Math252CalculusIII:Taylor,Jacobian,HessianandExtremeValues by ...
Math 252 Calculus III: Taylor, Jacobian, Hessian and Extreme Values
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by: javier
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−4
Repeat and Review
Repeat and Review Taylor Series in One Variable
Taylor Series in One Variable
Suppose f(x) = ln(x)
Taylor Series in One Variable
Suppose f(x) = ln(x) □ The Best Degree-0 approximation at c = 1 T0 (x) =
Taylor Series in One Variable
Suppose f(x) = ln(x) □ The Best Degree-0 approximation at c = 1 T0 (x) = f(1) □ The Best Degree-1 approximation at c = 1 T1 (x) =
Taylor Series in One Variable Suppose f(x) = ln(x) □ The Best Degree-0 approximation at c = 1 T0 (x) = f(1) □ The Best Degree-1 approximation at c = 1 T1 (x) = f(1) + f ′ (1)(x − 1) □ The Best Degree-2 approximation at c = 1 T2 (x) =
Taylor Series in One Variable Suppose f(x) = ln(x) □ The Best Degree-0 approximation at c = 1 T0 (x) = f(1) □ The Best Degree-1 approximation at c = 1 T1 (x) = f(1) + f ′ (1)(x − 1) □ The Best Degree-2 approximation at c = 1 T2 (x) = f(1) + f ′ (1)(x − 1) +
f ′′ (1) (x − 1)2 2!
Taylor in two variables
Taylor in two variables degree one approximation
degree one approximation
Suppose
f(x, y) = e−x −y 2
2
degree one approximation
Suppose
f(x, y) = e−x −y 2
2
□ The Best Degree-0 approximation at c = (1, 3) T0 (x, y) =
degree one approximation
Suppose
f(x, y) = e−x −y 2
2
□ The Best Degree-0 approximation at c = (1, 3) T0 (x, y) = f(1, 3) □ The Best Degree-1 approximation at c = (1, 3) T1 (x, y) =
degree one approximation Suppose
f(x, y) = e−x −y 2
2
□ The Best Degree-0 approximation at c = (1, 3) T0 (x, y) = f(1, 3) □ The Best Degree-1 approximation at c = (1, 3) T1 (x, y) = f(1, 3) + fx (x − 1) + fy (y − 3) □ The Best Degree-2 approximation at c = (1, 3) T2 (x, y) =
degree one approximation
degree one approximation
[] a ⃗u = ∆⃗x = b
degree one approximation
[] a ⃗u = ∆⃗x = b Du f
degree one approximation
[] a ⃗u = ∆⃗x = b Du f
D2u f
degree one approximation
[] a ⃗u = ∆⃗x = b Du f
D2u f
degree one approximation Suppose
f(x, y) = e−x −y 2
2
□ The Best Degree-0 approximation at c = (1, 3) T0 (x, y) = T0 (x, y) = f(1, 3) □ The Best Degree-1 approximation at c = (1, 3) T1 (x, y) = T1 (x, y) = f(1, 3) + fx (x − 1) + fy (y − 3) □ The Best Degree-2 approximation at c = (1, 3) [ ] [ ] [ ] [ ] x−1 ] fxx fyx 1[ x−1 T2 (x, y) = f(1, 3) + fx fy · · + x−1 y−3 · y−3 fxy fyy y−3 2
extreme values and optimization
extreme values and optimization finding critical points
finding critical points
What are critical points ?
finding critical points
What are critical points ? Find the critical points of f(x, y) = 11 − x2 − y2
finding critical points What are critical points ? □ points where fx or fy is undefined □ points where fx = 0 AND fy = 0
How to classify them? □ compute the Hessian □ if
H0
□ if
H>0
□ if
H=0
H = fxx fyy − (fxy )2
then (a, b) is a saddle point. & fxx > 0 then (a, b) is a relative minimum. & fxx < 0 then (a, b) is a relative max. try something else.
finding critical points Example, find local extrema for f(x, y) = 4 + x3 + y3 − 3xy
finding critical points Example, find the point on the line y = 1 − 2x closest to the origin
finding critical points Example, find the point on the plane 6x + 4y − 3z = 2 closest to (2, −2, 3)
finding critical points Example, find local extrema for f(x, y) = 3x2 y + y3 − 3x2 − 3y2 + 1
5
0 −4
−2
0 0
2
2 4
by: javier
4
−2
−4
Repeat and Review
Repeat and Review Taylor Series in One Variable
Taylor Series in One Variable
Suppose f(x) = ln(x)
Taylor Series in One Variable
Suppose f(x) = ln(x) □ The Best Degree-0 approximation at c = 1 T0 (x) =
Taylor Series in One Variable
Suppose f(x) = ln(x) □ The Best Degree-0 approximation at c = 1 T0 (x) = f(1) □ The Best Degree-1 approximation at c = 1 T1 (x) =
Taylor Series in One Variable Suppose f(x) = ln(x) □ The Best Degree-0 approximation at c = 1 T0 (x) = f(1) □ The Best Degree-1 approximation at c = 1 T1 (x) = f(1) + f ′ (1)(x − 1) □ The Best Degree-2 approximation at c = 1 T2 (x) =
Taylor Series in One Variable Suppose f(x) = ln(x) □ The Best Degree-0 approximation at c = 1 T0 (x) = f(1) □ The Best Degree-1 approximation at c = 1 T1 (x) = f(1) + f ′ (1)(x − 1) □ The Best Degree-2 approximation at c = 1 T2 (x) = f(1) + f ′ (1)(x − 1) +
f ′′ (1) (x − 1)2 2!
Taylor in two variables
Taylor in two variables degree one approximation
degree one approximation
Suppose
f(x, y) = e−x −y 2
2
degree one approximation
Suppose
f(x, y) = e−x −y 2
2
□ The Best Degree-0 approximation at c = (1, 3) T0 (x, y) =
degree one approximation
Suppose
f(x, y) = e−x −y 2
2
□ The Best Degree-0 approximation at c = (1, 3) T0 (x, y) = f(1, 3) □ The Best Degree-1 approximation at c = (1, 3) T1 (x, y) =
degree one approximation Suppose
f(x, y) = e−x −y 2
2
□ The Best Degree-0 approximation at c = (1, 3) T0 (x, y) = f(1, 3) □ The Best Degree-1 approximation at c = (1, 3) T1 (x, y) = f(1, 3) + fx (x − 1) + fy (y − 3) □ The Best Degree-2 approximation at c = (1, 3) T2 (x, y) =
degree one approximation
degree one approximation
[] a ⃗u = ∆⃗x = b
degree one approximation
[] a ⃗u = ∆⃗x = b Du f
degree one approximation
[] a ⃗u = ∆⃗x = b Du f
D2u f
degree one approximation
[] a ⃗u = ∆⃗x = b Du f
D2u f
degree one approximation Suppose
f(x, y) = e−x −y 2
2
□ The Best Degree-0 approximation at c = (1, 3) T0 (x, y) = T0 (x, y) = f(1, 3) □ The Best Degree-1 approximation at c = (1, 3) T1 (x, y) = T1 (x, y) = f(1, 3) + fx (x − 1) + fy (y − 3) □ The Best Degree-2 approximation at c = (1, 3) [ ] [ ] [ ] [ ] x−1 ] fxx fyx 1[ x−1 T2 (x, y) = f(1, 3) + fx fy · · + x−1 y−3 · y−3 fxy fyy y−3 2
extreme values and optimization
extreme values and optimization finding critical points
finding critical points
What are critical points ?
finding critical points
What are critical points ? Find the critical points of f(x, y) = 11 − x2 − y2
finding critical points What are critical points ? □ points where fx or fy is undefined □ points where fx = 0 AND fy = 0
How to classify them? □ compute the Hessian □ if
H0
□ if
H>0
□ if
H=0
H = fxx fyy − (fxy )2
then (a, b) is a saddle point. & fxx > 0 then (a, b) is a relative minimum. & fxx < 0 then (a, b) is a relative max. try something else.
finding critical points Example, find local extrema for f(x, y) = 4 + x3 + y3 − 3xy
finding critical points Example, find the point on the line y = 1 − 2x closest to the origin
finding critical points Example, find the point on the plane 6x + 4y − 3z = 2 closest to (2, −2, 3)
finding critical points Example, find local extrema for f(x, y) = 3x2 y + y3 − 3x2 − 3y2 + 1
