2. THE x-y PLANE
2.1. The Real Line
When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but
also fractions, like 3 or 4 . In fact we can, in principle, plot any real number. Roughly
speaking real numbers are positive or
Math 123 2008
Tutorial Exercises for Week 6
x+1
at the point where x = 0.
x2 + 1
2. Find equations for the lines through (3, 2) and tangent to the graph of y = x.
1. Find an equation for the line normal to the graph of y =
3. Find and classify the station
9. INTEGRATION
9.1. The Area Function
The area under a graph can be a very useful tool but counting squares is far too crude a
technique to use. Like differentiation theres a technique that can take us straight from the equation
of a function to the equat
8. NEWTONS METHOD
8.1 The Geometry Behind Newtons Method
Using various algebraic techniques you can solve equations such as linear and quadratic
equations. But for most equations theres no formula that you can apply to get the exact solutions.
There is, h
10. AREAS BETWEEN CURVES
10.1. Areas between curves
So areas above the x-axis are positive and areas below are negative, right? Wrong! We
lied! Well, when you first learn about integration its a convenient fiction thats true in a sense.
But now well level
Math 123 2008
Tutorial Exercises for Week 8
1. Find
dy
:
dx
a) y = x2 2x
b) y = (2x)3x
c) y = log(log x)
Answers: a) (2x)2x + x2 2x log 2, b) (2x)3x (3 log(2x) + 3), c)
1
x log x
2. Use Newtons Method to nd x such that, to 5 decimal places,
a) x3 + x = 11
11. NUMERICAL INTEGRATION
11.1. The Trapezium Rule
It often happens that we need to find the value of a definite integral for a function whose
indefinite integral cannot be found in terms of the elementary functions we know about already.
Since a definite
1. GRAPHS AND
THE STORIES THEY TELL
1.1. The Height of Points on a Graph
Were all familiar with graphs as a way of depicting information. You have two quantities
and the graph shows how one is related to the other. For example the weekly sales of a small
5. MAXIMA AND MINIMA
5.1 Local Maxima and Minima
A function y = f(x) has a local maximum at a point when the y-value at that point is greater
than at any other point in the immediate neighbourhood. There may be larger values somewhere
else but standing at
6. MAXIMA AND MINIMA
6.1 Local Maxima and Minima
A function y = f(x) has a local maximum at a point when the y-value at that point is greater
than at any other point in the immediate neighbourhood. There may be larger values somewhere
else but standing at
7. OPTIMISATION PROBLEMS
7.1 Applied Mathematics and Modelling
While it may be interesting to develop mathematics for its own sake, for most people the
whole point of studying the subject is to be able to apply it to the real world. The Babylonians took
t
3. DIFFERENTIATION
3.1 The Slope Graph
Suppose we have a smooth curve where theres a tangent at every point. We could find the
value of the slope at every point and plot this against the x-values. This new graph would be the
slope graph corresponding to t
Math 123 2008
Tutorial Exercises for Week 10
(1) At the beginning of each year a person deposits $8000 into a
savings account, which is earning interest at 6% per annum. Set
up the sum of the geometric progression and calculate the total value of the savi
4. TANGENTS AND NORMALS
4.1 Equation of the Tangent at a Point
Recall that the slope of a curve at a point is the slope of the tangent at that point. The
slope of the tangent is the value of the derivative at that point. In computing the equation of
the t