Min/Max + Lagrange
Long Problems
Surfaces + Contour Diagrams
Directional Derivative + The Gradient
Partial Derivatives + The Chain Rule
100

The critical points of the function f(x,y) = x2+ (y-1)?

What is the point (0,1) ?

100

The y partial derivative at the point (1,0.5) of the function

 f = y*sin-1(xy)

What is 1/sqrt(3) + pi/6 ?

100

The limit of the function f(x,y) = y5cos(xy + 7)  as (x,y) approaches (2,5)

(Don't need decimals)

What is 25cos(17) ?

100

The unit vector in the direction <4,3>

What is <4/5 , 3/5> ?

100

The interpretation of the x partial derivative for a function f(x,y) at the point (a,b)

What is the slope of f(x,y) at the point (a,b) in the direction (1,0) ?

200

Use the second derivative test to determine if the point (3,2) is a max/min or neither if fxx = 4  , fyy = -8   ,    fxy  = -3

What is the point is a saddle point ?
200

The local max, min, and saddle points of the function 

f(x,y) = x3 + y3 - 3x2 - 3y2 - 9x

What are the saddle points (3,0) , (-1,2) , the local maximum (-1,0) , and the local minimum (3,2) ?

200

The description of the surface x+ 2z2 = 1

What is a cylinder about the y-axis ?

200

This is perpendicular to the gradient at the point (a,b)

What is a level curve?

200

The first partial derivative of f(x,y) = x/y

What is fx = 1/y    fy = -x / y2

300

The process for finding the max/min/saddle points of a function (looking for a long answer)

What is finding the critical points where fx = fy = 0 and then using the second derivative test. If negative, it's a saddle point. If positive, it's a max/min. + fx = max     - fx = min  ?

300

The partial derivative dP/dx when P = sqrt(u2+v2+w2) for x = 0 , y = 2

 u = xe

v = yex 

w = exy

What is 2/sqrt(5) ? 

300
The description of the level surfaces of the function f(x,y,z) = x + 3y + 5z

What is a family of planes with the normal vector <1, 3, 5> ?

300

The gradient of the function: f(x,y) = y(2y - x2)2

What is < 4xy(2y-x2) , 4y(2y-x2) + (2y - x2)2 > ?

300

The expression for te partial derivative dF/dt. 

For the function of F(x,y) , where x and y are both functions of s and t, 

What is dF/dt = dF/dy * dy/dt  + dF/dx * dx/dt ?

400

The max/min values of F = x- y2 with the constraint x+ y2 = 1 (use lagrange)

What is the max value is f(+- 1, 0) = 1 and the min value is                 f(0, +-1) = -1 ?

400

The differential of the function z = e-2x * cos(2*pi*t)

What is dz = -2e-2x * cos(2*pi*t) dx - 2*pi*e-2x * sin(2*pi*t) dt ?

400

The domain and sketch of the domain of the function ln(9 - x2 - 9y2)

What is the interior of the ellipse x2/9 + y< 1 ?

400

 Find the maximum rate of change and the direction in which is occurs for f(x,y,z) = xln(yz) at the point (1,2,1/2)

What is sqrt(17) / 2 in the direction <0, 1/2, 2> ?

400

The equation of the tangent plane to z = x/y2 at the point (-4,2,-1)

What is z = 1/4 *x + y -2 ?


500

The minimum point and value of the function f(x,y) = x2 + y2

What is (0,0) and 0 ?

500

Show that every plane tangent to the cone x2 + y2 = z2 passes through the origin

What is : 

Cone is a level surface. Gradient is (2x,2y,-2z) is the normal vector. Tangent plane is x0x + yoy - zoz = 0. This always contains the origin

500
The contour map of the function f(x,y) = ln(x2+4y2) showing several level curves. Include a sketch and general form

What is a contour map of ellipses with the form x2 + 4y2 = e?

500

The directional derivative of f(x,y,z) = xy2z3 at P(2,1,1) in the direction of Q(0, -3, 5)

What is 1?

500

The x derivative of the function F = y2*ln(t2 - cos(y)) + tan(ty)

What is 0 ?