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'Weekly Challenge 7: Gradient Match' printed from http://nrich.maths.org/

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On 1cm paper neatly draw a pair of coordinate axes about 10cm by 10 cm and accurately mark the points  $(0, 0)$, $(8, 8)$  and $(4, 6)$.

Smoothly draw any curve between $(0, 0)$ and $(8, 8)$. Locate a point with gradient $1$

Smoothly draw any curve between $(0, 0)$ and $(4, 6)$. Locate a point with gradient $1.5$

Smoothly draw any curve between $(4, 6)$ and $(8, 8)$. Locate a point with gradient $0.5$  

How did I know that the curves you drew would necessarily have such points? Create a geometric proof.

Did you know ... ?

Geometric proofs are useful for gaining intuition concerning calculus, but concepts concerning 'smoothness' need to be made clear and fully mathematical before analytical proofs of such statements can be created. These ideas form the basis of first year undergraduate courses in calculus and analysis which include proof of the 'Mean Value Theorem'.