Calc III Exam 2

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Calc III Exam 2
2015-10-20 14:38:07
Calc III Exam
Calc III Exam 2
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
  5. Tangent plane of z(x,y)=
  6. f is differentiable at (a,b) if
    if  and  are defined and continuous at (a,b)
  7. If a function is differentiable at (a,b)
    it is continuous at (a,b)
  8. ΔZ=
  9. f(x,y)≃
    • or
    • ΔZ +f(a,b)

  10. f has its
    1) maximum rate of increase
    2) rate of zero increase
    3) maximum rate of decrease

    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
  11. What is the chain rule for z(x,y) with respect to t
  12. The implicit differentiation 
    can be rewritten as 
  13. 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)
  14. Critical points (on the interior of the domain of f) exist when
    or both do not exist
  15. Discriminant, D=
  16. if D(a,b)>0 and <0, there exists
    a local max
  17. if D(a,b)>0 and >0, there exists
    a local min
  18. if D(a,b,)<0, there exists
    a saddle point
  19. if D(x,y)=0, there exists
    inconclusive u hoe
  20. 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