Overview of MATH151 lectures
On this page I try to summarise the main
ideas of each lecture.
This is a guideline to the content of each lecture.
In any given year the pace of a lecture can vary, so it's possible that
either a little bit more or a little bit less is taught. Furthermore,
the schedule given below can be changed depending upon public holidays.
For example, if MATH151 has a lecture class on a Friday then a lecture
will be lost on Good Friday.
- Week 1.
- Week 2.
- Week 2:Suggested revision
- Week 3.
- Week 4.
- Week 5.
- Week 6.
- Week 7.
- Week 8.
- Week 9.
- Week 10.
- Week 11.
- Week 12.
- Week 13.
- Week 1,
Lecture 1 (Subject outline).
- You must read through the subject outline and the generic
subject outline.
The university (and I) will assume that you are familiar with the
content of these documents.
- Week 1, Lecture 2
(Chapter 0 - arithmetic of fractions).
- Additional resources for
Arithmetic of
fractions.
- You should understand how to add and subtract fractions.
2/4 + 3/5
2/7 - 5/14
Understanding the ideas of this process are important because (latter) we
will apply them to simplify expressions such as
1/(x-3) + x/(x-2)
x/(x^2+5) -1/(x-3).
- I mentioned in the lecture ``irrational numbers''. Here is a
youtube video about the
square root of 3,
which is an irrational number. (Video link supplied by
Lachlan Mitchell).
- Week 1, Lectures 3-4
(Chapter 1 : indices and surds).
- Additional resources on
Indices,
Surds.
- You need to know the rules for manipulation of indices and how
to use them.
- Remember that in a-b the minus sign
means `1 divided by' so that this expression is the same as
1/ab. (Useful Idea)
- In questions such as simplify
81/3 and 841/2
the key is to
first factorise the number. In particular, if you
are trying to simplify the square root of a number you want
to factorise using numbers such as 4, 9,
16...
- Is is often useful to write √x as
x0.5, i.e., to use fractional
indices. (Useful Idea).
- Evaluate (2m)0 +5.
- True or false? (Justify your answer).
(a2 +b2)1/2
= a +b.
- Sometimes there is more than one correct way to write an
answer. Don't assume that your answer is wrong if it is
different from the answer given in a textbook.
- (a-b)(a+b) = a2-b2.
(This is called the difference of two squares).
- Mathematicians don't like to see expressions which have
square root signs on the denominator of a fraction! To
eliminate the square root sign we have to multiply
both sides of the fraction by the conjugate.
(This reason why this is works is because we are using the
difference of two squares).
- Remove the surd from the denominator of the expression
2/(√5 +√3).
You are now able to do the practice questions at the end of chapter 1
for the topics indices and surds.
You should be doing these questions as part of your
weekly workload for this subject!
- Week 2, Lectures 1-2
(Chapter 1 : algebraic fractions, expansions, binomial expansion
and factorisation).
- Additional resources on
Algebraic Fractions.
- True or False? (Justify your answer).
x = qy^2/(q-y) = y^2/(1-y).
- Additional resources on
Expansion of algebraic factors.
- Expand (-x-1)(x-3).
- Additional resources on
Pascal's triangle.
- Write down the first four lines of Pascal's triangle and explain
how to use it.
- Expand (a+b)3.
- Expand (a-b)4.
- True or false? (Justify your answer).
(a+b)c = ac +bc.
- Additional resources on
Factorisation.
- Factorise x2 +2x -15.
- Factorise (c+1) -a(c+1).
- Factorise:
- a2 -b2.
- a2 +b2.
- a3 -b3.
- a3 +b3.
Week 2, Lecture 3 (Chapter 2 - Logarithms).
- Additional resources for
Logarithms
- Science Question. I started the lecture with a
science question.
The concentration of H+ ions in solution is related
to the pH of the solution by the formula
pH = -log10 H+.
If the pH of an aqueous solution is 3.12 what is the corresponding
concentration of H+ ions?
-
- x = logb N means that
bx = N.
- We cannot find logarithms of negative numbers
or zero!
- To solve questions such as
log3(1/9) they key is to factorise
the number. In this case
(1/9) = 3-2 so that
log3(1/9) = -2.
- You need to know the four rules for manipulation of logarithms and
how to use them. You will frequently use rules
(i)-(iii).
- logb xy = logbby.
- logb x/y =
logbby.
- logb xp = p*logbx.
Rule (iv) (the change of base rule) will only be
infrequently used.
- The following special cases are useful to know:
- logbb = 1.
- logb1 = 0.
- logbbp = p.
- logb(1/x) = -logbx.
- logbn√x =
(1/n)*logbx.
-
- Week 2, Lecture 4 (Practical Class).
- Week 2:
Suggested revision
- You should worked through practice questions on Indices, Surds
and Algebraic Fractions.
In your own time, you need to finish
the following.
- Complete the week two tutorial sheet.
(Worked solutions will be available on the e-learning site).
- Complete the week two practical class sheet.
(Look for solutions on the web forum).
- Problem Set A (section 1.6.3) on page 19.
- Problem Set B (section 1.6.4.3) on page 23.
- Problem Set C (section 1.6.5.2) on page 25.
- Multiple-Choice Questions (section 1.10.1) on page 29.
- Standard questions (section 1.10.2) on page 30.
- Practice questions (section 1.10.3) on page 32.
- Science questions (section 1.10.3) on page 32.
Anyone who finishes all these questions will have a very strong
foundation in the basic mathematics that will be used extensively
in the rest of the course.
Anyone who does none of these questions will be heading for
a fail on the course.
Chapter 2. You are strongly advised to.
- View the "Overview of Chapter 2" video. (Home Page > Unit 1).
- Work your way through the "Head Start Program"
(Home Page > Unit 1 > Chapter 2).
- Read through the "Maths for Vic 10" site.
(Look at the "Additional Resources" link on
Home Page > Unit 1 > Logarithms).
- You should make sure you can do all the problems in
Exercise 2.3.1 (page 38).
- Week 3,
Lecture 1 (Chapter 2 - logarithms).
- Additional resources for
Logarithms
- Science Question. I started the lecture with a
science question.
The concentration of H+ ions in solution is related
to the pH of the solution by the formula
pH = -log10 H+.
If the pH of an aqueous solution is 3.12 what is the corresponding
concentration of H+ ions?
- We reviewed the key properties about logarithms that are used
in solving equations.
- If logby = x then
y = bx.
- If x = y then
logbx = logb y.
- If logbx = logb y then
x = y.
- I reminded you that on the e-learning site you can generate
a multiple-choice quiz on logarithms. (You can do this for
many of the topics taught in this subject).
- We finished the lecture by using properties of logarithms to
solve the science problem that was stated at the start of the
lecture.
[H+] = 10-3.12 mol/l
= 0.00076 mol/l.
- Week 3,
Lecture 2 (Chapter 3 - Function Notation).
- Additional resources on
Functions.
- Think of a function as a rule
for converting an input into an output.
Celsius/Fahrenheit example.
- Composing functions. What does the notation
f(g(x)) mean?
- Week 3, Lecture 3 (Chapter 3 - Function Notation).
- Additional resources on
Functions.
- Application of function composition to converting scales.
- Finding zeros or roots of equations
- Definition of linear and quadratic functions.
- Roots of linear and quadratic functions - the
quadratic formula.
- Functions are not necessarily defined for all inputs.
"Bad" values for functions - reciprocals and square roots.
- Vertical line test for the graph of a function.
- Week 3, Lecture 4 (Practical Class).
- Additional resources on
Logarithms
and
Functions.
- Science Question. I started the practical class with the
following science question.
The concentration of H+ ions in solution is related
to the pH of the solution by the formula
pH = -log10 H+.
A solution of trifluoroethanoic acid has a concentration of
10-4 mol dm-3. What is the pH of the acid?
- You worked through practice questions on logarithms. If you
finish these then you worked through practice questions on
functions.
In your own time, you need to finish
the following exercises from chapter one of the MATH151 book.
- Multiple Choice Question - Logarithms -
in section 2.8.1 on page 41.
- Practice Questions - Logarithms - in section 2.8.2 on page 41.
- Standard Questions - Logarithms - in section 2.8.3 on page 43.
- Practice Questions - Functions - in section 3.10.1 on
page 56.
You should also
- Complete the week three tutorial sheet. (Worked solutions
will be available on the e-learning site.
- Science Question.
A science question that was given to you during the the practical class.
The Ideal Gas Law can be written
PV = nRT.
In this formula P is the pressure of the gas,
R is the ideal gas constant,
T is the temperature of the gas (in Kelvin),
V is the volume of the gas,
n is the number of moles of gas,
In a particular experiment the values for T and n are
kept constant and the volume is doubled. How does the pressure change?
(I was asked to give you this question by the co-ordinator of
first-year chemistry.)
- Week 4,
Lecture 1 (Chapter 3 - Function Notation).
- Additional resources on
Functions.
- I started the lecture by going over some basic ideas from this chapter:
- What is a function?
- Are functions valid for all inuts?
- Composition of functions [what does f(g(x)) mean?]
- Solving a quadratic equation.
We then finished going through:
- The vertical line test - when is a graph a function or not a
function?
- Two examples of graphs that are not functions are those give by:
- x=3.
- x2 +y2 = 1.
- Science Question. We finished the lecture with a science
question on the dissociation of an acid. The mathematical questions boil
down to the following.
- Rearrange the equation
K = x2/(1-x)
to obtain a quadratic equation for x.
- What is the value of x if K=0.6?
Hint. We are only interested in the positive value.
- Week 4,
Lecture 2.
- We start this lecture by going through some practice questions that
are not in the notes
- If 2x = 2.5 and 2y=3, then
2x+y is equal to
- 6.5.
- 7.5
5.5
- 1/5
- 25.5
- Simplify 2/(1-x) -1/(1+x).
- (1+3x)/(1-x2).
- (1+x)/(1-x2).
- (1+2x)/(1-x2).
- (3+3x)/(1-x2).
- 1/(1-x).
- We went through the solutions to some of the questions
on the practice version of the first class
test. The practice test paper and written solutions are available on the
e-learning page. Look in the folder
Home Page > Lecture Material > Assessment Tests in week 4 > In Class Tests
- Week 4,
Lecture 3 (Chapter 4 Mostly About Straight Lines).
- Additional resources on
Mostly About Straight
Lines.
- I started the lecture by going through some of the aims of the
chapter. These include
- The equation of a straight line is
y = mx +b
- What does the symbol m mean?
- Sketch an example of a straight line corresponding to:
- m<0
- m>0
- m=0
- What does the symbol b mean?
- Sketch an example of a straight line corresponding to:
- b<0
- b>0
- b=0
- Let y=m1x + b1 and
y=m2x + b2 be the
equations of two straight lines.
- What do you know if the two lines are
parallel?
- What do you know if the two lines are
perpendicular?
- Let y=m1x + b1 and
y=m2x + b2 be the
equations of two straight lines.
- What does it mean to ``solve'' these two equations?
- What does it mean geometrically to ``solve'' these
two equations?
- Under what conditions will there be:
- A single solution to the pair of equations?
- An infinite number of solutions to the pair of
equations?
- No solutions to their pair of equations?
- Consider an equation of the form Ax +By +C.
- Show that this equation can be transformed
into the equation of a straight line (y = mx + b).
- Sketch the equation when A=B=0. Is the resulting
graph a function?
- Sketch the equation when A=0. Is the resulting
graph a function?
- Given the points
P1 = (x1,y1) and
P2 = (x2,y2)
what is the gradient of the straight line connecting them?
- We learnt how to find the equation of a line given two
pieces of information. Typical questions are:
- Find the equation of a straight line with a specified
gradient going through a designated point.
- Find the equation of the straight line connecting
two points.
- Find the equation of the straight line that passes
through a designated point and is parallel to a
specified line.
- A useful idea is to try to sketch the straight line from
the specified information before you find its equation.
This is sometimes a useful way to discover if your equation
is incorrect (provided that your sketch was correct!).
- Some practice questions. In each case, rearrange to make x
the subject.
- ln(x) = 7.
- ln(x)2 = y.
- ln(xt) = y.
- ex = 4.
- e-6x = h.
- exp(x2) = y.
- Week 4,
Lecture 4 (Class Test 1).
- You did the first class test.
- In your own time, you should do the following.
- Mark sure that you have completed the Tutorial Sheet for week 4.
(solutions will go onto the e-learning page).
- In 2009 approximately 48% of students who scored 10.0 or lower
on the first test failed the course. (In 2010 the corresponding
figure was about 55%). If you scored 10.0 or lower
on the first test you are therefore at a high risk of failing this
course unless you start doing some serious work.
- Week 5
- Week 5
Lecture 1 (Chapter 4 - Mostly About Straight Lines).
- Additional resources on
Mostly About Straight
Lines.
-
-
- Find the equation of the straight line that passes
through a designated point and is perpendicular to a
specified line.
- We discussed the solutions of the system of linear equations
A1x +B1y +C1 = 0,
A2x +B2y +C2 = 0.
The important points are that this system has:
- no solution, when the two lines are parallel but
distinct
- one solution, when the two lines are not parallel
- an infinite number of solutions, when the two lines are
identical.
- There are two methods to solve a system of linear equations:
elimination and substitution. Which method is best? It depends upon
the particular problem that you are looking at.
- The distance between two points
P1(x1,y1) and
P2(x2,y2)
is given by
DP1P2 =
( (x2-x1)2
+(y2-y1)2)1/2
- Science Questions. We finished the lecture with two
science questions.
Question One dealt with the dissociation of an acid. The
mathematical questions boil
down to the following.
- Rearrange the equation
K = x2/(1-x)
to obtain a quadratic equation for x.
- What is the value of x if K=0.6?
Hint. We are only interested in the positive value.
We went through this question in week four and a similar question
was on the first assignment... but most of you can not do it!
Question Two.
An analytical chemist wants to prepare a calibration graph, relating
the amount of the natural pigment β-Carotene with its optical
absorbance when in solution. The analyst dissolves 0.01 g of
β-Carotene and obtains an optical absorbance of 0.8, then
weighs a mass of 0.03 g and obtains a higher absorbance of 2.0.
What is the relationship between the absorbance (the observed variable,
y) and the mass of β-Carotene (the controlled variable,
x)?
- Week 5
Lecture 2 (Chapter 5 - Trigonometry).
- I started the lecture by giving an overview of some of the aims
of this chapter. One of the more interesting aims is being
able to describe periodic experimental data using the formula
y = a*cos[w*(t-t0)]
where a is the amplitude of the oscillations,
w is the frequency of the oscillations
and t0 is the time to the first maximum.
- Pythagoras' Theorem.
- Definition of the trigonometric ratios:
sin, cos and tan.
- Right-angled triangles with 30, 45, 60 degree angles.
- Table of `standard' values of the basic trigonometric ratios.
- Week 5
Lecture 3.
-
- Radian measure of angle, converting degrees to radians and radians to
degrees.
2π radians = 360o
- Using the circle of radius one: a counterclockwise direction is
a positive angle whereas a clockwise direction is a
negative direction.
- In the Cartesian Plane where are:
Quadrant I, Quadrant II, Quadrant III, Quadrant IV, the
x-axis and the y-axis?
- Week 5
Lecture 4.
- You worked your way through the questions at the end of chapter 5.
In addition to doing these questions I gave you the following
science question.
- This week your first assignment was returned to you.
- Use the written solutions to mark the questions on the first
assignment that were not marked by your tutor.
- Look at the written solution for any questions on the first assignment
that you got wrong, both those marked by your tutor and those marked
by you.
- Week 6
Lecture 1 - (Chapter 5 Trigonometry).
- Let t be an angle measured in radiants.
To find the trigonometric ratio of sin(t),
cos(t) or tan(t):
- Locate the quadrant that the angle t is in.
Predict the algebraic sign of the answer using the `jingle':
All Stations To Central.
- Find the related angle θ.
Quadrant | Formula |
I | &theta = t |
II | &theta = π t |
III | &theta = t-π |
IV | &theta = 2π -t |
Note. It is always
true that: 0<= θ <= Π/2.
- Determine answer using steps (1) and (2).
(You will almost certainly need to use the table of exact values).
- Week 6
Lecture 2 (Chapter 5 - Trigonometry).
-
- Basic Facts about Trigonmetric Functions
- sin (-t) = - sin (t),
cos (-t) = cos (t),
tan (-t) = -tan (t).
- The periods of cos(x) and sin(x) are both
2π.
- The period of tan(x) is π.
- Solving Trigonometric Equations.
- How many solutions does the equation
sin(t) = 1/2 have?
- Which quadrant should I expect a solution in?
- Solve cos(x) = 1/2 where 0<= x<= 2π.
- Solve cos(x) = -1/2 where 0<= x<= 2π.
- Winner of the Dalek video competition
(Michelle Diab):
http://www.youtube.com/watch?v=w-d6vWMseVo.
- Week 6 Lecture 3 (Chapter 5 - Trigonometry)
-
- Graphs of sin and cos Functions
- The functions sin(x) and cos(x)
have period (P) P=2π.
- The functions sin(x) and cos(x)
have amplitude 1.
- The amplitude (a) of a periodic function
is defined by
a = (1/2)*(ymax-ymin)
where ymax and ymin are the maximum and
minimim of the function.
- The periodicity (P)
of the functions cos(ωx) and
sin(ωx) are given by
P = 2π/ω.
- Graphs of tan Functions
- The function tan(x)
has period (P) P=π.
Note the period!
- The function tan(x) does not have
amplitude.
- The periodicity (P)
of the function tan(ωx) is given by
P = π/ω.
- Revision Questions on functions.
For what values of x are the following functions
not defined?
- ln(x).
- (x-2)0.5.
- 1/(3x+2).
- ln (x-5).
- Revision Questions on Straight Lines.
Deduce the equations of the straight lines connecting the following
pairs of points.
- (1,2) and (2,4).
- (0,-2) and (3,11).
- (9,12) and (28,50).
- (7,9) and (6,10).
- (1,2) and (-3,-4).
- (-3,-3) and (-2,6).
- Week 6 Lecture 4 (Practical Class)
- You worked through
- 5.8 Practical Class - Evaluating Trigonometric Functions.
- Week 7
Lecture 1 (Chapter 5 - Trigonometry)
- Lecture material.
- The amplitude of a trig function
a*cos(b*x) or a*sin(b*x) is a.
- The period of a trig function
a*cos(b*x) or a*sin(b*x) is 2*π/b.
- We learnt how to plot the function
y = a*cos(b*x).
- Determine the amplitude of the function.
- Determine the period (P) of the function.
- Determine the critical values of the function.
These are the values for y when:
- x=0 (the start of the period).
- x=P/4 (one-quarter of the period).
- x=P/2 (one-half of the period).
- x=3P/4 (three-quarters of the period).
- x=P (the period).
Note. If the value for y is not the same
when x=0 as it is when x=P then you have
made a mistake!
- Plot the critical values.
- Week 7
Lecture 2 (Chapter 5 - Trigonometry).
- We practised sketching graphs of trigonometric functions!
- Week 7 Lecture 3 (Chapter 5 - Trigonometry)
- I started the lecture with some revision questions.
- Write down the exact values of:
- cos(11π/6),
- tan(2π/3),
- sin(5&pi/4).
- Write down all the solutions to:
- sin(t) = 1/2,
- cos(t) = -(2)0.5/2,
- tan(t) = -(3)0.5.
- We finished chapter 5.
- 5.10 Application of Trigonometric Graphs.
The key idea is to learn how to fit periodic experimental data
to a function of the form
y(t) = a +bcos(ω(t-t0)).
In this equation:
- a is the average value of the function over
one period. This is given by
a = (ymax+ymin)/2.
- b is the amplitude of the function. This
is given by
b = (ymax-ymin)/2.
- ω is the frequency of the oscillations.
This is given by
ω = 2π/P
where P is the period of the oscillations.
- t0 is the time at which the first
maximum is observed.
It's highly probably that there will be an exam question on
this topic.
- Week 7 Lecture 4 (Practical Class)
- I had some extra examples for you to work through
- Example One.
What is the period and amplitude of the function
y= f(x) = 5cos(x/8)?
- Example Two
A certain biological variable y is observed to vary approximately
in a sinusoidal manner, oscillating between the values y=1
and y=2 on a 24-hr cycle. It reaches its maximum at
3 o'clock P.M. and its minimum at 3 o'clock A.M. every day.
Find a formula for y as a function of t.
- Example Three
A variable y varies sinusoidally between the values -1 and +3
with a period of 2 sec. Taking t=0 to coincide with an instant
at which y=3, obtain a formula for y as a function of
t.
- Example Four
The respiratory cycle of a resting human being has a duration of
approximately 5 sec. Assuming that the rate of flow y of air
into the lungs varies sinusoidally as a function of time with a
maximum value of 0.5 liters/sec, express y as a function of
t. Take t=0 first to correspond to the instant when the
lungs are empty at the end of expiration, and then repeat the question with
t=0 corresponding to the instant when the lungs are full at the end
of inspiration.
- You finished the hour with a practical class on solving trigonometric
equations. (5.12 Practice Questions - Trigonometry).
Week 8
Lecture 1 (Chapter 5 - Trigonometry).
You did a practise session preparing for Test 2.
Week 8 Lecture 2 (Chapter 5 - Trigonometry)
You went through the solutions to the practise session.
Week 8 Lecture 3 (Chapter 5 - Trigonometry)
Week 8 Lecture 4 (Practical Class)
You did the second class test
Week 9
Lecture 1 (Chapter 5 - Trigonometry).
- Sketching the graph
y = y0 + acos[w(t-t0)].
- Applications to biology.
- Week 9
Lecture 2 (Chapter 6 - Exponential Growth & Decay).
-
- Relationship between the graph of f(x) and 1/f(x).
- Exponential Growth & Decay.
- Application to population growth.
- Exponential functions.
- Week 9 Lecture 3 (Chapter 6 - Exponential Growth &
Decay).
-
- Application of exponential functions to biological growth.
- Application of exponential functions to readioactive decay.
- Application of exponential functions to logistic growth.
- Week 9 Lecture 4
- Practical class.
- Week 10
Lectures 1 & 2 (Chapter 7 - Data Modelling)
- This is one of the most immediately useful lectures in the course.
The ideas in it are very important.
- The function y = f(x) = mx +b is a straight line,
slope is m and the y-intercept is b.
Given two points on the line (x1,y1)
and (x2,y2) the value for m
is given by
m = (y2-y1)/(x2-x1).
- The exponential function y = cekx can be
converted into the equation for a straight line by taking log's
of both sides of the equation. This leads to
ln(y) = ln(c) +kx.
Writing Y = ln (y) and C = log (c) we have
Y = kx + C.
Plotting log(y) against x is known as a
semi-log graph.
Given data we can then estimate the values for k
and C. We can then obtain the value for c
from the equation C = log (c).
- The power function y = cxp can be
converted into the equation for a straight line by taking log's
of both sides of the equation. This leads to
ln(y) = ln(c) +pln(x).
Writing Y = ln (y), C = ln(c)
and X = ln (x) we have
Y = px + C.
Plotting log(y) against log(x) is known as a
log-log graph.
Given data we can then estimate the values for p
and C. We can then obtain the value for c
from the equation C = log (c).
- Many scientific measurements are made using logarithmic scales.
This include: the Richter scale (for earthquake intensity),
pH (the acidity of a solution), and decibels (for sound).
Knowledge of log laws is essential to manipulate such equations.
- Week 10 Lecture 3 (Chapter 8 - limits).
-
- The idea of a limit involves what happens to the value of
a function f(x) as x gets closer and closer
to some limiting value a.
- The limit
limx→af(x)
is not necessarily
equal to f(a). This statement is only true for
sufficiently "nice" functions (continuous functions).
- A function is continuous at the point x=a if the graph
of the function is "unbroken" at the point x=a.
- If f(x)=p(x)/q(x) and
limx→af(x) = 0/0
then we should:
- factorise the numerator (p(x)) and
denominator (q(x));
- cancel the common factor;
- recalculate the limit.
Note that 0/0 is not necessary zero. Neither is it
necessarily ∞.
- Here are the values of some special limits:
- limx→01/x =
Does Not Exist or DNE.
- limx→∞1/x = 0
- limx→-∞1/x = 0
- If f(x)=p(x)/q(x) and
limx→∞f(x) =
∞/∞
then:
- identify the highest power of x in the
numerator and denominator;
- divide each term in the numerator and the denominator by
the highest power of x
(this is just multiplying by one!);
- recalculate the limit.
- limx→∞e-x = 0
limx→-∞e-x = ∞
- In limit questions involving square root functions, rationalise
the expression even if the square root functions appear in the
numerator.
- Week 10
Lecture 4 (Practical Class)
- Practical class.
-
- Week 11 Lecture 1 and Lecture 2
(Chapter 9 - Differentiation)
- Differentiation (9.1 & 9.2)
- Differentiation is about "rates of change". How does the
function y=f(x) chance as we change the value for
x.
- The value of the function at x is f(x) and
the value of the function at x+κ is
f(x+κ)).
The chance in the function (or y) is:
μ = f(x+κ) - f(x).
The corresponding change in x is:
κ.
- Geometrically, the value of μ is connected to the
slope of the graph of the function f.
- The average rate of change between the points x and
x+κ is the quotient:
μ/κ = [f(x+κ) - f(x)}κ.
Differentiation is considered with the value of this quantity
in the limit as κ→0.
- If the average rate of change is negative, then the quantity
is decreasing as x increases. If the average rate of
change is positive, then the quantity is increasing as
x increases.
- Don't forget that in scientific calculations the average rate
of change will have units.
- Week 11 Lecture 3 (differentiation)
- To find the instantaneous rate of change we
take the limit of the average rate of chance in the limit
lim κ→0.
(This is the key idea of the chapter).
- A point where the instantaneous derivative is equal to zero is
known as a critical point. Finding the location of
critical points is crucially in correctly sketching graphs.
- Critical points can be of three types:
- the slope of the function changes sign from positive to
negative (a relative maximum)
- the slope of the function changes sign from negative to
positive (a relative maximum)
- the slope does not change sign
- The degree of a polynomial is the highest power of x
that appears in the polynomial.
- Differentiation Rule 1: Power:
If y = cxp, where c and p are
real numbers then
dy/dx = pcxp-1.
- Differentiation Rule 2:. If
u = f(x) and v = g(x) are two differentiable
functions and c is a number, then u+v and
cu are differentiable functions. Further,
d(u+v)/dx = du/dx + dv/dx and
d(cu)/dx = cdu/dx.
- Differentiation Rule 3: The Product Rule. If
u = f(x) and v = g(x) are two differentiable
functions, then uv is a differentiable function and
d(uv)/dx = vdu/dx +udv/dx.
- Differentiation Rule 4: The Quotient Rule. If
u = f(x) and v = g(x) are two differentiable
functions, then u/v is a differentiable function and
d(u/v)/dx = (vdu/dx -udv/dx)/v2.
- Week 11, Lecture 4.
- You had a practical class.
- Week 12, Lecture 1 and 2
(differentiation).
-
- Differentiation Rule 5: The Chain Rule.
If y = f(x) and u = g(x) are two differentiable
functions, then y = f(u) is a differentiable function
and
d(y)/dx = dy/du×du/dx.
- Differentiation Rule 6: Trig, Exponential and Log Functions
.
- If y = sin(x) then dy/dx = cos(x).
- If y = cos(x) then dy/dx = -sin(x).
- If y = ex then dy/dx =
ex.
- If y = ln(x) (x>0) then dy/dx = 1/x.
- Note, by using the chain function you are expected to be able
to differentiate functions such as
y = sin(x2).
- Suppose that you are asked to sketch the graph
y = f(x). Finding dy/dx will give you
some good `pointers' for the sketch, e.g. identifying regions
where the function is increasing (dy/dx>0) and
decreasing (dy/dx<0).
- Week 12, Lecture 3 (integration).
-
- Antidifferentiation is the opposite to differentiation. The more
you understand differentiation, the easier you will find
antidifferentiation.
Given the function dy/dx = f(x) we are essentially asking
"What function y = g(x) did we differentiate to
obtain dy/dx = f(x)
- Suppose that when we differentiate y = g(x) we obtain
dy/dx = f(x). Does that mean that the answer to
the question `integrate f(x)' is g(x)?
No! The answer is g(x) +c, where c is a
constant. To see, differentiate g(x) +c: we obtain
f(x) because when we differentiate a constant we
get zero.
- The symbol ∫ is the
`antiderivative symbol'.
- Integration Rule 1: Powers. If
u = f(x) = xp, where p is a
real number (p ≠ -1), then
∫ xpdx |
= |
1/(p+1)xp+1 +c, |
p ≠ -1 |
C is a constant.
- Integration Rule 2: Powers. If
u = f(x) = x-1 then
C is a constant.
- Integration Rule 3: If u = f(x) and
v = g(x) are two functions such that
∫ f(x)dx and
∫ g(x)dx both exist,
and c is a number, then
∫ (f(x) +g(x))dx =
∫ f(x)dx +
∫ g(x)dx, and
∫ cf(x)dx =
c ∫ f(x)dx.
- Integration Rule 4:
∫ dx = x+c
- Integration Rules for exponentials and trigonometric
functions.
For u = ex, |
∫ exdx |
= ex +c
|
For u = sin(x), |
∫ sin(x)dx |
= -cos(x) +c,
|
For u = cos(x), |
∫ cos(x)dx |
= sin(x) +c,
|
- Week 13, lectures 1 & 2
(antidifferentiation)
-
- Antidifferentiation
- Antidifferentiation (or integration)
- The Definite Integral. If u = g(x) is any
antiderivative of u = f(x), then
∫ab f(x)dx
= [g(x)]ab
= g(b) -g(a).
- The definite integral given above finds the area bounded by
the graph of the function and the x-axis.
- More difficult types of antiderivatives can be solved using
the technique of substitution. For the purposes of this course,
if you see an integral involving the product or quotient of
functions then you should immediately think "substitution".
- Make a substitution (u=f(x)).
- Differentiate the substitution.
- Replace dx by du. (You might need to use
the `trick' of multiplying by 1).
- Integrate.
- Your final answer should only involve x: back-substitute
to remove u.
- Week 13, Lecture 3 (Revision)
-
- Integration Rule 3: Log Functions.
For u = ln(x), |
∫ ln(x)dx |
= xln(x) -x +c.
|
- That's all folks!
- Week 13, Lecture 4 (Sum-up).
- That's all folks!
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Page Created: 13th May 2008.
Last Updated: 19th March 2013.