Calculus 140, section 3.3 Derivatives of Combinations of Functions
notes prepared by Tim Pilachowski
What we have so far:
The (first) derivative of a function f is given by f ( x ) = lim
tx
f (t ) f ( x )
f (x + h ) f (x )
. [section 3.1]
= lim
h0
tx
h
Th
1. A for loop is when you tell the program how many times you want it to run through and will
continually increase/increment until the condition returns false. This is best to use when you
know when you know when the program should stop. A while loop is w
Prescription:
- Ethics never defined
- Underlying meaning to what you are doing
- Client tells you what to do - power dynamic
Driverless Cars:
- Cant act instinctually
- Who is responsible if there is an accident
- Cant code dont cause harm
- Needs to be
An information science degree focuses on the interaction between people and information
systems. As technology continues to play a big role in peoples lives, having a degree in this
field almost guarantees a career that will succeed. I eventually aim to h
Functions allow programmers to minimize the amount of code used in order to make
their code/program more efficient. Decomposition allows an individual to divide code into
smaller parts so that it makes it easier to solve a problem. Not only does decomposi
#include "library.h"
void backgroud(const int w, const int h)
cfw_
make_window(w, h);
int x = random_in_range(0, 24);
if (x<7) | (x>19)
cfw_
set_pen_color(color:black);
fill_rectangle(0, 0, w, h);
set_pen_color(color:white);
move_to(5 * w / 6, h / 7);
set
Probability, Statistics,
and Random Processes
for Electrical Engineering
Third Edition
Alberto Leon-Garcia
University of Toronto
Upper Saddle River, NJ 07458
Contents
Preface
ix
CHAPTER 1
1.1
1.2
1.3
1.4
1.5
1.6
CHAPTER 2
2.1
2.2
*2.3
2.4
2.5
2.6
*2.7
*2.
Math 562
Wed Mar 28: Exam 2
Spring 2012
Drew Armstrong
There are 3 problems and 4 pages. This is a closed book test. Any student caught cheating
will receive a score of zero.
1. Let R be a commutative ring with 1 and let I R be a maximal ideal.
(a) Given
Math 562
Exam 1 Fri Feb 17
Spring 2012
Drew Armstrong
There are 4 problems and 4 pages. This is a closed book test. Any student caught cheating will
receive a score of zero.
1. Let R be an integral domain and consider a, b, c, p R.
(a) [1 point] If a = 0,
Math 562
Exam 3
Friday, April 27
Drew Armstrong
There are 3 problems and 4 pages. This is a closed book test. Any student caught cheating
will receive a score of zero.
1. Let F K be a nite-dimensional (hence algebraic) extension of characteristic 0 elds.
Math 461
Impossible Constructions
Spring 2015
Drew Armstrong
In class we proved that 2 is not a rational number (the edge and diagonal of a square
are incommensurable). This crisis of incommensurables forced the Greeks to base their
mathematics on the con
Math 461
Exam 1 (Wed Feb 18)
Spring 2015
Drew Armstrong
There are 4 problems, each with 3 parts. Each part is worth 2 points, for a total of 24 points.
If any two exams are submitted with copied answers then both exams will receive 0 points.
1. Division W
Math 461
The Cubic Formula
Spring 2015
Drew Armstrong
The full cubic formula is too complicated to write on the board, so Ive typed it here.
Recall that the solution to the depressed cubic
x3 + px + q = 0
is given by Cardanos formula
p
q
q 2
+
+
2
2
3
Now
Math 461
Exam 2 (Wed Mar 25)
Spring 2015
Drew Armstrong
There are 3 problems with 12 parts. Each part is worth 2 points for a total of 24 points. If
two exams are submitted with copied answers then both exams will receive 0 points.
1. Complex Numbers.
(a)
Math 461
Rings and Fields
Spring 2015
Drew Armstrong
I will avoid stating the formal denition of rings and elds in the lecture, but here it is
in case you want to know.
Denition of a Ring. A ring is a set R together with two binary operations
+:RRR
and
:R
Math 562
Homework 6
Spring 2012
Drew Armstrong
Problems on Galois Connections
Let S, T be sets and let R S T be a relation (we will write aRb to denote the statement
(a, b) R). For all subsets A S and B T let us write
A := cfw_t T : aRt a A T,
B := cfw_s