Math 417 Final Exam
Name:
Instructions: This is a take-home nal exam. There are 9 questions, each
worth 20 points, for a total of 180 points. You may use any of our class
materials (textbook, notes, homework solutions, etc.), but do not discuss the
exam with others.
Bring the completed exam to my oce, Weber
Hall 221, on December 17 between 4:10 and 6:10pm.
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1. Let U = {(x, y, z) ∈ R3 : x + y + z 6= 0}. Prove that U is open in R3 .
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2. Let A = {(x, y) ∈ R2 : x2 + y 2 ≤ 1} and dene f : A → R by
p
f (x, y) = 1 − x2 − y 2 .
Prove that f is uniformly continuous but not Lipschitz continuous.
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3. Prove that
x3 y
= 0.
lim
(x,y)→(0,0) x2 + y 4
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4. Let P1 and P2 be the rst and second order approximations of
f (x, y) = e2x+3y ,
x, y ∈ R
at the point (0, 0). Compute P1 and P2 in the following ways:
i) by using the rst and second order approximation theorems; and
ii) by using Taylor's theorem in one dimension (Theorem 8.8).
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4. (continued) Use Taylor's theorem (Theorem 8.8) to show that
f (x, y) − P2 (x, y)
= 0.
k(x, y)k2
(x,y)→(0,0)
f (x, y) − P1 (x, y)
= 0,
k(x, y)k
(x,y)→(0,0)
lim
lim
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5. Let A be a symmetric n × n matrix and Q : Rn → R the quadratic form
Q(x) = xT Ax.
Let S = {x ∈ Rn : kxk = 1} be the unit sphere and suppose Q|S attains a
maximum or minimum at v . Use Lagrange multipliers to show that Av = λv
for some λ ∈ R.
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6. Recall that a symmetric matrix is positive denite if and only if all its
eigenvalues are positive. Use this and Problem 5 to give another proof1 of
Proposition 14.16. What is the constant c in the proposition?
1 Recall
that positive denite matrices are, by convention, symmetric.
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7. Let f : [a, b] → R be dierentiable with
f (a) < 0 < f (b),
0 < m < f 0 (x) ≤ M for all x ∈ [a, b],
and let x0 ∈ [a, b]. Prove that the sequence {xn } dened by
xn+1 = xn −
f (xn )
M
converges to the unique root x∗ of f .
(Hint: Use the contraction mapping theorem.)
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8. Let f : R → R be continuously dierentiable such that f (a) = b and
f 0 (a) 6= 0. Prove that there exist neighborhoods U and V of a and b,
respectively, such that given y∗ ∈ V , the sequence dened inductively by
x0 = a and
xn+1 = xn −
f (xn ) − y∗
f 0 (a)
converges to a unique point x∗ ∈ cl U such that f (x∗ ) = y∗ .
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9. Let f (x, y) = x2 − y 2 for (x, y) ∈ R2 . Show that the hypotheses of Dini's
theorem do not hold for f at (0, 0). Can the equation f (x, y) = 0 be solved
for y as a function of x (or x as a function of y ) in a neighborhood of (0, 0)?
Why or why not?
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