Question of the Week 13: Second Derivative

The following question is taken from my website Diagnostic Questions. Here you will find 1000s of free, high quality maths multiple choice diagnostic / hinge questions, ideal for Assessment for Learning, which have been created and shared by maths teachers all over the world.

Just like the resource on TES, or any other maths website, at Diagnostic Questions most of the uploads tend to be questions for younger students. This is perfectly logical, as not every school has a 6th form, and not every teacher teaches A Level.

However, in my completely biased way, I would argue that good quality diagnostic (hinge) questions are just as important for A Level students as for any other age or ability. As a teacher, I still need to know quickly and accurately what my students know and don’t know, and a well thought out maths multiple choice question is an ideal way to find out that crucial information.

Take the above question on the second derivative. Now, I find that many students are fairly comfortable and competent with calculus when it comes working out the second derivative of an expression, but when it comes to actually knowing what that means, they come unstuck. Specifically, I have found that students have particular difficulty relating the sign of the second derivative to the graph of the function.

So, what might we learn from students’ answers to the question above:

Answer a) is the correct answer. The rate of change of the gradient in increasing.

Answer b) might be given by students because the gradient is negative, or because the curve itself lies below the x-axis

Answer c) might imply that students are mistaking the point for a turning point, or are simply guessing!

Answer d) could be given because students think that it is impossible to tell the sign of d2y/dx2 at such an arbitrary looking point.

My prediction is that most students will go for option b). Have a go and see what happens.

And if that has whet your appetite for similar questions, well you are in luck. Here is a lovely quiz, jam-packed full of them 🙂

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