The Computer Laboratory

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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