- A contour plot is given with a constraint curve and a point on it. The gradient is given for the point.
- Label relative maximums, minimums, and saddle values on the graph.
- Use the curve as a constraint to relative extrema subject to the constraint.
- Start at the point and sketch the path of steepest increase (or decrease).

- Find the limits (if they exist). Two parts.
- Use partial implicit differentiation to find dy/dx.
- Use the chain rule to find the indicated ordinary or partial derivative.
- Use the total differential to estimate the change in z when from from one point to another.
- A function of two variables and a point are given. Parts d-m are to be evaluated at the given point. For parts i-m, let F(x,y,z) = f(x,y)-z.
- Find the partials with respect to x and y.
- Find all points on the surface where the tangent plane is horizontal.
- Classify each critical point as a relative maximum, relative minimum, saddle point, or the 2nd partial derivatives test fails.
- Find the direction of greatest increase at a point.
- Find the rate of the greatest increase at a point.
- Find the directional derivative in the indicated direction at a point as specified by an angle.
- Find the directional derivative in the indicated direction at a point as specified by a direction vector.
- Find a unit normal vector to the level curve containing a point.
- Find the gradient for the 3D extension at a point.
- Find the equation of the tangent plane at a point.
- Find the parametric or symmetric equations of a normal line to the surface at a point.
- Find the angle between the tangent plane and another plane at a point.
- Find the angle of inclination of the tangent plane containing the point

- Find and classify all relative extrema and saddle points of the function. A graph of the surface and its contour plot are given, but it's not detailed enough for you to answer the question. You will need to know how to find maximums and minimums (including the second partial derivatives test).
- Find the relative maximum or minimum for an applied problem. Although this is technically out of 13.9, it's really just an application of the concepts in 13.8.
- Use Lagrange multipliers to locate and classify any extrema of the function.

- This exam is made up of two parts.
- Questions 1-6 are to be done without computer technology (calculator is okay) and turned in before starting the second part.
- Maxima should be used for questions 7-9 (ie, some of the problems are going to be nearly impossible to work without it).
- There are 104 possible points.

# | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Total |
---|---|---|---|---|---|---|---|---|---|---|

Pts | 10 | 10 | 5 | 5 | 5 | 39 | 10 | 10 | 10 | 104 |