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