Math 417 Midterm 3
1. Find the matrix associated with the linear map
L(x, y) = (x + y, 3x + 5y, 2x − 4y),

Solution.

1 1
The matrix is A = 3 5 .
2 −4
1
(x, y) ∈ R2 .
2. Dene f : R3 → R3 by
f (x, y, z) = (x2 , x + 3y, xyz).
Is f invertible in a neighborhood of (1, 2, 3)? Justify your response.


2 0 0
Solution. Yes, since f is continuously dierentiable and Df (1, 2, 3) = 1 3 0 is
6 3 2
nonsingular: its determinant is 12 6= 0.
2
3. Let f : Rn → R be continuously dierentiable, let x, v ∈ Rn , and dene φ(t) = f (x+tv)
for t ∈ R. Using the chain rule, show that
φ0 (t) = h∇f (x + tv), vi.
Solution.
Let ψ(t) = x + tv . By the chain rule,
φ0 (t) = D(f ◦ ψ)(t) = Df (ψ(t))Dψ(t) = Df (x + tv)v = h∇f (x + tv), vi.
3
4. Let φ : R2 → R be continuously dierentiable. Explain why at each point (x, y) ∈ R2
the hypotheses of the inverse function theorem cannot hold for the function f (x, y) =
(φ(x, y), φ(x, y)2 ).
Solution.
The hypothesis cannot hold because the image of f is one-dimensional:
D
φ(x,
y)
D
φ(x,
y)
1
2
≡ 0.
det Df (x, y) = 2φ(x, y)D1 φ(x, y) 2φ(x, y)D2 φ(x, y)
4
5. Let f (x, y) = (x, 2y) for (x, y) ∈ R2 . Prove that f is stable.
Solution.
Note that
p
p
(x − r)2 + (2y − 2s)2 = (x − r)2 + 4(y − s)2
p
≥ (x − r)2 + (y − s)2 = k(x, y) − (r, s)k.
kf (x, y) − f (r, s)k =
Thus, f is stable with stability constant c = 1.
5