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🌿 The Illustrated Guide to Calculus

I. Limits

Example: Find \(\lim_{x \to 1} \frac{x^2 - 1}{x - 1}\)

Step 1: Recognize \(x^2 - 1 = (x - 1)(x + 1)\)
Step 2: Simplify to \(\frac{(x-1)(x+1)}{x-1} = x + 1\)
Step 3: \(\lim_{x \to 1} (x + 1) = 2\)

II. Derivatives

Example: Derivative of \(f(x) = x^2\)

\(f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h \to 2x\)

III. Applications of Derivatives

Example: Maximize \(f(x) = -x^2 + 4x\)

\(f'(x) = -2x + 4 = 0 \to x = 2\)
Second derivative \(f''(x) = -2 \to \text{Maximum}\)

IV. Integrals

Example: \(\int_0^2 (2x) \, dx\)

\(= [x^2]_0^2 = 4 - 0 = 4\)

V. Applications of Integrals

Example: Area between \(y = x\) and \(y = x^2\) from 0 to 1

\(\int_0^1 (x - x^2) \, dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}\)

VI. Differential Equations

Example: Solve \(\frac{dy}{dx} = 3y\)

Separate: \(\frac{1}{y} \, dy = 3 \, dx\)
Integrate: \(\ln|y| = 3x + C \Rightarrow y = Ce^{3x}\)

VII. Series

Example: Sum of geometric series \(\sum_{n=0}^{\infty} ar^n\) where \(|r| < 1\)

Result: \(\frac{a}{1 - r}\)
E.g., \(a = 1, r = 0.5 \Rightarrow \frac{1}{1 - 0.5} = 2\)