A core course unit for first year Mathematics, Mathematics with another subject and Joint Honours Mathematics students.
 

General Details

Credit Rating:	10
Level:	Level One
Delivery:	Semester One
Lecturer:	Prof. Reg Wood (Telephone 55849, email:reg@ma.man.ac.uk.)

General Description
The first part of the course unit is concerned with the exponential function, the log function, the hyperbolic functions and their inverses.  There follows a review of the basic techniques of differentiation with applications to graph sketching.  The trigonometric functions are introduced in terms of power series and their relationship to the exponential function explored.  The next major topic is a review of the basic techniques of integration.  Towards the end of the course a number of questions about power series are treated more fully and finite Taylor expansions are applied to approximation of numbers.  The last chapter contains some miscellaneous applications of calculus to variational problems, max-min problems, lengths of curves, areas and volumes of solids of revolution and the Newton-Raphson method of solving equations.

Note. The contribution of this course to the Study Skills course will be an exercise in the use of MATLAB for performing some task in calculus.

Aims  To raise the standard of basic calculus skills to that required by second semester and intermediate level course units.

Learning Outcomes On successful completion of the course unit students will be able to

• demonstrate a good understanding of the exponential function and the log function
• show an appreciation of the interconnections between hyperbolic, trigonometric functions and the exponential function
• develop skill in the processes of differentiation and integration
• demonstrate the ability to sketch graphs of functions
• achieve a level of confidence in applying calculus to elementary problems about approximation, rates of change, max-min phenomena, lengths, areas, volumes and solving equations.

Future topics requiring this course unit   Calculus will be used extensively in many later course units.

Content
The exponential function.  Review of basic notation and terminology.  Definition of the exponential series exp(x).  Partial sums. The number e. The fundamental property
exp (a+b) = exp(a) exp(b).  Binomial coefficients and Pascal's triangle.  The notation  e^x.  Properties  e^x > 0,  e^-x = 1/e^x.  The graph of  e^x.
The log function.  Definition of  ln(x)  as an integral.  The log function is the inverse of the exponential function.  The fundamental property  ln(ab) = ln(a) + ln(b).  Logs to other bases.  Graphs of an inverse function, in particular ln(x).  The hyperbolic functions cosh(x), sinh(x) in terms of  e^x: their infinite series expansions: derivatives:
formulae cosh(a + b) = cosh(a) cosh(b) + sinh(a) sinh(b)  etc.  Graphs of the hyperbolic functions.  Other hyperbolic functions tanh(x) etc.  The inverses of the hyperbolic functions in terms of  ln(x).
Differentiation. A review of five rules of differentiation: linearity, product rule, chain rule, reciprocal rule and inverse rule.  Graph sketching; stationary points, turning points, max and min, inflection points, asymptotic behaviour.  Derivatives of functions defined parametrically, implicitly and by inverse functions.
The trigonometric functions.  The basic trig functions cos(x) and sin(x) introduced as power series.  Derivatives and the fundamental property cos²(x) + sin² (x) = 1.  Other trig functions  tan(x) etc.  The relationship of trig functions to the complex exponential function.  Trig formulae and de Moivre's theorem.  The definition of p/2 as the smallest real number  q  such that e^iq = i: periodicity of the trig functions: navigating the unit circle and the evaluation of trig functions at various multiples of  p.  The solutions of sin(x) = 0 and cos(x) = 0.  The graphs of the trig functions.  Inverses of trig functions.
Integration. Indefinite and definite integration.  The general idea of the integral as a limit of a sum and the fundamental theorem of calculus.  Review of the basic rules of integration: linearity, integration by parts, integration by substitution, partial fractions, the general problem of integration in elementary terms.
Infinite series.  The radius of convergence.  Addition, subtraction and multiplication of infinite series.  Term by term differentiation of a series.  Term by term integration of a series over an interval in the domain of convergence.  Infinite Taylor series expansions about a point.  Finite Taylor expansions, error terms, the big O notation.  Approximation of numbers.  The mean value theorem. L'Hopital's rule.
Application of calculus.  Rates of change, max-min problems, lengths of curves, surfaces and solids of revolution.  Solving equations.

Teaching and learning methods   Two lectures each week and a weekly examples class.
 

Learning hours

Activity	Hours
Staff/student contact	30
Private study	66
Total hours	96

Assessment

Activity	Length	Weighted within unit
Course test		15%
Study skills classes		15%
End of semester examination	2 hours	70%

Core learning materials
Textbooks
The fifth edition of R.A. Adams book "Calculus, a Complete Course" was published this year (2003) by Addison Wesley Longman. ISBN 0-201-79131-5

Calculus has been around for over 300 years and textbooks on the subject are just too numerous to mention.  Browse the library, shelf number 517 for a sample.

Schaum outline series on any subject are good value.  Many books entitled "Mathematics for Engineers" contain a complete course on mathematics, covering everything from set theory, logic, differential equations, matrix theory, through to probability and statistics.  Such books seem to get fatter each year but they are good value, crammed with hundreds of exercises.
For your Study Skills in MATLAB, the book by the Higham brothers "MATLAB Guide" is recommended. ISBN 0-89871-469-9 published by SIAM 2000.