Quadric Surfaces and Level Curves
Describe and Sketch the Surface
y = z^2
Parabolic cylinder opening toward the y axis
Find the first partial derivatives of the function
z = x sin(xy)
dy/dx= x· [cos(xy)](y) + [sin(xy)]· 1 = xy cos(xy) + sin(xy), dz/dy = x [cos(xy)](x) = x^2 cos(xy)
Find an equation of the tangent plane at the specified point
z = (x+2)^2-2(y-1)^2-5 ; (2,3,3)
z-3 = f_x(2,3)(x-2)+f_y(2,3)(y-3) ; z=8x-8y+11
Use traces to sketch and identify the surface
3x^2+y+3z^2=0
Circular Parabaloid
Find first partial derivatives of the function
f(x, y) = x^y
f_x(x,y) = yx^(y−1) , f_y (x,y) = x^y lnx
Find the differential of the function
u=sqrt(x^2+3y^2)
x/sqrt(x^2+3y^2)dx+(3y)/(sqrt(x^2+3y^2))dy
Surprise! Limits Question
Prove the limit does not exist
lim_((x,y) -> (0,0))(xy-y)/((x-1)^2+y^2
Find the Indicated Partial Derivative of the function
f(x, y) = y tan^-1(xy); fy (1, 1/2)
1/(sqrt3) + pi/6
Find the directional derivative in the direction of v
u^2e^-v, (3,0), v=3i+4j
-18/5
Use traces to sketch and identify the surface
4x^2 + 9y^2 + 9z^2 = 36
Ellipsoid
Use implicit differentiation to find
dz/dx and dz/dy
yz +x ln y = z^2
I ain't copying all that check answers
Find the directional derivative of the function at the given point in the direction of v
f(x,y,z)=xy^2tan^-1z, (2,1,1), v=<1,1,1>
1/sqrt(3)((5pi)/4 + 1)
Use traces to sketch and identify the surface
z^2 − 4x^2 − y^2 = 4
Hyperboloid of 2 Sheets
P = sqrt(u^2 + v^2 + w^2), u=xe^y, v=ye^x, w=e^(xy); dP/dx, dP/dy when x=0, y=2
Check answers
Find the maximum rate of change f at the given point and the direction in which it occurs
f(s,t)=te^(st), (0,2)
Dir <4,1>
Mag=sqrt(17)