Graphs, approximation, and numerical sense

Taylor Approximations

A Taylor polynomial approximates a smooth function near an anchor point $a$:

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

A degree 1 polynomial is just the tangent line; higher degrees capture local curvature.

Common pitfall: Linear approximation is a local promise. $\sqrt{16.4} \approx 4.05$ works, but using that same tangent line for $\sqrt{25}$ fails. Always check your drift from $a$.

Euler and Graph Reading

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

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

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