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1

A nice class of examples that can be used for this purpose are functions that are differentiable but not continuously differentiable (since the "$\lim_{x \to x_0} f'(x)$ trick" doesn't work for them). A typical example is the following exercise: Exercise. Let $f: \mathbb{R} \to \mathbb{R}$ be given by $$ f(x) = \begin{cases} x^2\sin(\frac{...


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Here are some example questions. The graph of the function $f$ is given above. Evaluate the following limits. If the limit is infinite, write $\infty$ or $-\infty$ as appropriate. If the limit does not exist write "DNE". $$ \begin{align*} &\lim_{h \to 0^+} \frac{f(4+h)-f(4)}{h}\\ &\lim_{h \to 0} \frac{f(4+h)-f(4)}{h}\\ &\lim_{x \to 4^...


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I'll argue that this will be likely not feasible for a test question. There's several points in the calculus progression where there's a "hard bottleneck" of some sort, but once you get past that, you can leverage the new fact to easily solve previously-hard problems. One example: finding the derivative of a monomial (from the definition + binomial ...


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The following is not a foolproof solution, but one thing I've done for remote exams is have questions where you have students make up their own numbers, you can add some restrictions to make sure they don't make a problem too easy or find a problem done in the textbook. For example, I've asked things like here is $3 \times 3$ matrix with two or three ...


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A small example: Calculators were allowed on the 1983 (I took it) AP Calculus tests. There had been a long semi-political push to "get technology in". However, students using calculators performed worse than those that did not. Ha! So they pulled it out of the AP Calc. (And delayed putting it into SATs, GREs, etc., which was the real end game ...


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