Exam # 2 Practice
Math 2210
Fall 2025
Name:
Teacher:
Jonathan Tyler
Instructions: You may use a 3×5 note card and a non-graphing calculator. Write in the space provided.
1. Find the absolute maximum and minimum values of f (x, y) = x2 + 2xy + y 2 + 3 on the square region
enclosed by the lines x = ±1 and y = ±1.
2. Find fx , fy , and fz for the function f (x, y, z) = x −
p
y2 + z2.
3. Let w = xy + ln z, x = v 2 /u, y = u + v, and z = cos u. Find
∂w
in terms of u and v.
∂v
2
4. Let f (x, y, z) = xexy sin z. Find ∇f at P0 (−1, 0, π/2).
5. At the point P0 (2, 1, 31), find an equation for the tangent plane to the surface f (x, y) = 4x3 − y 2 .
6. Find any extreme values or saddle points of the function f (x, y) = x2 + 2xy.
7. By considering different paths of approach, show that
x2 y
does not exist.
(x,y)→(0,0) x4 + y 2
lim
8. Find all second-order partial derivatives of the function g(x, y) = ln(xy) + cos
y
x
.
9. Find the derivative of f (x, y) = 2x2 + y 2 at P0 (−1, 1) in the direction of the vector v = 3i − 4j.
10. Assuming that 2xy + ex+y − 2 = 0 defines y as a differentiable function of x, find the value of dy/dx
at P (0, ln 2).
3
4
t
−2 i+
− 3 j + cos(t − 2)k is tangent to the surface x3 + y 3 +
11. Show that the curve r(t) =
4
t
z 3 − xyz = 0 at (0, −1, 1).