Graphs, approximation, and numerical sense

Taylor polynomials approximate a smooth function near a point $a$:

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

Common Taylor expansions (around $a=0$)

$$e^{x} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$

$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$$

Euler's formula links exponentials to trigonometry in the complex plane $\mathbb{C}$:

$$e^{i\theta} = \cos\theta + i\sin\theta$$

A complex number $z = a + bi$ can be written in polar form as $z = r\,e^{i\theta}$, where $r=|z|$ and $\theta=\arg(z)$.

Graphical reading skills

• Concavity: $f''>0$ means the curve bends upward; $f''<0$ means it bends downward.
• Inflection point: Where concavity changes sign.
• Asymptotes: Horizontal ($x\to\pm\infty$), vertical (denominator $\to 0$).

Key insight: A Taylor polynomial of degree 1 is just the tangent line. Higher degrees capture curvature and beyond.

Common pitfall: Linear approximation is a local promise: $\sqrt{16.4} \approx 4.05$ works beautifully, while using the same tangent line for $\sqrt{25}$ fails badly. Always ask how far you have drifted from the anchor point.