Calculus and Optimization

Sheet 1

Limits, Differentiability & Optimization

5 formulas

Key formulas

Continuity at a Point ☆

\[ \lim_{x \to a} f(x) = f(a) \]

Differentiability at a Point ☆

\[ f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h} \]

L'Hopital's Rule ☆

\[ \lim \frac{f}{g} = \lim \frac{f'}{g'} \]

only for 0/0 or ∞/∞ forms

Taylor Series ☆

\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n \]

Maxima / Minima Test ☆

\[ f'(x)=0,\ f''(x)>0 \Rightarrow \text{min} \qquad f'(x)=0,\ f''(x)<0 \Rightarrow \text{max} \]

Test your recall

Can you recall the continuity at a point formula?

Reveal formula

\[ \lim_{x \to a} f(x) = f(a) \]

GATE traps 3 traps

Continuity doesn't imply differentiability

Differentiability implies continuity, but continuity does NOT imply differentiability -- the reverse direction is a frequently tested trick option. f(x)=|x| is the classic counterexample: continuous everywhere, not differentiable at x=0.

L'Hopital only fits 0/0 or ∞/∞

Applying L'Hopital to a form that isn't 0/0 or ∞/∞ is a classic trap -- check the indeterminate form first.

f'(x) = 0 alone doesn't guarantee an extremum

A zero first derivative can also mark an inflection point (e.g. f(x) = x^3 at x = 0) -- confirm with the second-derivative test or a sign change before calling it a max/min.