BitWithBite
BitWithBite / Applied Mathematics / Numerical Methods
📘 Lesson 2 of 7 · Applied Mathematics

🔢 Numerical Methods

Numerical methods find approximate solutions to problems that are too difficult or impossible to solve exactly — essential tools whenever an equation resists a clean algebraic answer.

Course progress: 29%

01 Key Concepts

Why Numerical Methods Matter

Many real-world equations (especially nonlinear ones) have no exact algebraic solution, so we approximate the answer instead, often using a computer.

Bisection Method

Repeatedly halves an interval known to contain a root, narrowing in on the answer by checking which half still contains a sign change.

Newton's Method

Uses a function's derivative to rapidly converge on a root: x_(n+1) = x_n - f(x_n)/f'(x_n), starting from an initial guess.

Numerical Integration (Trapezoidal Rule)

Approximates the area under a curve by dividing it into trapezoids instead of computing an exact integral.

Error and Convergence

Numerical methods produce approximations, not exact answers; the 'error' shrinks as you do more iterations or use smaller step sizes, until the method 'converges' close enough to the true answer.

02 Key Formulas

03 Solved Examples

Example 1 Using the bisection method, a root is known to lie between x=1 and x=2. What is the first midpoint tested?
  1. The bisection method starts by testing the midpoint of the interval.
  2. Midpoint = (1+2)/2.
Answer: x = 1.5
Example 2 Apply one step of Newton's method to f(x)=x^2-2, starting at x_0=1.5 (f'(x)=2x).
  1. f(1.5) = 1.5^2-2 = 0.25. f'(1.5) = 2*1.5 = 3.
  2. x_1 = 1.5 - 0.25/3 = 1.5 - 0.0833.
Answer: x_1 ≈ 1.4167 (getting closer to sqrt(2) ≈ 1.4142)
Example 3 Why can't every equation be solved with simple algebra, requiring numerical methods instead?
  1. Some equations, especially those mixing polynomials with trig, exponential, or log terms, have no algebraic formula for their solution.
  2. Numerical methods provide a practical way to approximate the answer to any desired precision.
Answer: Because such equations lack a closed-form algebraic solution, numerical approximation becomes necessary

04 Practice Questions

1A root lies between x=0 and x=4. Find the first bisection midpoint.
2
2What information does Newton's method require besides the function itself?
Its derivative
3What does 'convergence' mean in a numerical method?
The approximations get closer and closer to the true answer with more iterations
4Why might numerical methods be preferred over algebra for some equations?
Because those equations have no exact algebraic solution
5What does the trapezoidal rule approximate?
The area under a curve (a definite integral)

📄 Numerical Methods — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet