Mathematics 41, section 1 -- Analysis 3

Review Sheet -- Exam 2

March 12, 1998

General Information

The second exam for the course will be given next Friday, March 20, as announced in the course syllabus. It will cover the material discussed in class since immediately before the first exam, through the material from class on Monday, March 16:

• a) Graphs, contour plots, and density plots for f(x,y)
• b) Directional and partial derivatives (know the definition of D_u f(x₀,y₀) and how to compute it using the definition).
• c) Limits and continuity for f(x,y)
• d) Differentiability for f(x,y) (the ``total derivative'' and the tangent plane)
• e) The gradient vector field grad f(x,y); the geometric meaning of the vector grad f(x₀,y₀); the relation between grad f(x₀,y₀) and directional derivatives.
• f) Critical points of f(x,y) and their relation with critical points of grad f(x,y) -- the "First Derivative Test"

There will again be 4 or 5 problems, some possibly with several parts. Some may ask you to analyze a graph (e.g. for the result of a calculation; others may ask for a short description or explanation of some phenomenon (similar to the questions from Lab assignments).

I will be happy to schedule an problem session before the exam. (Is Wednesday afternoon or evening good?)

Suggested Review Problems

From the text:

• Section 3.1/3, 9 a,c,e,g, 14 a,c,f, 18
• Section 3.2/10, 13
• Section 3.3/3, 4
• Section 3.4/1, 8,5
• Section 3.5/1, 4 a,b,d, 7 a, b, problems like 8, 9, 10
• Section 4.1/3, 5

You should also review the problems from the labs and problem set on this material.

Practice Exam

• A) Verify that the curve alpha(t) = (2e^4t + e^-t, -e^4t + 2e^-t) is a flow line of the vector field F(x,y) = (3x - 2y, -2x).
• B) Show that the vector field F from part A is equal to grad f(x,y) for the function f(x,y) = 3x²/2 - 2xy.
• C) The initial point of the flow line alpha from part A is alpha(0) = (3,1), which lies on the contour curve f(x,y) = 5/2. If you move along alpha(t) will the values of f increase or decrease? Explain.

II. All parts of this question refer to the function defined by f(x,y) = (2x⁵+y⁴)/(x⁴ + y⁴) if (x,y) <> (0,0) and f(0,0) = 0

• A) Compute the limits lim _{x -> 0} f(x,0) and lim _{y -> 0} f(0,y). Does lim_{(x,y) -> (0,0)} f(x,y) exist?
• B) Find the partial derivatives of f with respect to x and y at (0,0) by using the definition.
• C) Examine the graph of z = f(x,y) using Maple. From that evidence, does f appear to be differentiable at (0,0)? Explain. (On the actual exam, I would give you the Maple plot.)

III. All parts of this question refer to the function g(x,y) = sin(x - y)

• A) Determine the directional derivative D_u g(0,pi) for a general unit vector u.
• B) In which direction u is the directional derivative largest?

IV.

• A) Find all critical points of the function h(x,y) = xe^-x2-y2. (Note: e^-x2-y2<> 0 for all (x,y).)
• B) From a plot of the gradient vector field grad h(x,y) on the region [-1,1] x [-1,1], using the First Derivative Test, identify each critical point of h(x,y) as either a local maximum, a local minimum, or a saddle point. (On on the actual exam, if I asked a question like this, I would provide the Maple plot.)