Area L

Draw the graph of a continuous increasing function in the first quadrant and horizontal and vertical lines through two points. The areas in your sketch lead to a useful formula for finding integrals.

Integral Equation

Solve this integral equation.

Integral Sandwich

Generalise this inequality involving integrals.

Calculus Countdown

Why do this problem?

This problem gives a great engaging context in which to practise calculus. It is very useful for getting across important ideas concerning integration and differentiation as operations, which will be useful at university in both science and maths courses.

Possible approach

Initially play as a straightforward game to try to hit the targets.

Students could then try to create new targets for their friends to try to hit using the same initial functions.

You could then give the students the task of creating their own game of Calculus Countdown.

Once students have hit a target they will need to communicate their answers clearly. How might they write down an answer in a clear and unambiguous way that can easily be interpreted by someone else?

Key questions

What happens when you integrate or differentiate the starting functions?
What would you have to integrate or differentiate in one step to hit the targets? Does this help?

Possible extension

Interesting side questions which might emerge are: What targets are possible? Can you prove that certain targets (such as $\sqrt{x}$ or $3$) are impossible to hit? Clearly, proving the impossibility of a target will require some clear thinking; be sure that students concentrate on clearly writing down their arguments.

Possible support

Focus on the first 3 targets.