Differentiation, rigorously

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   Let $f(x) = \alpha x^{2} + \beta x + \gamma$ with $\alpha = 3$, $\beta = 5$ and $\gamma = 9$. Using the limit definition $f'(c) = \lim_{h \to 0} \dfrac{f(c + h) - f(c)}{h}$, compute $f'(-1)$.

2. Warm-up

   Let $f(x) = \alpha x^{2} + \beta x + \gamma$ with $\alpha = 4$, $\beta = 5$ and $\gamma = 8$. Using the limit definition $f'(c) = \lim_{h \to 0} \dfrac{f(c + h) - f(c)}{h}$, compute $f'(-2)$.

3. Warm-up

   Let $f(x) = \alpha x^{2} + \beta x + \gamma$ with $\alpha = 2$, $\beta = 4$ and $\gamma = 5$. Using the limit definition $f'(c) = \lim_{h \to 0} \dfrac{f(c + h) - f(c)}{h}$, compute $f'(0)$.

Problems shown under their original terms: Generated (template, solver-computed).