Determinants
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
narr
DERIVATIVE
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
appears straigh
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.
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5
4? [email protected]
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6? "[email protected] &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 follo
Row Exchanges
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 c
More Boxes
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 o
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 valu
Implicit Differentiation
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 referr
A Menagerie of Curves
Classified
by the
Signs
of their
First and Second
Derivatives
0 w 0 ww
0 ww !
0 w 0 ww
1
0
3
0
0
2
-1
-2
2
w
0
0
2
1
-1
-2
0
0
-1
2
0w !
0
-0.5
0
1.5
min
1
-1
0
0.5
1
-2
-1
1
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 t
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,
Hlccr.h5
Homogeneous Linear Differential Equations
with Constant Coefficients
We start off with the simplest homogeneous linear differential equations with constant
coefficients, those which are first
The Exponential Function
Exponential functions are functions like #B $B . You get the idea, the general form is
C +B
You should realise that #B , for example, is very different from B# This reflects a
Deeper into Non-homogeneous Equations
Well start right off with an example:
Find a solution to
Cw
#C $/#B
Normally we would try a solution like the right hand side, C E/#B ,
and plug this into our equ
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
li
Curve Plotting
HOW TO PLOT THE GRAPH OF y f x
Example: f x
B# &B"
x 1 # = "
$
B"
$
B"#
1)
1) Asymptotes
a. Horizontal: If 0 L c, a standard constant, then the line
y c is a horizontal asymptote.
Dra
Matrix, Matrices
As is common in multidimensional calculus, points in space are represented by vectors. For
example
5
0
F
9
is the three-dimensional column vector that represents the point (5, 0,
th
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
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 gra
Fall 2016
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 addr
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
Topics
1 Sept. 14
BB Not
Differential Equations:
first steps
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 #