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# Practice Paper 3 Question 1

Produce a sketch of $$|x|^n + |y|^n =1$$ for each $$n \in \{1, 2, 1000\}.$$

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## Hints

• Hint 1
Try to consider the case when both $$x$$ and $$y$$ are positive, for $$n=1.$$
• Hint 2
Can the behaviour in the first quadrant be reproduced for the other three quadrants, given we take absolute values of both $$x$$ and $$y?$$
• Hint 3
What is the formula for the distance from a given point to the origin? Does it look familiar?
• Hint 4
How small is $$0.1^{1000}?$$ What about $$0.5^{1000}$$ or even $$0.99^{1000}?$$

## Solution

The equation $$|x|^n+|y|^n=1$$ will yield a symmetric shape with respect to both $$x$$ and $$y$$ axes (why?). Thus we can focus only on the first quadrant where $$x$$ and $$y$$ are both positive i.e. where $$x^n+y^n=1,$$ and then reflect the curve onto the other three quadrants.

For $$n=1,$$ we have $$y=1-x.$$ Reflecting this onto other quadrants forms a diamond with corners at $$(\pm1,0)$$ and $$(0,\pm1).$$

For $$n=2,$$ we obtain an equation of the unit circle. This is true, because $$x^2+y^2$$ is the formula for the squared distance from a given point to the origin.

For $$n=1000,$$ think about what happens to any number strictly less than 1 when raised to a very big power. It tends to 0. Now consider all values of $$x$$ that are sufficiently less than 1, even up to 0.999; these will cause $$x^n$$ to decrease sharply towards $$0,$$ and $$y^n$$ to raise sharply towards $$1,$$ for all such $$x.$$ Thus the graph tends to become a square.

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