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Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Fractional Calculus I

You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.

Curve Hunter

Age 14 to 18
Challenge Level

 

Why do this problem?

This problem is well suited for those who are about to begin to learn the concepts of calculus. It is easy to access, yet offers many insights into the relationships between functions and their derivatives. The language of calculus - change, derivative, turning points, maximum, minimum, curve, functions, equations, axes, zeros, continuity etc. - should naturally arise in the exploration of this task and it should provide a natural framework onto which to build the formality of calculus at a later date.

As with most NRICH tasks, this problem is low threshold high ceiling, so it also will prove an interesting exploration for the more sophisticated thinker.
 

Possible approach

You can use this task before the words 'turning points' and 'derivative' have been introduced to the students, although the concept of 'continuous' might need explaining. This condition is quite restrictive but does allow for curves which are not differentiable, such as the modulus function. Throughout the task, encourage discussion and explanation of results.
 
This problem works well as a group task. 
 
Start by suggesting that students draw a pair of coordinate axes and roughly sketch a curve with one point of zero gradient. Ask them to locate the places on the x-axis where either the sign of their curve or the sign of its gradient changes. Collect responses concerning the various numbers of zeros in the class, and sketch on the board those with exactly two zeros. Discuss whether there are any other possibilities, with students sketching any other possibilities found on the board. 
 
Once the group is convinced that all possibilities have been found, move on to the next question where students are asked to sketch a curve with exactly two points of zero gradient. Ask them to try to make the most interesting curve they can with exactly two points of zero gradient.  Also discuss what the "simplest" case is.  Does a straight line count as a "curve"?
 
Discuss the interesting curves arising and sketch on the board those with exactly 2 zeros. Discuss whether there are any other possibilities, with students sketching any other possibilities found on the board. It might be useful to discuss curves which are continuous, but are not smooth (i.e. curves which can change direction abruptly, such as a sawtooth function, or the modulus function $y=|x|$).
 
Once this part is complete, allow students to continue with the full task individually or in small groups.
 
It is expected that questions concerning the notion of differentiability, turning points, points of inflection and asymptotes will emerge. These are the key geometric concepts of A-level calculus. Higher levels of continuity will also emerge if students create curves with vertical asymptotes, which are not continuous at the asymptotes.
 

Key questions

  • Can you describe to a friend what a continuous curve is?
  • What can you say about a curve which alternates in sign?
  • What happens to your results if you shift the axes up or down?
  • What is the possible behaviour of a curve as it tends to infinity and minus infinity?
  • What is the simplest curve you can find that fits this description? What is the most complicated?
  • Does a repeated root count as one or two zeros?
 

Possible extension

Students can be encouraged to construct particularly interesting examples: pushing the boundaries of the problem is a good thing.  Students should be encouraged to come up with clear explanations as to why they can create curves with any number of zeros and any number of points with zero gradient. Students can also consider what happens if they are restricted to smooth curves.  
 
You can also extend the task to involve asymptotes and ask, "Explain, with justification, for which whole number values of $A, B, C$ can you create a continuous (except at the asymptotes) curve with $A$ zeros, $B$ turning points and $C$ vertical asymptotes".
 

Possible support

If student are struggling to start, you can sketch a few arbitrary curves on the board and ask students to locate the points of zero gradient and the zeros. Then suggest that they work in pairs and do the same to get a feel for the problem.
 
Some precise and neat students might be resistant to making rough sketches. Encourage them and make it clear that it is allowed and expected in this task! Conversely, some rough and messy students might be resistant to neatly sketching the results of their experiments. Encourage them to understand that writing up neatly is an essential part of this task.
 
Students who are algebra-focussed might not perceive some of this task to be 'maths'. Reassure them that mathematical reasoning is taking place and that constructing clear, precise examples and explanations is highly mathematical and will provide a very solid foundation for the algebraic calculus which will follow.