Real Analysis Visualizer

The Weierstrass Function

Continuous everywhere, differentiable nowhere.

f(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x)

a (0 < a < 1): 0.5

b (odd integer): 7

Zoom: 1x

This function is a "fractal curve". No matter how much you zoom in, it never becomes a straight line (smooth). Standard calculus breaks down here. Try increasing b to add more "noise", or zoom in to see the self-similarity.

The Epsilon-Delta Game

Visualizing the formal definition of a limit.

\forall \epsilon > 0, \exists \delta > 0 \text{ s.t. } 0 < |x - c| < \delta \implies |f(x) - L| < \epsilon

Target Point (c): 2

Epsilon ($\epsilon$): 0.5

Delta ($\delta$): 0.5

Status: Checking...

Goal: For a given error margin $\epsilon$ (the horizontal band), find a input range $\delta$ (the vertical band) such that the function graph stays entirely within the rectangle. If the graph exits the top or bottom of the box before exiting the sides, your $\delta$ is too big!

Riemann Sums

Approximating the area under a curve.

\int_a^b f(x) dx \approx \sum_{i=1}^n f(x_i^*) \Delta x

Rectangles (n): 10

Method:

Left Midpoint Right

Actual Area: ...

Approx Area: ...

Error: ...

As $n \to \infty$, the sum of the rectangle areas converges to the exact area under the curve. Notice how the Midpoint rule usually converges faster than Left or Right!

Taylor Series

Approximating functions with polynomials.

f(x) \approx P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x-a)^k

Degree (n): 1

Center (a): 0

Function:

A Taylor Series builds a polynomial that matches the function's value, slope, curvature, etc., at a single point $a$. Increase the Degree to see the polynomial "hug" the function further away from the center.