Calculus with Functions II: MATH 232, Spring 2014


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$$\lim_{x \to c} f(x) = L \mbox{ if } \forall \epsilon>0, \exists \delta>0 \mid x \in (c-\delta,c+\delta) \Rightarrow f(x) \in (L-\epsilon,L+\epsilon)$$ $$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ $$\int_a^b \!\!\!f(x) \; dx = \lim_{n \to \infty} \sum_{k=1}^{\infty} f(x_k^*) \; \Delta x, \mbox{ where } \Delta x = \tfrac{b-a}{n}, x_k = a+k\,\Delta x, x_k^* \in [x_{k-1},x_k]$$