Calc 4 Midterm Review!

Min/Max + Lagrange

100

The critical points of the function f(x,y) = x²+ (y-1)² ?

What is the point (0,1) ?

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 ?

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 = min     - fx = max ?

400

The critical points for the function 

g(x,y) = 26 − 27x^2 − 18y^2 + y^4 

and whether they are min/max/saddle

What is the points saddle (0, -3), max (0,0), and saddle (0,3) ?

500

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

What is (0,0) and 0 ?

Long Problems

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 ?

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) ?

300

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

 u = xe^y 

v = ye^x 

w = e^xy

What is 6/sqrt(5) ? 

400

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

What is 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 + y_oy - z_oz = 0. This always contains the origin

Surfaces + Contour Diagrams

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) ?

200

The description of the surface x² + 2z² = 1

What is a cylinder about the y-axis ?

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> ?

400

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

What is the interior of the ellipse x²/9 + y² < 1 ?

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^k ?

Directional Derivative + The Gradient

100

The unit vector in the direction <4,3>

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

200

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

What is a level curve?

300

A vector normal to the graph 

f(x,y) = x^3 - 3yx^2 + 6xy^2 - y^3

at the point (1,1,3)

What is < 3, 6, -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> ?

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?

Partial Derivatives + The Chain Rule

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

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

What is fx = 1/y    fy = -x / y²

300

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

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

500

Assuming that the equation x^3 + y^3 + z^2 = 0 defines z as a function of the independent variables x and y, find the partial derivatives ∂z/∂x and ∂z/∂y

What is ∂z/∂x = − Fx/Fz= − 3x^2 / 2z   and
∂z/∂y = − Fy/Fz = − 3y^2 / 2z ?