Determinants have been known and studied for 300 years. Today, however, there is far less
emphasis on them than in the past. In modern mathematics, determinants play an important but
narrow role in theory and almost no role at all in computat
In this lesson we're going to try to understand the meaning of the mysterious phrase
A point is a smooth point of a graph
if an infinite zoom-in (at that point) of the graph
clooks like a straight lined.
That is well translate
1. Gaussian Elimination
The central problem of linear algebra is the solution of simultaneous linear equations.
We begin with a simple system of 3 equations and 3 unknowns.
2? @ A
6? "7@ &A $
The problem is to find the unknown values of ?, @, a
Non-homogeneous Linear Differential Equations
with Constant Coefficients
To define what a NON-homogeneous linear differential equation is well first need
some technical terminology. Consider the following differential equation
&Cw %C $B (
now everything t
We now return to the question of what happens when we run into zero pivots.
Example 1: We first consider the system
u # v $w "
2u NULL *w 5
2u 'v (w 4
Using Gaussian elimination on the corresponding array
The one compartment diagrams are nice as far as they go but, as Ive said before,
usually there are many things happening at the same time, often interacting with each other.
This leads us on to two compartment problems which might look somethin
To Infinity and Beyond!
Here are our new numbers , they will be called Non-Standard Numbers and the old numbers
are called standard numbers.
Infinite numbers- These numbers are bigger in absolute value than any standard number
that is, the positive Infini
Here is the graph of
B# C# "
which you should recognize as
the circle of radius 1 centered
at the origin. Its a common
curve something you see every
day, familiarly referred to as
The Unit Circle. Notice that
although the curv
Systems of Differential Equations
Up to now we have considered only differential equations with one variable function with respect
to time, but in real life there are usually many things going on at the same time. This leads us to
consider differential eq
Systems with Many Solutions
Consider the single linear equation in one unknown ax b. Most people would
immediately say the solution is x b/a. But in fact there are three possibilities:
Case 1: If a 0, then there is one solution. For example 2x 6, so x 3.
Homogeneous Linear Differential Equations
with Constant Coefficients
We start off with the simplest homogeneous linear differential equations with constant
coefficients, those which are first order. What are first order linear homogeneous equatio
The Exponential Function
Exponential functions are functions like #B $B . You get the idea, the general form is
You should realise that #B , for example, is very different from B# This reflects a major
difference between Calculus and all the mathemat
Deeper into Non-homogeneous Equations
Well start right off with an example:
Find a solution to
Normally we would try a solution like the right hand side, C E/#B ,
and plug this into our equation to find E:
That's the Limit
You will see ( or you have already seen ) in many calculus books references to "limits."
Roughly these are the usual calculus version of standard parts. They occur in expressions
This is really something quite simple. All it
HOW TO PLOT THE GRAPH OF y f x
Example: f x
x 1 # = "
a. Horizontal: If 0 L c, a standard constant, then the line
y c is a horizontal asymptote.
Draw this line
L # &L"
a. f L
L # &L"
H # 2H 1
As is common in multidimensional calculus, points in space are represented by vectors. For
is the three-dimensional column vector that represents the point (5, 0,
We can also write our vectors in
Matrices and Differential Equations
Since were going to write down differential equations involving matrices we have to define
the derivative of a matrix. Actually all we're going to differentiate are vectors.
Suppose \ is a vector whose entries are funct
a) A drug distributed throughout the blood plasma is eliminated at a rate proportional to the
amount in the blood :
ans: Bw 5B
b)Radioactive iodine is absorbed by the thyroid at .01 mg/hr where it gradually decays.
ans: Bw !" 5B
c) A drug is transfused in
MATH 1251 Calculus and Differential Equations for Biology I
Instructor Name: David P. Lang
Office Location: 538 NIghtingale Hall
Office Hours: MWR 12:00 noon 1:15 pm, 3:00 4:15 pm
Email address: [email protected]
Calculus & Its A
Calculus and Differential Equations for Biology I
8 Quizzes and Topics for Coverage from Blackboard Notes and/or Textbook
Quiz# Date (tentative) Sections of Textbook and/or BB
1 Sept. 14
BB Notes 1-2
Hyperreal Numbers and Standard Parts
2 Sept. 22
The time has come to learn to solve differential equations. What is a differential
equation? Simply an equation involving a derivative, Cw . Like, for example, Cw #B. And
what, exactly, is a solution? A solution is not,