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'A Function of Gradient' printed from

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Why do this problem?

This problem involves an exploration of the gradient function and will make or reinforce the notion that 'the gradient at each point' is itself a function. It either paves the way for calculus or allows those who already know about the calculus of polynomials to apply their knowledge in an unusual context. In either case, the reasoning is sophisticated and suitable for the highly interested student.
The first parts of this problem are well suited for those about to learn calculus and the second parts (involving the actual curve fitting) well suited to those who already know about integration and might be about to learn more involved aspects of calculus. The problem draws together elements of coordinate geometry, curve sketching, curve fitting and basic integration of polynomials.
The result is both visually beautiful and perhaps surprising.

Possible approach

This problem is suitable for quiet, reflective work, particularly the algebraic parts. 
To begin, start with case $A$ on the board. Discuss how a curve might get smoothly from $(1,1)$ to $(3,4)$ whilst having a gradient $1$ at each of these points. The aim of this discussion is for students to realise that the gradient MUST increase at some point and the MUST decrease as some other point in the hidden region.
You might then continue with case $E$. In this case the goal of the discussion is to realise that the gradient MUST be zero are some point in the hidden region.
Once the problem is understood, students might wish to continue working alone or in small groups.
For the first parts demand clear examples and explanations.
For the second parts students can plot their functions on the same axes to produce a pleasing mathematical image.

Key questions

What are the constraints on the curves of gradient vs $x$?
Do the curves need to go up or down between the start and end points? What does this imply about the gradients?

Possible extension

Extension is suggestion in the problem. You might ask the explicit question: 'Can you find two curves which satisfy condition $B$? If so, what is its equation? If not, what categories of curve fail to work?

Possible support

If students are struggling to get started, suggest that they draw curves freehand which match the conditions and then sketch the gradient function for these examples.
For the part involving the curve fitting, suggest explicitly that students try fitting a gradient function and then integrating. Quadratics are the easiest to fit and all that are needed in the problem.