Unit 10 of 10

Numerical Methods and Mathematical Computing

  • 7 lessons
  • Parts 243 to 256

Most real problems do not have a neat formula as their answer, so we approximate. Numerical methods turn calculus into step by step procedures a computer can carry out, together with a way to judge how large the error is.

This unit covers reliable use of software, root finding with bisection and Newton's method, approximate derivatives and integrals, Taylor approximation, and Euler and Runge-Kutta methods for differential equations.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Reliable Computation

  1. Graphing Software and Numerical Reliability

    How a computer graph is made from finitely many rounded samples, how it can hide or invent features, and how the derivative theorems turn what a graph suggests into a proof.

    Theory part 243Practice part 244

Root Finding

  1. Bisection and Newton’s Method

    How to solve f(x) = 0 when no formula gives the root, with bisection as the slow method that cannot fail and Newton's method as the fast one that needs a good start.

    Theory part 245Practice part 246

Numerical Differentiation

  1. Finite Differences and Derivative Approximation

    How to estimate a derivative from function values or measured data with forward, backward and central differences, and why making the step size smaller helps only up to a point.

    Theory part 247Practice part 248

Numerical Integration

  1. Integral Approximation Rules

    How the midpoint, trapezoidal and Simpson's rules approximate a definite integral, how derivative bounds tell you how many subintervals are enough, and how adaptive methods put the work where it is needed.

    Theory part 249Practice part 250

  2. Approximating Improper Integrals

    How to approximate an integral over an infinite interval or with an unbounded integrand by cutting off a tail you can bound, removing singularities, and keeping track of each source of error.

    Theory part 251Practice part 252

Approximation with Series

  1. Approximating Using Taylor’s Formula

    How Taylor polynomials with a remainder bound give function values and integrals to a guaranteed accuracy, and how to pick the center and the degree to meet a given tolerance.

    Theory part 253Practice part 254

Numerical Differential Equations

  1. Euler and Runge-Kutta Methods

    How Euler's method and its improved and Runge-Kutta versions step along a solution of y' = f(x, y), how their errors shrink as the step gets smaller, and when a step is too large to be stable.

    Theory part 255Practice part 256