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On the Shoulders of Giants A Course in Single Variable Calculus by G. Smith, G. Mclelland



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On the Shoulders of Giants A Course in Single Variable Calculus by G. Smith, G. Mclelland . This book presents an innovative treatment of single variable calculus designed as an introductory mathematics textbook for engineering and science students. The subject material is developed by modelling physical problems, some of which would normally be encountered by students as experiments in a first year physics course. The solutions of these problems provide a means of introducing mathematical concepts as they are needed. The book presents all of the material from a traditional first year calculus course, but it will appear for different purposes and in a different order from standard treatments.
The rationale of the book is that the mathematics should be introduced in a context tailored to the needs of the audience. Each mathematical concept is introduced only when it is needed to solve a particular practical problem, so at all stages, the student should be able to connect the mathematical concept with a particular physical idea or problem. For various reasons, notions such as relevance or just in time mathematics are common catchcries. We have responded to these in a way which maintains the professional integrity of the courses we teach.
The book begins with a collection of problems. A discussion of these problems leads to the idea of a function, which in the first instance will be regarded as a rule for numerical calculation. In some cases, real or hypothetical results will be presented, from which the function can be deduced. Part of the purpose of the book is to assist students in learning how to define the rules for calculating functions and to understand why such rules are needed. The most common way of expressing a rule is by means of an algebraic formula and this is the way in which most students first encounter functions. Unfortunately, many of them are unable to progress beyond the functions as formulas concept. Our stance in this book is that functions are rules for numerical calculation and so must be presented in a form which allows function values to be calculated in decimal form to an arbitrary degree of accuracy. For this reason, trigonometric functions first appear as power series solutions to differential equations, rather than through the common definitions in terms of triangles. The latter definitions may be intuitively simpler, but they are of little use in calculating function values or preparing the student for later work. We begin with simple functions defined by algebraic formulas and move on to functions defined by power series and integrals. As we progress through the book, different physical problems give rise to various functions and if the calculation of function values requires the numerical evaluation of an integral, then this simply has to be accepted as an inconvenient but unavoidable property of the problem. We would like students to appreciate the fact that some problems, such as the nonlinear pendulum, require sophisticated mathematical methods for their analysis and difficult mathematics is unavoidable if we wish to solve the problem. It is not introduced simply to provide an intellectual challenge or to filter out the weaker students.

On the Shoulders of Giants A Course in Single Variable Calculus by G. Smith, G. Mclelland cover the following topics.

  • 1. Terror, tragedy and bad vibrations
    1.1 Introduction
    1.2 The Tower of Terror
    1.3 Intothinair
    1.4 Music andthebridge
    1.5 Discussion
    1.6 Rulesofcalculation

  • 2. Functions
    2.1 Rulesofcalculation
    2.2 Intervalsonthe real line
    2.3 Graphsof functions
    2.4 Examplesof functions

  • 3. Continuity and smoothness
    3.1 Smoothfunctions
    3.2 Continuity

  • 4. Differentiation
    4.1 Thederivative
    4.2 Rules fordifferentiation
    4.3 Velocity, acceleration and rates of change

  • 5. Falling bodies
    5.1 The Tower of Terror
    5.2 Solving differential equations
    5.3 General remarks
    5.4 Increasinganddecreasingfunctions
    5.5 Extremevalues

  • 6. Series and the exponential function
    6.1 The airpressureproblem
    6.2 Infiniteseries
    6.3 Convergenceof series
    6.4 Radiusof convergence
    6.5 Differentiationofpower series
    6.6 The chainrule
    6.7 Properties of the exponential function
    6.8 Solution of the air pressure problem

  • 7. Trigonometric functions
    7.1 Vibratingstrings andcables
    7.2 Trigonometric functions
    7.3 More on the sine and cosine functions
    7.4 Triangles, circles and the number 
    7.5 Exact values of the sine and cosine functions
    7.6 Other trigonometric functions

  • 8. Oscillation problems
    8.1 Second order linear differential equations
    8.2 Complexnumbers
    8.3 Complexseries
    8.4 Complex roots of the auxiliary equation
    8.5 Simple harmonic motion and damping
    8.6 Forced oscillations

  • 9. Integration
    9.1 Another problem on the Tower of Terror
    9.2 Moreonairpressure
    9.3 Integralsandprimitivefunctions
    9.4 Areas under curves
    9.5 Area functions
    9.6 Integration
    9.7 Evaluationof integrals
    9.8 The fundamental theorem of the calculus
    9.9 The logarithmfunction

  • 10. Inverse functions
    10.1 The existenceof inverses
    10.2 Calculatingfunctionvalues for inverses
    10.3 The oscillation problem again
    10.4 Inverse trigonometric functions
    10.5 Other inverse trigonometric functions

  • 11. Hyperbolic functions
    11.1 Hyperbolic functions
    11.2 Properties of the hyperbolic functions
    11.3 Inverse hyperbolic functions

  • 12. Methods of integration
    12.1 Introduction
    12.2 Calculationofdefinite integrals
    12.3 Integration by substitution
    12.4 Integrationbyparts
    12.5 The method of partial fractions
    12.6 Integrals with a quadratic denominator
    12.7 Concludingremarks

  • 13. A nonlinear differential equation
    13.1 The energyequation
    13.2 Conclusion

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