Guides
Each guide explains one method, then points to the topic where you can practice it. Guides that belong together are grouped below, in the order to read them.
Bernoulli’s count of successes in repeated trials, and the normal and Poisson approximations.
Polynomials orthogonal on the whole line, weighted by a Gaussian.
Polynomials orthogonal on the half-line from 0 to infinity, with an exponential weight.
Average speed, the limit that turns it into an instant rate, and the tangent whose slope is that rate.
2The tangent rise dy stands in for the curve’s real change, with a formula for the error.
3A second derivative is acceleration, a higher one measures bend, and Leibniz’s rule differentiates a product.
4Rolle and Lagrange: between two points the chord has a parallel tangent.
5The 0/0 and ∞/∞ cases, settled by comparing how fast the numerator and the denominator change.
6A polynomial that matches the derivatives at one point, with an explicit bound on the remainder.
7Velocity and acceleration along a path, and the same derivative taken with respect to an angle.
8Where the graph rises or falls, its peaks and valleys, and where the bend changes direction.
9The derivative gives the tangent line, and the normal is the line perpendicular to it.
10Vertical, horizontal, and slant lines that a curve approaches, each found from a limit.
Thin strips under a curve, between two curves, and in polar coordinates.
2A curve measured by adding short chords, which turns Pythagoras into an integral.
3A curve spun about an axis. The area is the integral of circumference times arc length.
An integral that still depends on an extra variable, and when that function is continuous.
2Convergence for every value of the parameter can still leave the result badly behaved.
3Differentiating under the integral sign, used on the Gaussian and the Dirichlet integral.
4Gamma extends the factorial past the integers, and Beta turns a family of integrals into arithmetic.
5A function that does not repeat, built as an integral over every frequency.
Comparison, ratio, root, and integral tests for a sum of positive terms.
2Leibniz’s test, absolute versus conditional convergence, and what reordering the terms can do.
3An infinite table summed by rows, by rectangles, or along the diagonals.
4When the error shrinks everywhere at once, the sum stays continuous and can be integrated.
5The interval where the series converges, and the Taylor series used inside it.
6A periodic shape, corners and jumps included, built from sines and cosines.
7Multiplying forever, turned into an ordinary series by taking logarithms.
A plane as the points perpendicular to one vector, and the distance from a point to that plane.
2A line fixed by a point and a direction, or cut out as the intersection of two planes.
3The angle between a line and a plane, and whether the line cuts it, misses it, or lies in it.
4Spheres, cylinders, cones, and conoids, each given an equation by eliminating the motion that builds it.
In the plane a limit has to agree along every path, not only along the axes.
2The slope along each coordinate, added up into the total differential and the chain rule.
3The second-order Taylor formula, used for a local extremum, a constraint, and a singular point.
Gamma in the complex plane, with poles at 0 and the negative integers, and Beta beside it.
2One series that specializes to many familiar functions, and the zeta function’s product over the primes.
3sn, cn, and dn: the functions with one real period and one imaginary period.