Calc III Exam 2

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  1. A function f(x,y) is continuous at a point (a,b) if (3 conditions)
    • a) f(x,y) is defined at (a,b)
    • b) lim(x,y)->(a,b) f(x,y) exists
    • c) lim(x,y)->(a,b) f(x,y) = f(a,b)
  2. Interior point P of shape R
    Has a disk centered at point P which contains only points inside R
  3. Boundary point Q of R
    Every disk centered at Q has a point inside of R and a point outside of R
  4. Clairaut's theorem
    chart?chf=bg,s,00000000&cht=tx&chl=f_%7Bxy%7D%3Df_%7Byx%7D&chs=140x42
  5. Tangent plane of z(x,y)=
    chart?chf=bg,s,00000000&cht=tx&chl=L(x%2Cy)%3Df_x(a%2Cb)(x-a)%2Bf_y(a%2Cb)(y-b)%2Bf(a%2Cb)&chs=708x44
  6. f is differentiable at (a,b) if
    if chart?chf=bg,s,00000000&cht=tx&chl=f_x&chs=36x42 and chart?chf=bg,s,00000000&cht=tx&chl=f_y&chs=36x42 are defined and continuous at (a,b)
  7. If a function is differentiable at (a,b)
    it is continuous at (a,b)
  8. ΔZ=
    chart?chf=bg,s,00000000&cht=tx&chl=f_x(a%2Cb)(x-a)%2Bf_y(a%2Cb)(y-b)&chs=450x44
  9. f(x,y)≃
    • chart?chf=bg,s,00000000&cht=tx&chl=f_x(a%2Cb)(x-a)%2Bf_y(a%2Cb)(y-b)%20%2Bf(a%2Cb)&chs=572x44
    • or
    • ΔZ +f(a,b)
  10. chart?chf=bg,s,00000000&cht=tx&chl=D_uf(a%2Cb)%3D&chs=166x38
    chart?chf=bg,s,00000000&cht=tx&chl=%5Cnabla%20f(x%2Cy)%20%5Ccdot%20%5Cvec%7Bu%7D&chs=158x38

    • chart?chf=bg,s,00000000&cht=tx&chl=%5Cvec%7Bu%7D%20%3D%5Cfrac%7B%5Cvec%7Bw%7D%7D%7B%5Cmid%5Cvec%7Bw%7D%5Cmid%7D&chs=106x78
    • chart?chf=bg,s,00000000&cht=tx&chl=%5Cvec%7Bw%7D%3Ddirection%20vector&chs=296x28

    chart?chf=bg,s,00000000&cht=tx&chl=%5Cmid%5Cnabla%20f(a%2Cb)%5Cmid%20cos(%5Ctheta)&chs=224x40
  11. f has its
    1) maximum rate of increase
    2) rate of zero increase
    3) maximum rate of decrease

    when:
    1) in the direction of the gradient. The rate of increase is the magnitude of the gradient

    2) in any direction orthogonal to the gradient

    3) in the direction of the negative gradient. The rate of decrease is the negative of the magnitude of the gradient
  12. What is the chain rule for z(x,y) with respect to t
    chart?chf=bg,s,00000000&cht=tx&chl=%5Cfrac%7Bdz%7D%7Bdx%7D%5Cfrac%7Bdx%7D%7Bdt%7D%2B%5Cfrac%7Bdz%7D%7Bdy%7D%5Cfrac%7Bdy%7D%7Bdt%7D&chs=194x80
  13. The implicit differentiation chart?chf=bg,s,00000000&cht=tx&chl=%5Cfrac%7Bdy%7D%7Bdx%7D&chs=40x74
    can be rewritten as chart?chf=bg,s,00000000&cht=tx&chl=%5Cfrac%7B-f_x%7D%7Bf_y%7D&chs=62x90
  14. if F(x,y,z)=K and ⊽F(a,b,c)≠0 then ⊽F(a,b,c) is
    orthogonal to the tangent plane to the level surface at (a,b,c)
  15. Critical points (on the interior of the domain of f) exist when
    chart?chf=bg,s,00000000&cht=tx&chl=f_x%3Df_y%3D0&chs=168x42or both do not exist
  16. Discriminant, D=
    chart?chf=bg,s,00000000&cht=tx&chl=D(x%2Cy)%3Df_%7Bxx%7D(x%2Cy)f_%7Byy%7D(x%2Cy)-(f_%7Bxy%7D(x%2Cy))%5E2&chs=588x50
  17. if D(a,b)>0 and chart?chf=bg,s,00000000&cht=tx&chl=f_%7Bxx%7D&chs=52x42<0, there exists
    a local max
  18. if D(a,b)>0 and chart?chf=bg,s,00000000&cht=tx&chl=f_%7Bxx%7D&chs=52x42>0, there exists
    a local min
  19. if D(a,b,)<0, there exists
    a saddle point
  20. if D(x,y)=0, there exists
    inconclusive u hoe
  21. What are the steps for finding the absolute max and min on a closed and bounded region of a function with two variables (not lagrange)?
    • 1) Determine value of f at all critical points
    • 2) Find all extrema of f on the boundary
    • 3) The greatest value is the absolute max, least value is the absolute min

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Author:
stpierrewm
ID:
309811
Filename:
Calc III Exam 2
Updated:
2015-10-20 18:38:07
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Calc III Exam
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Calc III Exam 2
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Calc III Exam 2
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