Loch Ness

Draw graphs of the sine and modulus functions and explain the humps.

Squareness

The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

Complex Countdown

Play a more cerebral countdown using complex numbers.

Slide

Stage: 5 Challenge Level:

Plot the graph of the function $y=f(x)$ where $f(x) = x(x+|x|)$. Find the first and second derivatives of the function. Show that the first derivative exists at $x=0$ but that the second derivative does not exist at $x = 0$.

NOTES AND BACKGROUND
This is a very simple question but it requires an understanding of how to handle functions that are defined differently on different parts of their domains.

Turning points. Functions and their inverses. Limits of Sequences. Iteration. Differentiation. Graph sketching. Asymptotes. Families of Graphs. Modulus function. Domain/codomain/range. Symmetry.

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Loch Ness

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Stage: 5 Challenge Level: