Finding the Balance in a Math Question (Teaching)

Question

As we try to work and teach in the midst of this pandemic, some problems arise when making online math exams. My questions is simple: What could be an interesting basic differentiation question such that students doing the online exam still have to work a bit to get it, rather than simply Googling it or using WolframAlpha/Mathematica to solve it for them?

Everything I can think of seems to be either too easy (meaning they can easily solve it online) or too demanding. I'm struggling to find a good balance, if there is any. This is a first-year math module I'm teaching (non-math students).

Any ideas?

Answer 1

How about proving a more obscure result from first principles?

For example:

Prove using first principles that $$\frac{d}{dx}(x\sin x)=\sin x +x\cos x$$ You may assume that $\lim\limits_{h\to0}\frac{\cos h-1}{h}=0,~~\lim\limits_{h\to0}\frac{\sin h}{h}=1.$

As a high school student myself I'd say that is potentially the most challenging part of differentiation (at least at high school level).

However, I think it also depends on what stage your students are at. Have they learnt the chain rule, product rule, quotient rule? Can they differentiate any logarithm? Can they differentiate any exponential and basic trigonometric functions? Can they differentiate inverse trigonometric and hyperbolic functions? Can they derive Maclaurin series for a given function? Can they use l'Hopital's rule?

Additionally, do you want them to use other skills in this question as well, such as knowledge of trigonometric identities?

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Alternatively, perhaps you could ask them to prove certain results using differentiation. For example, if they are proficient in sigma notation and algebra:

Given the standard formula for the value of a geometric series, use differentiation to prove that $$\sum_{r=1}^n ra^r=\frac{a(na^{n+1}-(n+1)a^n+1)}{(a-1)^2}$$ Hence evaluate $$\sum_{r=1}^{13} r2^r$$

This may be too challenging, but it will hopefully be interesting and stimulating for your students.

Another example may be:

Use differentiation to prove that $$\frac{1-\cos x}{\sin x}\equiv\tan\frac{x}{2}+A$$ for some constant $A$. Now find the value of $A$.