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
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 exp
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
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 tha
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
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
Solut
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
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 t
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 ci
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 i
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