Math 425: Homework 2
Mary Radclie
due 29 Jan 2014
In Rudin: Page 114, problems 18, 25; Page 196, problems 4, 5, 6b
Also, complete the following.
1. Let f and g be analytic at 0, with radii of converge
Worksheet 24
1. Find the derivative, Df , of f (x, y) = (3 sin(x), y 2 + x).
2. Find the tangent plane to the surface z = ln(x2 + y 2 ) at the point (2, 1, ln 5).
3. Find a vector is perpendicular to
Worksheet 4
1. Match the surfaces with the verbal description of the level curves by placing the letter
of the verbal description to the left of the letter of the surface.
p
z = x2 + y 2
z = 2x + 3y
Worksheet 5
1. Find the point P where the line x = 1 + t, y = 2t, z = 3t intersects the plane
x+yz =2
2. Find the equation of a sphere if one of its diameters has endpoints: (12, 6, 10) and
(4, 14, 2)
Worksheet 7
1. Find the derivative of the vector function r(t) = ln(9 t2 )i +
13 + tj + 6e1t k
2. Find the derivative of the vector function:
r(t) = ta (b + tc)
where
a = h4, 3, 4i, b = h3, 2, 2i, c =
Worksheet 6
1. Use three different methods for determining if the following points are colinear: (1, 2, 3),
(3, 4, 4), and (5, 4, 0).
2. Find the shortest distance between the point (1, 1, 1) and the
Worksheet 25
1. What is the minimum/maximum distance from the origin to the ellipse x2 + 2y 2 +
3z 2 = 6
2. Find the linearization of f (x, y) = xy at the point (1, 2).
3. Find a unit tangent vector t
Worksheet 9
1. The position of a particle at time t is
(t, t2 , t3 ) = (x(t), y(t), z(t)
(a) Find the speed of the particle at time t = 1
(b) Find the angle between the x-axis and the line connecting
Worksheet 8
1. A plane climbs at a steady rate. Its airspeed is 400 mph. Its ground speed is 360 mph.
It moves over the ground in a true Northerly direction (so for example, it flies directly
over a N
Worksheet 11
1. Let A =
4 5 9
8 8 4
Define
T : R3 R2 as T (x) = Ax. Find the images of
the linear transformation
4
a
u = 3 and v = b under T .
4
c
T (u) =
T (v) =
1 4
2. A =
Find A1 .
5 1
4 4 5
15
Recitation 3
1. Let L be the line passing through the origin and the point (2, 5), and let M be the line
passing through the points (3, 2) and (5, 3). Find the smaller angle between L and
M.
2. Find a
The Three Possibilities
Summary
No solution
Infinitely many solutions
Unique solution
Existence of Infinitely Many Solutions
More unknowns than equations
Missing variable
Zero equations
Exam
Math 425: Homework 1
Mary Radclie
due 13 Jan 2014
In Rudin: Page 114, problems 1, 2, 5, 8, 9, 11, 14
1. Let f be dened for all real x, with |f (x) f (y)| (x y)2 for all real x
and y. Prove that f is c
Math 425: Homework 2
Mary Radclie
due 22 Jan 2014
In Rudin: Page 114, problems 22, 27, 28
Also, complete the following.
1. Let
f (x) =
e1/x
0
2
x=0
x=0
Prove that f has derivatives of all orders at x
Name:
Math 425/575: Midterm
7 February 2014
Turn o and put away your cell phone.
No notes or books are permitted during this exam.
No calculators or any other devices are permitted during this exam.
R
Math 425: Homework 5 Solutions
Mary Radclie
due 19 Feb 2014
In Rudin:
16. Show that the continuity of f at the point a is needed in the inverse
function theorem, even in the case n = 1: if f (t) = t +
Math 425: Homework 4
Mary Radclie
due 5 Feb 2014
In Rudin: Page 239, problems 5, 6, 7, 10, 13
Also, complete the following:
1. Let A be an m n matrix. Prove that A
i,j
a2
ij
1/2
.
2. (a) Let fi : (a,
Math 425: Homework 1
Mary Radclie
due 13 Jan 2014
In Rudin: Page 114, problems 1, 2, 5, 8, 9, 11, 14
Also, complete the following.
1. Let f be a dierentiable function on (a, b), where f is bounded. Pr
Math 425: Homework 6
Mary Radclie
due 26 Feb 2014
In Rudin: Page 239, problems 20, 24, 25, 26, 29
Also, complete the following:
1. Suppose that f : R3 R, and a = (a1 , a2 , a3 ) has f (a) = 0. Suppose
Math 425: Homework 3
Mary Radclie
due 29 Jan 2014
In Rudin: Page 114
18. Suppose f is a real function on [a, b], n is a positive integer, and f (n1)
exists for every t [a, b]. Let , , P be as in Taylo
Math 425: Homework 5
Mary Radclie
due 19 Feb 2014
In Rudin: Page 239, problems 16, 17, 21, 23
Also, complete the following:
1. (a) Let f = (f1 , f2 , . . . , fn ) be a vector valued function dened on
Math 425: Homework 7
Mary Radclie
due 5 Mar 2014
In Rudin:
4. if f (x) = 0 for all irrational x, f (x) = 1 for all rational x, prove that
f R on [a, b] for any [a, b].
/
Proof. Note that for any parti
Math 425: Homework 2
Mary Radclie
due 22 Jan 2014
In Rudin: Page 114, problems 22, 27, 28
22. Suppose f is a real function on (, ). Call x a xed point of f if
f (x) = x.
(a) If f is dierentiable and f
Math 425: Homework 6
Mary Radclie
due 26 Feb 2014
In Rudin: Page 239
20. Take n = m = 1 in the implicit function theorem, and interpret the
theorem (as well as its proof) graphically.
Solution. Here w
Math 425: Homework 7
Mary Radclie
due 5 Mar 2014
In Rudin: Page 138: 4, 5, 8, 11, 12
Also, complete the following:
1. (a) Let f : [0, 1] R have the property that for all > 0, cfw_x | |f (x)| >
is nit
Math 425: Homework 4 Solutions
Mary Radclie
due 5 Feb 2014
In Rudin: Page 239
5. Prove that to every A L(Rn , R1 ) corresponds a unique y Rn such that
Ax = x y. Prove also that A = y 2 .
Proof. Let yi
Discrete Dynamical Systems
Suppose that A is an n n matrix and suppose that x 0 is a vector in n . Then
x 1 Ax 0
n
is a vector in . Likewise,
x 2 Ax 1
is a vector in n , and we can in fact generate an