Calculus
Different kinds of numbers; What is a function? Sums and products of functions; Composite functions (function of a function); Inverse function.
Scales and axes; What is a graph? Common examples; Slope of a straight line; Plotting 1/x and related examples.
Piecewise defined functions; Trigonometric functions; The curious case of sin(1/x); Plotting an unplottable function.
An intuitive approach to limits; Limits of various functions; Defining limits properly; An important theorem.
Continuous and discontinuous functions; Left and right limits; Some deep consequences of continuity; Solved examples.
Gradients and tangents; Differentiation; Examples from daily life; Differentiable and non-differentiable functions.
Derivative of a sum of functions; Derivative of product of functions; derivative of ratio of functions; Derivative of composition of functions; Chain rule.
Rates of change; When the first derivative vanishes; Inflection points; Higher derivatives; Solved examples from daily life.
Rolle's theorem; Mean Value Theorem; Applications of MVT; L'Hopital's theorem; Applications of L'Hopital's theorem.
Dealing with exponents; The exponential function; Some obvious properties; Applications; The exponential series..
Monotonic functions; When does a function have an inverse?; Differentiating inverse functions; The logarithm as the inverse of exponential.
Approximating functions with polynomials; Taylor's theorem; Estimating the remainder; Examples of failure.
Integral as anti-derivative; Some common integrals; Integration as area under the curve; Solved examples.
Integration as area; Upper and lower sums; Functions that can be integrated; Functions that cannot be integrated.
Integration - it's best to guess the answer; Integration by parts; Integration by change of variables.
Partial fractions; Completing the square; Differentiating under the integral sign.
Integrals of trigonometric functions; Hyperbolic functions; Applications.
Integrals of discontinuous functions; Integrals with singular integrands; Integrals with infinite limits.
When do you need numerical integration? Trapezoidal rule; Simpson's rule; Solved examples.
Length of a curve - general formula; Worked examples; Implicit functions; Length of curves that are defined implicitly.
Surface area of a solid; Volume of a solid; Examples: cone, sphere, ellipsoid; toroid.
What is a parameter?; Parametrized curves; Derivatives of a parameterized curve; Length of a parametrized curve.
Definition of polar coordinates; Polar plots; Slopes and lengths in polar coordinates; Area calculation.
Scalars and vectors; Adding vectors; Dot product and cross product; Towards abstract vectors.
Vector fields; Differentiating vectors; Tangents and planes; Integration of vectors.
How DE's arise; Solving a DE numerically; Easy examples; Separable DE's.
Linear DE's. Integrating factor; Bernoulli equation; Autonomous DE's.
Imaginary and complex numbers; Complex conjugation; Argand diagram; Euler's representation.
Infinite sequences and functions; Convergence of a sequence; Testing for convergence; Sequences of partial sums.
Summation of series; When does a series diverge?; When does a series converge; Alternating series.
