Practice Problems
Math 2500
March 19, 2008
These problems are not in any particular order. The exam will be shorter (about or 5 problems).
1. Consider the function f(x, y) = x3y−yx2+ 3xy.
(a) Write down the equation of the tangent plane to the graph of fat (0,1).
(b) At which points (x, y) is the tangent plane horizontal?
2. Consider
f(x, y) = Zy
x
et2dt.
(a) Compute ∂f
∂x and ∂f
∂y .
(b) Does fhave any critical points? Explain your answer.
3. Let some of the level sets of the function fbe given by the figure below.
(−2,1)
f=10 f=8
f=3
f=6
f=7
f=11
f=15
f=9
(3,4)
(2,0)
(3,5)
(3,6)
(3,0)
(a) Estimate ∂f
∂x (2,0)
(b) Estimate ∂f
∂y (3,4)
(c) In which direction does ∇f(−2,1) point?
(d) Suppose you know (1,0) is a critical point. Would you guess it’s a local maximum, a local minimum, or
neither? Explain your answer.
4. Consider f(x, y) = xex2+y2and let (x0, y0) = (1,1).
(a) Compute ∇f(1,1).
(b) Find the directional derivative ∇f·~u(1,1), where ~u = (1/2,√3/2).
(c) What is the direction of steepest increase for f, starting at (1,1). (Be sure to write down a unit vector!)
(d) Notice f(1,1) = e. What is the equation of the tangent line to the level set {f=e}, at the point (1,1)?
5. Consider f(x, y) = cos xcosy.
(a) Classify all the critical points of f.
(b) Find the absolute maximum and minimum of fon the square π/4≤x≤3π/4, π/4≤y≤3π/4.
6. Consider the composition F(t) = f(x(t), y(t)), where fis a function of the two variables xand y, while xand
yare both functions of t.
(a) Suppose x(1) = 0, y(1) = 2, x0(1) = 2, y0(1) = −1, ∂f
∂x (0,2) = 5, and ∂ f
∂y (0,2) = −3. Compute F0(1).
(b) If x0(0) = 0 and y0(0) = 0, is it true that F0(0) = 0? Explain your answer.
(c) If F0(3) = 0, is it true that x0(3) = 0 and y0(3) = 0? Explain your answer.
