Calculus

IB Higher Level Mathematics : Core

8. Calculus

8.1

Informal areas of limit, and convergence.
The result

justified by geometric demonstration.

See The Definition of Continuity, from CalculusQuest, a comprehensive interactive online textbook from Oregon State University.

Continuous functions, as well as many other topics, are dealt with at Analysis WebNotes. The epsilon-delta definition is given (below), and an interactive demonstration can be used to explore the consequences of using different values of d for a given e.

The definition of a continuous function, from Continuous functions.
Used with the permission of John Orr.

A definition and JAVA applet illustrating the concept can be found at the Continuity section of Interactive Real Analysis.

An illustration of continuity using e and d notation. In the words of G.H. Hardy : If we draw two such horizontal lines, no matter how close together, we can always cut off a vertical strip of the plane by two vertical lines in such a way that all that part of the curve which is contained in the strip lies between the two horizontal lines.

The diagram is taken from Calculus . . . , and is used with the permission of Eric Gumtow. The quotation is from A Course of Pure Mathematics, published by the Cambridge University Press as part of their Cambridge Mathematical Library. Used with permission.

8.2

Differentiation from first principles as the limit of the difference quotient:

Differentiation of: x xⁿ, n ; x

sin x; x

cos x; x

tan x; x

e^x; x

ln x.

The derivative of the sine function, based on first principles, is demonstrated and explained at The Derivative of the Sine.

An animation of the tangent line, from a tutorial on Derivatives of Powers and Polynomials, part of Hofstra University's Web resource for Finite Math & Applied Calculus. Used with permission.

For a historical approach to many aspects of calculus, see Basic Calculus : From Archimedes to Newton to its Role in Science. This site, from Notre Dame University, attempts to teach elementary calculus, as well as the necessary trigonometry and analytic geometry, from within the 'historical flow' of the subject. Recommended.

See the Differentiable Functions entry from the on-line interactive textbook Interactive Real Analysis, from Seton Hall University.

Calculus . . . provides a definition of the derivative and other fundamental terms.

At Java Projects, click on SecantCalc, "a program to demonstrate the definition of the derivative in terms of the slope of approximating secant lines."

See Karl's Calculus Tutor for tips on how to do calculus questions. Note that tips only are provided, not complete solutions.

Finally, a Mathematica-based supplement for any calculus course can be found at
Calculus Explorations using Mathematica. The site has Sample Labs that are available for downloading. Recommended for users of Mathematica.

8.3

Differentiation of sums of functions and real multiples of functions. The chain rule for composite functions.

See Tutorial for the Product & Quotient Rules, from Hofstra University's Tutorials.

The slope and derivative of a rational function are animated at Calculus Animations.

See Tutorial for the Chain Rule, from Hofstra University's Tutorials.

The chain rule is explained at Calculus . . . .

8.4

Further differentiation: the product and quotient rules; the second derivative; differentiation of a^x and log_ax.
The chain rule for composite functions.

The number e is explained at The Natural Logarithmic Base, and is one of many numbers that make up Favorite Mathematical Constants.

Exponential Functions are fully discussed at the Analysis WebNotes site.

The same site introduces logarithmic functions in a problem: see Homework Assignment 06.

8.5

Graphical behaviour of functions; tangents, normals and singularities, behaviour for large | x |; asymptotes.
The significance of the second derivative; distinction between maximum and minimum points and points of inflexion.

8.6

Applications of the first and second derivative to maximum and minimum problems. Kinematic problems involving displacement, s, velocity,

, and acceleration,

See Concavity and the Second Derivative, from CalculusQuest.

8.7

Implicit differentiation. Differentiation of the inverse trigonometric functions.

Implicit differentiation is explained at Calculus . . . .

SuperGraph is a downloadable shareware program for the Mac that can be used to graph explicit and implicit functions.

See Parametric Equations, from The Interactive Textbook of PFP 96. This site, from UPenn, demonstrates the role of mathematics in science. Recommended.

Implicit differentiation of x² + y² with respect to x, using Mathematica. The y is expressed as a function of x; the dy/dx is expressed as the derivative of a function of x.

See Related Rates, and Related Rates Problems, both from CalculusQuest.

8.8

Indefinite integration as anti-differentiation. Indefinite integrals of: xⁿ; n ; sin x; cos x; e^x. Composites of these with x ax + b. Application to acceleration and velocity.

Visual Calculus provides a graphic example of the fundamental theorem of calculus at
Fundamental Theorem of Calculus - 1 .

Information on integrating the trig functions can be found at Trigonometric Integrals .
The site provides exercises and attractive graphics :

An illustration from Trigonometric Integrals, used with the permission of David Hart.

8.9

Anti-differentiation with a boundary condition to determine the constant term. Definite integrals. Areas under curves.

There are several demos that calculate the area under a curve. At the Numerical Integration Tutorial, created by Joseph Zachary, from the University of Utah, a JAVA applet makes it possible to display any one of a number of functions, and to calculate the area between the function, two moveable white lines, and the x-axis. Various numerical methods of calculation can be used, including the trapezeum method. The tutorial is a resource for the book Introduction to Scientific Programming. Recommended.
At Java Projects, click on IntegCalc, "a program to demonstrate the definition of the integral in terms of successive approximations by Riemann sums. A similar demo can be found at The Lower Riemann Sums, where the true area under a curve can be approximated by adding rectangles. Recommended.

At Volume of a Solid of Revolution (1), a JAVA applet allows a curve to be rotated about the x-axis. Cross-sections can be highlighted and shifted, and the number of disks making up the volume increased or decreased. The cross-sectional area and the volume are calculated. This Manipula Math with Java site is from Japan. Recommended.

See Numerical Integration : Accumulating Rates of Change, at the Gallery of Interactive Geometry, from the University of Minnesota's Geometry Center. At this site it is possible to explore the numerical integration of data sets. If the velocity of an object has been measured at certain instants of time, is it possible to "integrate" this discrete data to estimate the change in the object's position? Recommended.

8.10

Further integration; integration by substitution; integration by parts; definite integrals.

See Techniques of Integration : Substitution and Techniques of Integration : Integration by Parts, from the extensive S.O.S. MATHematics site.

Also see Review of Integration Techniques from Oregon State University's extensive Web Study Guide for Vector Calculus site. Substitution, parts, and the integration of rational functions are discussed. There is also a table of common integrals.

See the table of Common Integrals, from OSU's Web Study Guide for Vector Calculus.

8.11

Solution of first order differential equations by separation of variables.

See Web Study Guide for Ordinary Differential Equations from Oregon State University. This site also contains a clickable Roadmap for Solving First-Order ODE.

Also see Differential Equations, from The Interactive Textbook of PFP 96, "an inter-
disciplinary course in chemistry, mathematics and physics." There are several versions of this textbook, from UPenn, one of which is JAVA-enabled. Recommended.

There is a Mathematica lab involving differential equations at Search for Problem Sets.
Click the Differential Equations link.

See Some Applications for examples involving differential equations : Radioactive Decay and Newton's Law of Cooling.

For material related to differential equations and their representation using Mathematica, Maple, and the TI-92 calculator, see The Language of Differential Equations, from The Connected Curriculum Project.

See Separable Equations, and Homogeneous Equations, both from the S.O.S. Mathematics Site. Also see Separable First-Order ODE, and Homogeneous First-Order ODE, from OSU's Web Study Guide for Ordinary Differential Equations.

See First Order Linear Equations.

Also see Linear First-Order Differential Equations, from Oregon State University.