# Calc III

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1. 14.4
Explain the process to find an eq. for the tangent plane.
Given: f(x,y), pt.
• 1. Find the gradient vector.
• 2. Plug the point into the
• 3. Insert given pt. and gradient vector into the equation of a plane (where z=-1).
2. 14.6
Find the directional derivative.
Given: f(x,y), vector, and pt.
• 1. Find the gradient vector
• 2. Find the unit vector of the given pt.
• 3. dot product of 1. and 2.
3. 14.7
In which direction is the rate of change of f(x,y) maximized.
unit vector of the gradient vector
4. 14.7
Find the max rate of change.
5. 14.8
Find the max value of f(x,y) subjected to g(x,y).
• 1. Find the gradient vector with respect to f(x,y) and g(x,y).
• 2. Solve of λ.
• 3. Solve for x or y.
• 4. Substitute to find y or x.
• 5. Find the max by substituting x and y into g(x,y).
6. 12.5
Find and eq. for the plane that passes through a pt. and contains a parametric equation.
• 1. Dot product of Q and R.
• 2. QP X V.
• 3. answer in eq. form.
7. 14.4
Find the linearization of f(x,y) at pt.
• 1. Find the gradient vector.
• 2. Find z using pts in f(x,y).
• 3. answer in eq. form
8. 15.5
What is the mass m using the applications of double integrals?
p(x,y) dA; where the domain is D
9. 15.5
What is the moment of the entire lamina about the x-axis using the applications of double integrals?
Mx=y * p(x,y) dA; where the domain is D
10. 15.5
What is the moment of the entire lamina about the y-axis using the applications of double integrals?
My=x * p(x,y) dA; where the domain is D
11. 15.5
What is the polar moment of inertia using the applications of double integrals?
Io(x2+y2) p(x,y) dA; where the domain is D
12. 15.5
What is the moment of inertia of the lamina about the x-axis using the application of double integrals?
Ix(y2)p(x,y) dA; where the domain is D
13. 15.5
What is the moment of inertia of the lamina about the y-axis using the application of double integrals?
Iy=(x2) p(x,y) dA; where the domain is D
14. 15.9
What is the formula for triple integration in spherical coordinates?
• E= f(psinΦcosθ, psinΦsinθ, pcosΦ)p2sinΦ dpdθdΦ;
• where E{(p,θ,Φ)|a≤p≤b, α≤θ≤β, c≤Φ≤d}
15. 15.9
Explain the triple integrals with spiracle coordinates using a diagram:
16. 15.9
In spherical coordinates, what is x? y? z?
• x= psinΦcosθ
• y= psinΦsinθ
• z=pcosΦ
17. What is C?
• A smooth space curve given by the parametric equation:
• x=x(t)  y=y(t)  z=z(t)

or my a vecor:

r(t)=x(t)+y(t)j +z(t)k
18. 16.2
Evaluate a line integral, where C is the given curve.
Given: f(x,y,z),x,y,z, domain of t.
• 1. f(x(t),y(t),z(t))√(dx/dt)2 +(dy/dt)2 +(dz/dt)2)dt
19. 16.2
What is the representative of the line segment that starts at r0 and ends at r1?

r(t)=(1-t)r+ tr1; where rand rare point on the line.
20. 16.2
Evaluate the line integral  F dr, where C is given by the vector functions r(t).
Given: F(x,y), r(t)
• 1. Remember that r(t) goes into F.
• *F is the normal vector of the curve
• 2. Find r'(t)
• 3. Plug x and y of r into eq. F(r(x,y))
• 4. Combine x's with x's and y's with y's
• 5. use W=  F(r(t)) ⋅ r'(t)dt
21. 16.3
How do you determine if the vector field is a conservative function?
δP/δy =δQ/δx; throughout D
22. 16.3
If F(x,y)= some i+j, what are the steps to find a function such that F=
• 1. Find δP/δy =δQ/δx to be conservative.
• 2. Find the partial with respect to x and y. i.e.<i,j>
• 3. with respect to x.
• 4.  Take the partial with respect to y of 3..
• 5.  g'(y) with respect to y.
• 6 .Combine 3. and 5..
23. 16.3
What is the fundamental theorem of line integrals?
F⋅dr = f(end)-f(start)
24. 16.4
What is the Green's Theorem?
Pdx+Qdy=
25. ?
"del" =
26. 16.5
What is the curl?
;where = <i,j,k> and F= P,Q,R
27. 16.5
curl(f)=
0
28. 16.5
div F=
 Author: jesperez10 ID: 246322 Card Set: Calc III Updated: 2013-11-19 03:04:07 Tags: Calculus III Folders: Description: Chapter 12: Vectors and the Geometry of Space (and more) Show Answers: