The CSAT and Practice[+] are designed by the Climb Foundation to help candidates. We are advocates for more opportunity to shine and less opportunity to fail, and we strive to level the playing field. 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. If you have queries or suggestions about the content on this page or the CSAT Practice Platform then you can write to us at . Please do not write to this address regarding general admissions or course queries.