calc3 pt 2

Question 1

finding domain

Answer 1

sqrt >| 0
ln() > 0
1/() =! 0
{(x,y) | …}

Question 2

lim (x,y,z) -> …

Answer 2

solve for the limit
do not forget to write lim everytime

Question 3

find fx fy fxx fyy

Answer 3

easy but be mindful of the product rule

Question 4

equation of tangent plane

Answer 4

1) plug in x and y into equation to find z
2) find the gradient f(x,y,z)
3) f(x,y,z) sub in x y and z values
4) x(x-Px)+y(y-Py)+z(z-Px)
5) use the cd to cancel out fractions

Question 5

dz /dt

Answer 5

dz/dt = dz/dx * dx/dt +dz/dy * dy/dt
sub and simplify

Question 6

compute the directional derivative

Answer 6

1) f(x,y) find the gradient
2) plug in points in gradient and simplify
3) find u using vector : v/|v|
4) Duf P(x,y) = gradient (dot product) vector u

Question 7

find and classify critical points

Answer 7

1) fx =0
2) fy =0
3) list critical points
4) fxx =
5) fyy =
6) fxy = fyx =
7) D = fxx * fyy -fxy^2
8) check each critical point by substitution
9) D(a,b) > 0 , fxx (a,b) < 0 : local max
D(a,b) > 0 , fxx (a,b) > 0 : local min
D(a,b) < 0 : saddle point

Question 8

lagrange

Answer 8

1) set the constraint to g(x,y) … = 0
2) find the gradient of f(x,y) and g(x,y)
3) delta f = lambda *delta g
4) solve for lambda for f and g gradient
5) set lambdas equal to each other
6) plug y ‘s into g
7) find points
8) check points by plugging them into f
9) decide your max and min
10) f has an abs max of .. @ (,) on g and f has an abs min of .. @ (,) on g