Module title & Course code
Mathematics with Maple
MAS100
Lecturer
Dr Simon Willerton
Course description
University course content
0. Introduction to Maple
Simple examples of the main capabilities of
Maple. Expansion, factoring, collection of similar
terms, simplification of algebraic fractions.
Plotting graphs, controlling scales and axes,
combining plots. Polar coordinates and curves
given parametrically. Graphical solution of
various problems.
1. Solving equations
Various methods of solution, by hand or using
Maple. Symbolic and numerical solutions.
Problems with no solutions or many solutions.
Using solutions in further calculations.
In this course we learn to use a program called Maple, which is a very powerful tool for solving
problems in mathematics. Maple will also be used, to varying extents, in many subsequent courses. In
parallel with learning Maple, we will review and extend some topics from A-level. Using Maple we will
be able to treat complex examples painlessly, look systematically for patterns, visualize our results
graphically, and so gain new insights.
A Level course content
Module
Further
module
only?
C4
No
FP2
Yes
FP1
Yes
It is unlikely students will have seen Maple before they will however have
seen:Curves given parametrically
Polar Coordinates
Further maths students should be familiar with several different methods of
solving equations:
Solving equations using interval bisection, linear interpolation and the
Newton-Raphson process.
2. Special functions
Exponentials and logarithms. Trigonometric and
hyperbolic functions. Identities between such
functions, and how to prove them.
Primary special functions
 sinx, cosx and tanx
C2
No
C3
No
C3
No
FP3
Yes
FP2
No
FP1/FP2
Yes
C2
No
tan2x + 1 = sec2x, 1 + cot2x = cosec2x, double angle for sin,cos and tan and
addition formulae for sin,cos and tan
C3
No
Special Values for Trig Functions
All students should be familiar with these.
C2
No

x
lnx, e , arcsin, arccos and arctan
Secondary special functions
 sec, csc, cot

sinh, cosh, tanh, sech, csch, coth, arcsinh, arccosh, arctanh.
𝑒𝑥 = 1 + 𝑥 +
𝑥2
+⋯
2!
Complex numbers
Trig Identities
Sin2x + cos2x = 1
tanx=sinx/cosx
Students will probably not be familiar with Fourier Series as it does not feature
on any A Level Maths module.
3. Differentiation
The geometric and numerical meaning of
differentiation. Derivatives of some standard
functions. Rules for calculating derivatives.
Implicit derivatives and higher-order derivatives.
Students should have seen at least x2 differentiated from first principles but
they do not need to prove this in any exam and so generally not much
emphasis is placed on it.
The product rule, the quotient rule, the chain rule, the logarithmic rule,
inverse function rule, implicit differentiation and parametric differentiation
should be covered by all students.
C3/C4
No
FP2
Yes
FP3
Yes
The power rule is covered in C1.
𝑒𝑥 = 1 + 𝑥 +
𝑥2
+⋯
2!
Special Functions – Students should be familiar with all the derivatives by the
end of A level maths with the exception of:
Differentiating hyperbolic functions – The derivatives for sinhx, coshx and
tanhx are given to students in their formula books.
It will be unlikely that any students will have seen differentiation with a
complex variable such as cosx=cosh(ix) seen in Lecture 5.
4. Integration
The meaning of integration. Integrals of some
standard functions. Methods for finding integrals
(by parts and by substitution).
Students who have done A level should be familiar with all the standard
integrals mentioned with the exception of:
Hyperbolic functions and Inverse Trig Functions – This will only have been seen FP3
by students who have done further maths.
All of the standard integrals listed in Lecture 7 are given to students in their
formula books with the exception of:
exp(x), sin(x), cos(x) which students learn
sin2(x), cos2(x) which students have to show how to integrate using the double
angle formulae for cos2x
ln(x) which students can be asked to show using integration by parts.
Students are asked to differentiate ax in C3 but are not actually expected to be
able to show they can integrate ax.
Lecture 8 – It is unlikely students will have be familiar with the curve shape of
exponential oscillation and will be unfamiliar with the term PEO.
5. Taylor series
Approximation by polynomials and the relation
with higher derivatives. Calculation of Taylor
series.
FP2
This will only have been seen by students who have done further maths.
Students are given the Taylor’s series for:
f(x), f(x + a), ex , ln(1 + x), sinx, cosx, arctanx, sinhx,coshx and arctanhx in their
formula books.
Students may not have considered the subtleties as to what functions have a
Taylor’s series.
Note:
The notation O(x7) will probably be unfamiliar to students.
Even and Odd functions is not covered by every exam board
(Hyperbolics
are FP3)