Introduction to Differential Calculus.pdf

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Mathematics Learning Centre
Introduction to Differential Calculus
Christopher Thomas
c 1997
University of Sydney
Acknowledgements
Some parts of this booklet appeared in a similar form in the booklet
Review of Differen-
tiation Techniques
published by the Mathematics Learning Centre.
I should like to thank Mary Barnes, Jackie Nicholas and Collin Phillips for their helpful
comments.
Christopher Thomas
December 1996
Contents
1 Introduction
1.1
An example of a rate of change: velocity . . . . . . . . . . . . . . . . . . .
1.1.1
1.1.2
1.2
Constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Non-constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
1
3
4
6
6
7
9
9
Other rates of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 What is the derivative?
2.1
2.2
Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The derivative: the slope of a tangent to a graph
. . . . . . . . . . . . . .
3 How do we find derivatives (in practice)?
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Derivatives of constant functions and powers . . . . . . . . . . . . . . . . .
Adding, subtracting, and multiplying by a constant . . . . . . . . . . . . . 12
The product rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
The Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
The composite function rule (also known as the chain rule) . . . . . . . . . 15
Derivatives of exponential and logarithmic functions . . . . . . . . . . . . . 18
Derivatives of trigonometric functions . . . . . . . . . . . . . . . . . . . . . 21
24
4 What is differential calculus used for?
4.1
4.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Optimisation problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2.1
4.2.2
4.2.3
Stationary points - the idea behind optimisation . . . . . . . . . . . 24
Types of stationary points . . . . . . . . . . . . . . . . . . . . . . . 25
Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 The clever idea behind differential calculus (also known as differentiation
from first principles)
31
6 Solutions to exercises
35
Mathematics Learning Centre, University of Sydney
1
1
Introduction
In day to day life we are often interested in the extent to which a change in one quantity
affects a change in another related quantity. This is called a
rate of change.
For example,
if you own a motor car you might be interested in how much a change in the amount of
fuel used affects how far you have travelled. This rate of change is called
fuel consumption.
If your car has high fuel consumption then a large change in the amount of fuel in your
tank is accompanied by a small change in the distance you have travelled. Sprinters are
interested in how a change in time is related to a change in their position. This rate
of change is called
velocity.
Other rates of change may not have special names like fuel
consumption or velocity, but are nonetheless important. For example, an agronomist
might be interested in the extent to which a change in the amount of fertiliser used on a
particular crop affects the yield of the crop. Economists want to know how a change in
the price of a product affects the demand for that product.
Differential calculus is about describing in a precise fashion the ways in which related
quantities change.
To proceed with this booklet you will need to be familiar with the concept of the
slope
(also called the
gradient)
of a straight line. You may need to revise this concept before
continuing.
1.1
1.1.1
An example of a rate of change: velocity
Constant velocity
Figure 1 shows the graph of part of a motorist’s journey along a straight road. The
vertical axis represents the distance of the motorist from some fixed reference point on
the road, which could for example be the motorist’s home. Time is represented along the
horizontal axis and is measured from some convenient instant (for example the instant an
observer starts a stopwatch).
Di stance
(metres)
300
200
100
2.00
4.00
6.00
8.00
t ime (seconds)
Figure 1:
Distance versus time graph for a motorist’s journey.
Mathematics Learning Centre, University of Sydney
2
Exercise 1.1
How far is the motorist in Figure 1 away from home at time
t
= 0 and at time
t
= 6?
Exercise 1.2
How far does the motorist travel in the first two seconds (ie from time
t
= 0 to time
t
= 2)?
How far does the motorist travel in the two second interval from time
t
= 3 to
t
= 5? How far
do you think the motorist would travel in any two second interval of time?
The shape of the graph in Figure 1 tells us something special about the type of motion
that the motorist is undergoing.
The fact that the graph is a straight line tells us that the
motorist is travelling at a constant velocity.
•
At a constant velocity equal increments in time result in equal changes in distance.
•
For a straight line graph equal increments in the horizontal direction result in the
same change in the vertical direction.
In Exercise 1.2 for example, you should have found that in the first two seconds the
motorist travels 50 metres and that the motorist also travels 50 metres in the two seconds
between time
t
= 3 and
t
= 5.
Because the graph is a straight line we know that the motorist is travelling at a constant
velocity. What is this velocity? How can we calculate it from the graph? Well, in this
situation, velocity is calculated by dividing distance travelled by the time taken to travel
that distance. At time
t
= 6 the motorist was 250 metres from home and at time
t
= 2
the motorist was 150 metres away from home. The distance travelled over the four second
interval from time
t
= 2 to
t
= 6 was
distance travelled = 250
−
150 = 100
and the time taken was
time taken = 6
−
2 = 4
and so the velocity of the motorist is
velocity =
distance travelled
250
−
150
100
=
=
= 25 metres per second.
time taken
6
−
2
4
But this is exactly how we would calculate the slope of the line in Figure 1. Take a look
at Figure 2 where the above calculation of velocity is shown diagramatically.
The slope of a line is calculated by vertical rise divided by horizontal run and if we were
to use the two points (2, 150) and (6, 250) to calculate the slope we would get
slope =
To summarise:
The fact that the car is travelling at a constant velocity is reflected in the fact that the
distance-time graph is a straight line. The velocity of the car is given by the slope of this
line.
rise
250
−
150
=
= 25.
run
6
−
2

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