Calc 4 Midterm Review

Quadric Surfaces and Level Curves

100

Describe and Sketch the Surface

y = z^2

Parabolic cylinder opening toward the y axis

200

Use traces to sketch and identify the surface

3x^2+y+3z^2=0

Circular Parabaloid

300

Surprise! Limits Question

Prove the limit does not exist

lim_((x,y) -> (0,0))(xy-y)/((x-1)^2+y^2

(x, 0) , (x, x-1) -> DNE

400

Use traces to sketch and identify the surface

4x^2 + 9y^2 + 9z^2 = 36

Ellipsoid

500

Use traces to sketch and identify the surface

z^2 − 4x^2 − y^2 = 4

Hyperboloid of 2 Sheets

Partial Derivatives and Chain Rule

100

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)

200

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

300

Find the Indicated Partial Derivative of the function

f(x, y) = y tan^-1(xy); fy (1, 1/2)

1/(sqrt3) + pi/6

400

Use implicit differentiation to find 

dz/dx and dz/dy

yz +x ln y = z^2

I ain't copying all that check answers

500

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

Tangent Approximations and Directional Derivatives

100

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

200

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

300

Find the directional derivative in the direction of v

u^2e^-v, (3,0), v=3i+4j

-18/5

400

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)

500

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)