Math 411Fall 2011Boyle Exam 1
1. (10 points) A metric space is a set X together with a realvalued function d dened on pairs of points (x, y ) from X . State the
axioms that d must satisfy for X to be a metric space.
2. (5 points) What is the Cauchy-Schwar
Math 411 Exam 2 Sample
1. Suppose the temperature of the plane at (x, y ) is given by the function f (x, y ) = xy + x2 y y .
At a given instant you are at x = (1, 3) and travelling in the direction h = (1, 1). Calculate
and describe the physical relevence
Math 411 Exam 1 Sample
1. Prove from the -K denition that cfw_(2 + 1/k, 4 1/k 2 ) converges to (2, 4).
2. Suppose that cfw_uk and cfw_vk are sequences which converge to to u and v respectively.
(a) Prove that if k, uk vk then u v .
(b) Give an example w
Math 411 Spring 2012 Boyle Exam 1
1. (10 points) A metric space is a set X together with a real-valued
function d (the distance function) dened on pairs of points (x, y ) from X .
1. State the axioms that d must satisfy for X to be a metric space.
2. For
Math 411 Spring 2012 Boyle Final Exam
Points in Rn are column vectors (even if they are typed horizontally).
1. (a) (15 pts.) Suppose F : Rn Rm . Dene what it means for F to be
dierentiable.
(b) (30 pts.) Suppose F : Rn R all partial derivatives are conti
Math 411Fall 2011Boyle Exam 2
1. (10 points) For a positive integer n of your choice, give an
example of a function f : Rn Rn which is dierentiable but is not
C 1. You do not need to give a proof, just a correct example.
2. (20 points) Consider the follow
Math 411Fall 2011Boyle Exam 1
1. (10 points) A metric space is a set X together with a realvalued function d dened on pairs of points (x, y ) from X . State the
axioms that d must satisfy for X to be a metric space.
2. (5 points) What is the Cauchy-Schwar
Math 411Spring 2012Boyle Exam 4
1. (20 points) Suppose v1 , v2 , . . . is a convergent sequence of points in Rn
and E = cfw_xn : n N. Prove that E has Jordan content zero in Rn .
2. (20 points) For 1 i n, suppose Ii is an interval [ai , bi ] and I is
the
Math 411 Exam 3 Sample
1. Dene f : R2 R2 by f (x, y ) = (xy x2 , 4 xy ).
(a) Find all points (x0 , y0 ) for which the IFT guarantees the local invertibility of f .
(b) For every such (x0 , y0 ) nd a linear approximation to the inverse.
2. Show the set S =
Math 411: Homework for Chapter 10 Solutions
1. In R2 it is not necessarily true that |u v | |u|. Draw two pictures, one in which it is true and
the other in which it is false. Give some measurements for verication. You can use inches as vector
lengths, fo
Math 411: Homework for Chapter 19 Solutions/Hints
Note: No problems will be dropped.
1. Let R = [1, 3] [2, 7] R2 and suppose you wish to evaluate
x + 2y xy dA.
R
(a) Explain why the hypotheses for Fubinis Theorem (in the plane) are satised.
(b) Calculate
Math 411: Homework for Chapter 18 Solutions
1. (a) We know f is integrable over R because f is continuous on R.
j
i
(b) Due to the denition of f , for any small rectangle Rij = [xi1 , xi ] [yj 1 , yj ] = [ i1 , k ] [ j 1 , k ],
k
k
the minimum occurs in t
Math 411: Homework for Chapter 17 Solutions
1. The InvFT applies if the derivative (matrix) at a point is invertible. For a function R R we have
Dt(x) = t (x) = 3x2 4x and so t (5) = 55 which is nonzero, hence invertible (the reciprocal). The
linear appro
Math 411: Homework for Chapter 16 Solutions
1. (a) We have Df (x) = f (x) = 3x2 3. This equals 0 only when x = 1, so it is at these points that
the InvFT fails. Hence it applies everywhere else.
(b) Preliminary Note: It may be useful to think in nonfamili
Math 411: Homework for Chapter 15 Solutions
1. (a) Suppose we wish to know where the point (x0 , y0 , z0 ) gets mapped. We nd the line
through this point and perpendicular to the plane and then we see where it hits the
plane. Since the direction vector fo
Math 411: Homework for Chapter 14 Solutions
1. (a) We know that the tangent plane is the rst-order approximation. The rst-order approximation at x0 is
given by
g () = f (0 ) +
x
x
f (0 ), ( x0 ) .
x
x
In this situation we have x = (x, y ) and x0 = (0, 0).
Math 411: Homework for Chapter 13 Solutions
1.
(By contradiction.) Assume rst that m + n > 2. Since m and n are positive integers, this means
theyre both equal to 1, so were then examining the limit
lim
f (x, y ) where f (x, y ) = x2xyy2 .
+
(x,y )(0,0)
Math 411: Homework for Chapter 11 Solutions
1. Observe that
f () = Projv u
u
uv
=
v
vv
1 ()1 () + . + n ()n ()
u
v
u
v
=
v
1 ()1 () + . + n ()n ()
v
v
v
v
= (1 (), ., n ()
v
v
= (1 (), ., 1 ()
v
v
where =
1 ()1 ()+.+n ()n ()
u
v
u
v
1 ()1 ()+.+n ()n () .
Math 411Spring 2012Boyle Exam 3
Points in Rn are column vectors (even if they are typed horizontally).
1. (25 points)
1. (5 points) State the Implicit Function Theorem.
2. (20 points) Prove the nal statement of the theorem (this is the equation
relating c
Math 411Spring 2012Boyle Exam 2
1. (10 points) Compute
x2 y + y 2 x
lim
(x,y )(0,0) x2 + y 2
or show the limit doesnt exist.
2. (20 points) Dene F : R2 R by the rule
F (x1, x2) = 3x2 + (x1)2 + (x2)2 sin(x1/x2)
=
0
if x2 = 0
if x2 = 0 .
Determine whether F
Math 411 Spring 2012 Boyle Final Exam Solutions
Points in Rn are column vectors (even if they are typed horizontally).
1. (a) (15 pts.) Suppose F : Rn Rm . Dene what it means for F to be
dierentiable.
SOLUTION.
For every x in Rn , there exists an m n matr
Math 411 Spring 2012 Boyle Final Exam
Points in Rn are column vectors (even if they are typed horizontally).
1. (a) (15 pts.) Suppose F : Rn Rm . Dene what it means for F to be
dierentiable.
(b) (30 pts.) Suppose F : Rn R all partial derivatives are conti
Math 411Spring 2012Boyle Exam 2 Solutions
1. Compute
x2 y + y 2 x
lim
(x,y )(0,0) x2 + y 2
or show the limit doesnt exist.
Solution.
Substituting polar coordinates, we get
x2y + y 2x (r cos()2(r sin() + (r sin()2(r cos()]
0 2
=
x + y2
r2
= r cos2() sin()
Math 411Spring 2012Boyle Exam 4
1. (20 points) Suppose v1 , v2 , . . . is a convergent sequence of points in Rn
and E = cfw_xn : n N. Prove that E has Jordan content zero in Rn .
2. (20 points) For 1 i n, suppose Ii is an interval [ai , bi ] and I is
the
Math 411Spring 2012Boyle Exam 3
Points in Rn are column vectors (even if they are typed horizontally).
1. (25 points)
1. (5 points) State the Implicit Function Theorem.
2. (20 points) Prove the nal statement of the theorem (this is the equation
relating c
Math 411Spring 2012Boyle Exam 2
1. (10 points) Compute
x2 y + y 2 x
lim
(x,y )(0,0) x2 + y 2
or show the limit doesnt exist.
2. (20 points) Dene F : R2 R by the rule
F (x1, x2) = 3x2 + (x1)2 + (x2)2 sin(x1/x2)
=
0
if x2 = 0
if x2 = 0 .
Determine whether F
Math 411Fall 2011Boyle Exam 2
1. (10 points) For a positive integer n of your choice, give an
example of a function f : Rn Rn which is dierentiable but is not
C 1. You do not need to give a proof, just a correct example.
2. (20 points) Consider the follow