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MTH 254H Homework 3 Solutions
Section 2.3 # 2: Prove that the function f : Rn R given by
f (x = x is continuous.
Strategy: I will use the denition of continuity to prove this
statement. Since the domain of f is all of Rn , I begin by considering
an arbitr
MTH 254H Homework 4 Solutions
Section 3.1, # 1 (b,f): Calculate the partial derivatives of each function.
Note: I will use coordinates rather than vectors, so I will write f (x, y )
rather than f ([x, y ]T ). I will also denote f by fx . These conventions
Solutions to Homework # 5
Section 2.2 # 12a: Prove that every convergent sequence is a Cauchy
sequence.
Proof: Suppose that cfw_xn is a sequence which converges to a Rk . Let
> 0. Choose N so that if n > N , then xn a < /2. Then, by the
triangle inequali
Solutions to Homework 6
Exercise 3.2 # 17: Prove that f (x, y ) = y if x = 0, f (0, y ) = 0 is
continuous at (0, 0). Is f dierentiable at (0, 0)?
Solution: f is continuous at (0, 0) if and only if the limit of f as
(x, y ) (0, 0) is equal to f (0, 0) = 1.
Solutions to Homework 7
Exercise #3 in section 5.2: A rectangular box is inscribed in a hemisphere
of radius r. Find the dimensions of the box of maximum volume.
Solution: The base of the rectangular box lies in the plane that contains
the base of the hem
Solutions to Homework 8
Express the number 12 as the sum of three positive real numbers x, y, z so
that xy 2 z 2 shall be maximum. Is there a maximum if each of x, y, z is
permitted to be any real number?
Solution: Let g (x, y, z ) = x + y + z 12 and let
Solutions to Homework 9
Read the proof of proposition 1.7 on p. 271 (section 7.1). Write a more
detailed proof. In particular, state the dention of uniformly continuous
and explain the comment whose sidelengths are less than / n .
Solution: Here are some
Solutions to Homework 10
Section 7.2, exercise # 1 (b,d):
(b) Compute the value of
R = [1, 3] [2, 4].
Rf
dV , where f (x, y ) = y/x and
Solution: Since f is continuous over R, f is integrable over R. Let
x [1, 3]. Since f is continuous over R, the functio
Solutions to Homework 11
Read the statement of Proposition 5.4 of Chapter 3, Section 5. Write a
summary of the proof. Comment on the following details:
Does the proof work if g is piecewise C 1 ? Or did the proof assume that
g C1?
The short answer is, yes