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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.

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.

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?

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?

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.