From Math 2220 Class 15 V1cc HW Absolute Extrema From Math 2220 Class 15 1 Variable Taylor Series Multivariable Dr. Allen Back Taylor Polynomials More than 2 Variables Multivariable Oct. 1, 2014 Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
Prelim and HW From Math 2220 Class 15 V1cc HW Absolute Extrema 1 Variable Taylor Series Your next homework on 3.2, 3.3, and 3.4 will be due Fri Oct. Multivariable 10. Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
Absolute Extrema From Math 2220 Class 15 V1cc HW Absolute Closed Set contains all its boundary points. Extrema 1 Variable Taylor Series Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
Absolute Extrema From Math 2220 Class 15 V1cc HW Absolute Bounded Set lies inside some single ball of some radius. Extrema 1 Variable Taylor Series Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
Absolute Extrema From Math 2220 Class 15 V1cc HW Absolute Compact Set both closed and bounded. Extrema 1 Variable Taylor Series Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
Absolute Extrema From Math 2220 Class 15 V1cc HW Theorem A continuous functions whose domain is a compact Absolute Extrema set always attains (i.e. “has”) an absolute maximum and an 1 Variable absolute minimum. Taylor Series Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
Absolute Extrema From Math 2220 Class 15 V1cc HW Absolute Counterexamples when not compact: Extrema 1 Variable Taylor Series Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
Absolute Extrema From Math Procedure for finding an absolute extremum for a cont. fcn. on 2220 Class 15 a compact domain: V1cc 1 Above theorem guarantees existence since the domain is HW compact. Absolute Extrema 2 Look for critical points (in the interior of the domain.) 1 Variable Taylor Series 3 Investigate the boundary either using lower dimensional Multivariable calculus or Lagrange multipliers. Taylor Polynomials 4 Compare values at all candidates to find the absolute max More than 2 Variables and min. Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
Absolute Extrema From Math 2220 Class 15 V1cc Find the absolute max and absolute min of HW Absolute Extrema f ( x , y ) = xy (2 − x − y ) 1 Variable Taylor Series on the square | x | ≤ 1 , | y | ≤ 1 . Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
Absolute Extrema From Math 2220 Class 15 V1cc Find the absolute max and absolute min of HW Absolute f ( x , y ) = xy Extrema 1 Variable Taylor Series on (and inside) the triangle with vertices (2 , 0) , (10 , 0) , and Multivariable (4 , 1) . Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series From Math 2220 Class 15 V1cc HW Multivariable Taylor approximation follows (using the chain Absolute Extrema rule) easily from the 1-variable case, so we’ll start by reviewing 1 Variable how the results there work out. Taylor Series Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series From Math 2220 Class 15 V1cc Taylor Series of f ( x ) about x = a : HW Absolute f n ( a ) Extrema Σ ∞ ( x − a ) n . n =0 1 Variable n ! Taylor Series Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series From Math 2220 Class 15 V1cc Partial sums (the k + 1’st) give the k ’th Taylor Polynomial. HW f n ( a ) Absolute P k ( x ) = Σ k ( x − a ) n . Extrema n =0 n ! 1 Variable Taylor Series ( P k is of degree k but has k + 1 terms because we start with Multivariable Taylor 0.) Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series From Math 2220 Class 15 V1cc HW Absolute So P 1 ( x ) is the linear approximation to f ( x ) at x = a . Extrema 1 Variable Taylor Series Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series From Math 2220 Class 15 V1cc Examples: HW Absolute ( a ) e x about x = 0 ( b ) e x about x = 2 Extrema 1 Variable Taylor Series Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series From Math 2220 Class 15 V1cc HW Absolute Where does the formula for P k come from heuristically? Extrema 1 Variable Taylor Series Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series From Math 2220 Class 15 V1cc HW Where does the formula for P k come from heuristically? Absolute The k’th Taylor polynomial P k ( x ) is the unique polynomial of Extrema degree k whose value, derivative, 2nd derivative, . . . k ’th 1 Variable Taylor Series derivative all agree with those of f ( x ) at x = a . Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series Where does the formula for P k come from heuristically? From Math 2220 Class 15 Or integration by parts: u = f ′ ( x ) , v ′ = 1 , v = ( x − b ) V1cc � b HW f ′ ( x ) dx f ( b ) = f ( a ) + Absolute a Extrema � b � � � b ( x − b ) f ′ ( x ) � ( x − b ) f ′′ ( x ) dx 1 Variable f ( b ) = f ( a ) + a − Taylor Series a Multivariable � b Taylor f ( a ) + ( b − a ) f ′ ( a )+ ( b − x ) f ′′ ( x ) dx f ( b ) = Polynomials a More than 2 Variables The last term is one form of the “remainder” in approximating Multivariable f ( b ) by the first Taylor polynomial (aka linear approximant) Taylor Pictures P 1 ( b ) . Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series Where does the formula for P k come from heuristically? From Math 2220 Class 15 Or integration by parts: u = f ′′ ( x ) , v ′ = ( b − x ) , v = − ( b − x ) 2 V1cc 2 � b HW � f ( a ) + ( b − a ) f ′ ( a )+ ( b − x ) f ′′ ( x ) dx f ( b ) = Absolute Extrema a 1 Variable � b − ( b − x ) 2 � Taylor Series f ( a ) + ( b − a ) f ′ ( a )+ f ′′ ( x ) � f ( b ) = � 2 Multivariable � a Taylor � b Polynomials − ( b − x ) 2 � f ′′′ ( x ) dx − More than 2 2 Variables a Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series Where does the formula for P k come from heuristically? From Math 2220 Class 15 Or integration by parts: V1cc � b − ( b − x ) 2 � HW f ( a ) + ( b − a ) f ′ ( a )+ f ′′ ( x ) � f ( b ) = � 2 Absolute � a Extrema � b − ( b − x ) 2 � 1 Variable f ′′′ ( x ) dx − Taylor Series 2 a Multivariable Taylor ( b − a ) 2 Polynomials f ( a ) + ( b − a ) f ′ ( a )+ f ′′ ( a ) f ( b ) = 2 More than 2 � b Variables ( b − x ) 2 f ′′′ ( x ) dx Multivariable + Taylor 2 a Pictures Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series From Math 2220 Class 15 V1cc The above “integral formula” for the remainder can be turned HW into something simpler to remember but sufficient for most Absolute Extrema applications: 1 Variable The next Taylor term except with the higher derivative Taylor Series evaluated at some unknown point c between a and b . Multivariable Taylor Polynomials More than 2 Variables Multivariable Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
1 Variable Taylor Series Taylor’s Theorem with Remainder From Math 2220 Class 15 If f ( x ) , f ”( x ) , . . . f ( k ) ( x ) all exist and are continuous on [ a , b ] V1cc and f ( k +1) ( x ) exists on ( a , b ), then there is a c ∈ ( a , b ) so that HW Absolute f ( b ) = P k ( b ) + R k ( b ) Extrema 1 Variable where P k is the k’th Taylor polynomial of f about x = a and Taylor Series Multivariable Taylor R k ( b ) = f k +1 ( c ) Polynomials ( k + 1)!( b − a ) k +1 . More than 2 Variables Multivariable Please remember this! Taylor Pictures Higher Partials in Polar Cordinates Method of Characteristics
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