2.3 Fitting the tide
Find a vertical pier at the edge of the sea, put a scale down the side, and measure the height of the water
against time and youll get a graph something like that belowdata taken in the Bay of Fundy over a 4-day
period. This region is
1.3 The log laws
10 x
Example 1. (The product law) Take the addition law for exponents:
a
b
a+b
10a+b = 10a 10b
What log law can you extract from that?
A
B
AB
log(x)
a
A
b
B
a+b
AB
Example 2. (The exponent law). If we take the product law for logs.
and se
2.5. The equation of the spring
I mount a spring with a weight on it. I tie the top of
the spring to a stick projecting out the top end of a
cupboard door, and fasten a ruler down the edge of
the door, so that as the spring oscillates, the weight
moves up
1.4 Exponential and logarithm graphs.
Example 1. Recall that
b = 2a if and only if
a = log2(b)
That tells us that the functions
f(x) = 2x and g(x) = log2(x)
are inverse functions. It also tells us that the graphs
y = 2x and y = log2(x)
are reflections of
1.1 Exponential growth and decay.
t
0
1
2
Example 1. A year ago I bought a mint condition Paul Anka 45 rpm
vinyl for $100. I went back to the dealer the other day and was told it
was worth $120. What do you think its value might be after another
year? Des
1.2 The logarithm is the index.
Example 1. Write 874 as a power of 10. That is, find x such that
874 = 10x .
Solution 1. The slow way: calculator guess and check. Start with
100 = 102
1000 = 103
Solution 2. The fast way: the LOG button.
Definition of the
2.4 Bicycle Wheel
The graph
We mount a bicycle wheel so that it is free to rotate in a vertical plane. In fact, what works easily is to put an extension on
one of the axles, and get a student to stand on one side and
hold the wheel steady while another st
2.2 The graphs of sin(x) and cos(x).
Now I am going to define the two basic trig functions:
sin(x)
and
cos(x).
P
1
x
O
sin(x) the distance from P to the horizontal axis
cos(x) the distance from P to the vertical axis
sin(x)
cos(x)
Study the diagram at the
2.7 Two wheels
y
In the diagram at the right the big circle has centre at the origin and radius 2, and the small circle has centre at (3,0) and
radius 1. Thus the small circle is tangent to the big circle at
the point (2,0). Now heres what happens. The sm
2. Trigonometric Functions
2.1 Radians
1
1
Definition
A radian is a measure of angle size. It is defined by the diagram at the right, that is, 1 radian is the angle which subtends
an arc of length 1 in a circle of radius 1.
1
1
Example 1.How big is a radi
1.6 Sunflower
In 1919 a pair of scientists in the United States grew a sunflower. Over a 12week period they made careful weekly measurements of its height, and attempted to fit their data to a simple model. Then they published their results in
the Proceed