IMPORTANT UNITS
NEED TO REMEMBER Time: hour (h), minute (min), second (s) Length: metre (m), centimetre (cm), milimetre (mm), foot (ft), inch (in), yard (yd), mile (mile) Mass: kilogram (kg), gram (g), ounce (oz), pound (lb) Time 1 h = 3,600 s = 60 min 1
Chapter 5
Exponential and Logarithmic Functions
Chapter 5 Prerequisite Skills
Chapter 5 Prerequisite Skills
Question 1 Page 250
a)
b)
c) Answers may vary. For example:
The equation of the inverse is y = log 2 x since 2
Chapter 5 Prerequisite Skills
log 2
Unit 3 Rates of Change
Day 8 Properties of Limits
Properties of Limits
For any real number a, suppose f and g both have limits at x = a.
1. lim k = k for any constant k
x a
2. lim x = a
x a
3. lim[ f ( x ) g( x )] = lim f ( x ) lim g( x )
x a
x a
(
x a
)
Unit 4 Derivatives
Day 4 The Product Rule
How might you differentiate f ( x ) = ( 3x 2 1)( x 3 + 8) ? Do you simply differentiate the contents
of each bracket and then multiply? Lets consider an example to see if this could be so.
Example 1
Let h ( x ) =
Unit 3 Rates of Change
Day 9 Properties of Limits (contd)
Consider the following example.
In the example above, we have to consider the limit from two sides: negative and positive
(alternatively, left and right).
Left-hand Limit
Right-hand Limit
lim f ( x
Unit 4 Derivatives
Day 2 Derivatives of Polynomial Functions
Recall: the derivative of a function at a certain point is the slope of the tangent to the graph of the
function at that point. Consider the function f(x) = a, where a is a constant.
Ask yoursel
Unit 3 Rates of Change
Day 1 The Slope of a Tangent
What is calculus?
Calculus relies heavily upon two techniques: differentiation and integration. Our first step
on the calculus journey is to investigate the role of the slope of the tangent at a given po
Unit 4 Derivatives
Day 6 The Chain Rule
The Chain Rule allows us to find the derivative of a composite function h ( x ) = f ( g( x ) in terms
of the derivatives f and g .
The Chain Rule
If f and g are functions having derivatives, then the composite funct
Unit 4 Derivatives
Day 5 The Quotient Rule
There are two ways to differentiate a rational.
Example 1
2x
If h ( x ) =
, find h' ( x ) .
x+3
Solution 1
1
Rewrite h () = (2x )( x + 3) , and then use the Product Rule:
x
1
h'( x) = 2( x + 3) + (1)( x + 3)
=
=
Unit 3 Rates of Change
Day 2 The Slope of a Tangent (contd)
Today we consider the slope of a tangent at an
arbitrary point.
In the diagram on the left, the slope of secant PQ is:
y f ( a + h ) f ( a)
=
x
a+ ha
f ( a + h ) f ( a)
=
h
Our goal is to make th
Unit 3 Rates of Change
Day 6 The Limit of a Function
x3
as x
x 4x + 3
approaches a value of 3. To help you visualize
the scenario, a sketch of this function is shown
on the right.
Consider the function f ( x ) =
x<3
2.5
2.9
2.99
2.999
2.9999
2
f(x)
0.6666
Math1LS3Intro
Sept.11th2016
Material covered in the course (selection from the following chapters):
* Models and Functions (Chapter 1)
* Elementary functions and models (Chapter 2)
* Discrete-time dynamical systems (Chapter 3)
* Limits and basic notions o
9/3/14
Life Science
is a systematic study of living organisms
Botany
Zoology
Microbiology
Biochemistry
Biophysics
Molecular and
Cellular Biology
[no need to copy theses slides; they will be posted on Math
1LS3 web page]
Math 1LS3/1LT3
There are three sect
Assignment 5.5
Section 1.2 (geese) 0.1 (elephants) Conversion between units
For conversion factors, consult the last page of this assignment
1. (a) The average volume of an adult human eyeball is 5.5 cm3 . What is its volume
expressed in in3 (cubic inches
9/3/14
Ebola virus (EBOV) outbreak in West Africa
Research needed to understand the spread of infection,
so that it can be controlled effectively
new
research
since
previous
models
are not
adequate
Research goal: study infections in three countries
Guinea