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
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 norm
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 u
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 tha
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 conve
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
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 eleme
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
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. Th
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. Th
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 abl
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. Thi
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 geometri
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 deriv