maclaurin series of sin x converges to sin x forall x .The point here is that all derivatives are bounded functionstaking values between −1 and +1. So you can show theremainder Rn(x) goes to 0 as n→∞.Taylor polynomials of f (x) = 1
1+x+x2 about x = 1.Typically higher order approximate well for a greater distance.
(Bad behavior beyond the radius of convergence.)
![Page 34: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/34.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
It is more advanced than our course, but there is an amazingconnection between when Taylor series converge to a functionand differentiability of f(x) for x viewed as a complex variable.
![Page 35: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/35.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
It is more advanced than our course, but there is an amazingconnection between when Taylor series converge to a functionand differentiability of f(x) for x viewed as a complex variable.For example, consider the MacLaurin series of f (x) = 1
1+x2 .
This fcn. is complex differentiable when x 6= ±i = ±√−1. And
if one identifies the complex number a + bi with the point(a, b) ∈ R2, the nearest bad point of ±i ∼ (0,±1) is distance 1from 0 = 0 + 0i ∼ (0, 0). So from this advanced point of view,the radius of convergence is 1!
![Page 36: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/36.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
1 Variable Taylor Series
Similarly, consider the above picture for f (x) = 11+x+x2 about
x = 1. This fcn. is bad when 1 + x + x2 = 0, or x = −1±i√
32 .
The distance from 1 + 0i ∼ (1, 0) to −(12 ,√
32 ) is
√3, so from
this advanced point of view, that is the radius of convergencefor this Taylor series.
![Page 37: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/37.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Polynomials
Given f : U ⊂ Rn → R, to find the formula for the k’th Taylorpolynomial Pk of a C k function about the pointp0 = (a1, a2, . . . , an) ∈ U, we consider the 1-variable functiong(t) defined by
g(t) = f (p0 + th)
where h = (h1, h2, . . . , hn) ∈ Rn.If we are interested in relating the Taylor polynomial Pk to thebehavior of f at the point p = (x1, x2, . . . , xn) ∈ U we mightchoose
h = (x1 − a1, x2 − a2, . . . , xn − an)
as long as the entire line segment between p0 and p lies in U.
![Page 38: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/38.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Polynomials
g(t) = f (p0 + th)
![Page 39: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/39.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Polynomials
Problem: Find the first and second Taylor polynomials of
f (x , y) = cos (x + 3y) + sin x
about (x , y) = (π
2, π).
![Page 40: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/40.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Polynomials
Problem: Find the first and second Taylor polynomials of
f (x , y) = cos (x + 3y) + sin x
about (x , y) = (π
2, π).
Estimate the error in approximating f by P1 for |x − π
2| < .1
and |y − π| < .1.
![Page 41: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/41.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Polynomials
Problem: Find the first and second Taylor polynomials of
f (x , y) = cos (x + 3y) + sin x
about (x , y) = (π
2, π).
Estimate the error in approximating f by P1 for |x − π
2| < .01
and |y − π| < .01.
![Page 42: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/42.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Polynomials
Problem: Find the first and second Taylor polynomials of
f (x , y) = cos (x + 3y) + sin x
about (x , y) = (π
2, π).
Find an ε so that the error in approximating f by P1 for
|x − π
2| < ε and |y − π| < ε is at most .01.
Or at most 10−6.
![Page 43: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/43.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Pictures
Graph of f (x , y) = 1x+y2 for .5 ≤ x , y ≤ 1.5
![Page 44: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/44.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Pictures
P2 about (1, 1) for f (x , y) = 1x+y2 for .5 ≤ x , y ≤ 1.5
![Page 45: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/45.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Pictures
Both superimposed for .5 ≤ x , y ≤ 1.5
![Page 46: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/46.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Pictures
Error in using P2 for f for .5 ≤ x , y ≤ 1.5
![Page 47: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/47.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Pictures
Error in using P2 for f for .75 ≤ x , y ≤ 1.25
![Page 48: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/48.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Multivariable Taylor Pictures
Error in using P2 for f for 0 ≤ x , y ≤ 3
![Page 49: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/49.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Higher Partial Derivatives
A typical application of mixed partials arises out of yourhomework problem 3.1 #22 where you show that the waveequation
∂2w
∂x2=∂2w
∂y2
becomes after a change of coordinates
x = u + v , y + u − v
∂2w
∂u∂v= 0
![Page 50: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/50.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Higher Partial Derivatives
A typical application of mixed partials arises out of yourhomework problem 3.1 #22 where you show that the waveequation
∂2w
∂x2=∂2w
∂y2
becomes after a change of coordinates
x = u + v , y + u − v
∂2w
∂u∂v= 0
![Page 51: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/51.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Higher Partial Derivatives
A typical application of mixed partials arises out of yourhomework problem 3.1 #22 where you show that the waveequation
∂2w
∂x2=∂2w
∂y2
x = u + v , y + u − v∂2w
∂u∂v= 0
implying the general solution
w = f (u) + g(v) = f (x − y
2) + g(
x + y
2)
for any C 2 functions f and g .
![Page 52: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/52.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
A Counterexample
When f is not C 2, fxy need not equal fyx even when both exist.
![Page 53: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/53.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
A Counterexample
An example illustrating this is
f (x , y) =
{xy(x2−y2)
x2+y2 if (x , y) 6= (0, 0)
0 if (x , y) = (0, 0)
This function
![Page 54: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/54.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
A Counterexample
An example illustrating this is
f (x , y) =
{xy(x2−y2)
x2+y2 if (x , y) 6= (0, 0)
0 if (x , y) = (0, 0)
This functionis clearly C 2 for (x , y) 6= (0, 0) since it is a rational function(quotient of polynomials) with nonzero denominator.
![Page 55: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/55.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
A Counterexample
An example illustrating this is
f (x , y) =
{xy(x2−y2)
x2+y2 if (x , y) 6= (0, 0)
0 if (x , y) = (0, 0)
This functionsatisfies f (x , y) = −f (y , x) implying
fxy = fyx
whenever x = y . (A good chain rule exercise . . . )
![Page 56: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/56.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
A Counterexample
An example illustrating this is
f (x , y) =
{xy(x2−y2)
x2+y2 if (x , y) 6= (0, 0)
0 if (x , y) = (0, 0)
This functionsatisfies fx(0, 0) = 0 and fy (0, 0) = 0 since f vanishes along theaxes.
![Page 57: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/57.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
A Counterexample
An example illustrating this is
f (x , y) =
{xy(x2−y2)
x2+y2 if (x , y) 6= (0, 0)
0 if (x , y) = (0, 0)
This function
satisfies fx =y(x2 − y2)
x2 + y2+
2x2y
x2 + y2− 2x2y(x2 − y2)
(x2 + y2)2.
implying
fx(0, y) = −y (the last two terms vanish)
and∂2f
∂y∂x(0, 0) = −1.
![Page 58: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/58.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
A Counterexample
An example illustrating this is
f (x , y) =
{xy(x2−y2)
x2+y2 if (x , y) 6= (0, 0)
0 if (x , y) = (0, 0)
This function
fx(0, y) = −y (the last two terms vanish)
and∂2f
∂y∂x(0, 0) = −1.
By fxy (x , x) = −fyx(x , x) (mentioned above) or directly we see
∂2f
∂x∂y(0, 0) = 1
and the mixed partials really differ!
![Page 59: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/59.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Higher Partials in Polar Cordinates
Let x = r cos θ, y = r sin θ. Given f (x , y), express
∂2f
∂r2
in terms of partials of f with respect to x and y .
![Page 60: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/60.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Higher Partials in Polar Cordinates
Really there are two functions here; namely f (x , y) and
g(r , θ) = f (r cos θ, r sin θ).
But many applied fields routinely use the same letter f for bothfunctions.
![Page 61: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/61.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Higher Partials in Polar Cordinates
The key step is realizing that the task (via the chain rule) of
relating∂f
∂rto
∂f
∂xand
∂f
∂yis just like the task of doing the
analagous thing with expressions like
∂
∂r
(∂f
∂x
).
(The right hand side “operator notation” means partially
differentiate the function∂f
∂xof (x , y) with respect to r .)
![Page 62: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/62.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Higher Partials in Polar Cordinates
x = r cos θ y = r sin θ
∂f
∂r=
∂f
∂x
∂x
∂r+∂f
∂y
∂y
∂r
=∂f
∂xcos θ +
∂f
∂ysin θ
∂
∂r
(∂f
∂x
)=
(∂
∂x
(∂f
∂x
))∂x
∂r+
(∂
∂y
(∂f
∂x
))∂y
∂r
=
(∂2f
∂x2
)cos θ +
(∂2f
∂y∂x
)sin θ
![Page 63: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/63.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Higher Partials in Polar Cordinates
A similar computation with∂2f
∂θ2would show that the laplacian
∂2f
∂x2+∂2f
∂y2
is given in polar coordinates by
∂2f
∂r2+
1
r
∂f
∂r+
1
r2
(∂2f
∂θ2
).
![Page 64: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/64.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Can you estimate the value of the gradient?
![Page 65: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/65.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Method of Characteristics
Problem 33 on page 146 from your homework asks you to show
using the chain rule that if z = f (y
x) for a 1-variable
differentiable function f , then
x∂z
∂x+ y
∂z
∂y= 0.
![Page 66: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/66.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Method of Characteristics
Problem 33 on page 146 from your homework asks you to show
using the chain rule that if z = f (y
x) for a 1-variable
differentiable function f , then
x∂z
∂x+ y
∂z
∂y= 0.
You don’t need it to do this homework problem, but a naturalquestion is
Does x∂z
∂x+ y
∂z
∂y= 0 imply z = f (
y
x)
for some 1-variable differentiable function f ?
![Page 67: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/67.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Method of Characteristics
The answer is yes, and we can easily show it using what welearned in section 2.6!(i.e. essentially chain rule ideas applied to curves.)
![Page 68: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/68.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Method of Characteristics
The beautiful idea for the equation
a(x , y)∂z
∂x+ b(x , y)
∂z
∂y= c(x , y , z)
is to consider curves (x(t), y(t) called characteristics) satisfying
dx
dt= a(x(t), y(t))
dy
dt= b(x(t), y(t)).
![Page 69: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/69.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Method of Characteristics
Here for
x∂z
∂x+ y
∂z
∂y= 0
we havedx
dt= x
dy
dt= y
with solutions x(t) = x0et and y(t) = y0et .
![Page 70: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/70.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Method of Characteristics
Here for
x∂z
∂x+ y
∂z
∂y= 0
we havedx
dt= x
dy
dt= y
with solutions x(t) = x0et and y(t) = y0et .The key observation is that for a solution z(x , y)
d
dt[z(x(t), y(t))] =
∂z
∂x
dx
dt+
∂z
∂y
dy
dt
=∂z
∂xx+
∂z
∂yy
= 0 !
![Page 71: From Math 2220 Class 13pi.math.cornell.edu/~back/m222_f14/slides/sep26_v2c.pdfSep. 26, 2014 From Math 2220 Class 13 V2c Prelim and HW Absolute Extrema 1 Variable Taylor Series Multivariable](https://reader034.vdocuments.site/reader034/viewer/2022052010/601f68c974bf217c83032aae/html5/thumbnails/71.jpg)
From Math2220 Class 13
V2c
Prelim andHW
AbsoluteExtrema
1 VariableTaylor Series
MultivariableTaylorPolynomials
MultivariableTaylorPictures
Higher PartialDerivatives
A Counterex-ample
Higher Partialsin PolarCordinates
GradientEstimate fromGraph
Method ofCharacteristics
Method of Characteristics
Here for
x∂z
∂x+ y
∂z
∂y= 0
with solutions x(t) = x0et and y(t) = y0et .The key observation is that for a solution z(x , y)
d
dt[z(x(t), y(t))] =
∂z
∂x
dx
dt+
∂z
∂y
dy
dt
=∂z
∂xx+
∂z
∂yy
= 0 !
So z(x , y) is constant along linesy
x= k for any k and thus
z(x , y) = z(1,y
x) = f (
y
x)
as we wanted to show.