Index for Calculus Notes.
These are notes for a first semester first year calculus course. The notes refer to Adams: Calculus - A Complete Course, Addison Wesley Fourth edition
ApproxFixed.doc
Fixed point iteration through examples. Cobweb diagram
ApproxFunctions.doc
Linear and quadratic approximations to a function near a point
ApproxNewton.doc
Newton-Raphson approximation. Examples. No error analysis
ApproxPoly.doc
Polynomial approximations - examples
Composite Functions.doc
Definition and examples of composite functions. Graph sketching using composition
DiffInverse.doc
Differentiation of inverse functions. Choice of domain for inverse trig functions
Differentiation.doc
Mainly revision of differentiating composite functions, using the one variable chain rule
ExpLogFns.doc
Discussion of exponential and logarithm as inverses. relationship between laws of logarithms and indices. General powers
Fn2Var.doc
Plotting functions of two variables, cross sections and contours. Examples with MAPLE plots
Fn2VarLim.doc
Examples where limiting properties are more complicated than for one variable
Functions.doc
Basic properties of functions. Domain & Range. Modulus, square root, odd & even, increasing & decreasing
HigherPDiff.doc
Second partial derivatives, verifying Laplace's equation as an example
HyperbolicFns.doc
Definition and basic properties of hyperbolic functions. Logarithmic form of inverses
ImplicitDiff.doc
Introduction to the technique of implicit differentiation
IntApplic.doc
Volume of revolution (disc and shell methods). Arc length, surface of revolution
IntLog.doc
A discussion of the integral of 1/x, explaining what happens when x is negative, and the reasons for ln|x|
IntParts.doc
Revision of basic integration by parts. Example giving an equation for the integral on integration by parts twice
IntRational.doc
Step-by step method for integrating rational functions, with examples
IntReduction.doc
Integration by parts giving reduction formulae
IntRevision.doc
Revision of basic integration. Using odd and even properties with definite integrals. Example with trig identities for a product of trig functions
IntSub.doc
Revision of basic integration by substitution
IntSubFurther.doc
Particular types. Square roots of quadratics, square roots of sums and differences of squares, square root of linear function, rational functions of cos and sin (half-angle substitution)
InverseFns.doc
Inverse functions, domain and range, graphical properties, restricting domains to obtain 1-1 functions.
InverseTrigFns.doc
Discussion of restricting domain for inverse trig functions.
LeibnizThm.doc
Proof and example of Leibniz theorem for the n-th derivative of a product
LimitTechniques.doc
Finding limits by squeezing, algebraic manipulation, change of variable and l'Hopital's rule
Limits.doc
Brief discussion of intuitive ideas about limits, introducing terminology
LogarithmicDiff.doc
The technique of logarithmic differentiation through examples
PDiff.doc
Basic partial differentiation
PDiffChRule.doc
The chain rule. Proof (not completely rigorous). Examples. using tree diagrams as an aid.
ParametricDiff.doc
Discussion and example of parametric differentiation, including the second derivative
Polynomialdivision.doc
A worked example of polynomial division needed as the first step for integrating rational functions
TaylorPoly.doc
Derivation of Taylor polynomials with examples
TaylorTheorem.doc
Statement of Taylor's Theorem (no proof) and an example on approximation
Tplanes.doc
Deriving equations of tangent plane and normal line to a surface z=f(x,y)
Trig.doc
Very brief revision of some properties of trig functions