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Weekly Challenge 43: A Close Match

Can you massage the parameters of these curves to make them match as closely as possible?

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Weekly Challenge 44: Prime Counter

A weekly challenge concerning prime numbers.

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Weekly Challenge 28: the Right Volume

Can you rotate a curve to make a volume of 1?

Weekly Challenge 30: the Sign of the Times

Stage: 5 Short Challenge Level: Challenge Level:2 Challenge Level:2
Draw some sketches of familiar curves and note that the condition in the question can be interpreted geometrically as the function and the gradient having opposite signs at each point along the curve.

So, the question can be rephrased as: the sign of a curve and the sign of its gradient be opposite everywhere?