Math 425, Spring 08
Midterm Practice Problems
You are expected to give careful explanations/proofs.
1. Let
X={(x, y) : x2+y2= 1, x > 0}.
Is Xopen or closed?
2. Let
X={(x, y) : x2+y2= 1, x ≥0}.
Is Xopen or closed?
3. Give an example of a function f:R2→Rsuch that its partial
derivatives exist everywhere but fis not differentiable at the origin.
4. Set
f(x) = (x2sin 1
xif x6= 0
0 if x= 0
Show that f(x) differentiable everywhere but its derivative is not con-
tinuous at zero.
5. Is there exist a function f:R2→Rsuch that all its partial
derivatives are polynomials but fis not differentiable at the origin?
Set F(x, y) = (exsin y, excos y) and A={(x, y ) : (x, y)6= (0,0)}.
6. (cont.) At what points does Fsatisfy the conditions of Inverse
Function Theorem?
7. (cont.) Show that the image of Fis A.
8. (cont.) Does F−1:A→R2exist?
Consider a system of equations
u=x+y+z
v=x2+y2+z2
w=x3+y3+z3
9. Show that the map (x, y, z)7→ (u, v , w) is invertible near (0,1,2).
10. (cont.) From the previous problem it follows that the inverse map
x=x(u, v, w)
y=y(u, v, w)
z=z(u, v, w)
is defined near (3,5,9). Calculate ∂x
∂u at (3,5,9).
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