# Calc III Exam 2

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 Author: stpierrewm ID: 309811 Filename: Calc III Exam 2 Updated: 2015-10-20 14:38:07 Tags: Calc III Exam Folders: Calc III Exam 2 Description: Calc III Exam 2 Show Answers:

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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

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
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