Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

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?

What Do Functions Do for Tiny X?

Looking at small values of functions. Motivating the existence of the Taylor expansion.

Loch Ness

Stage: 5 Challenge Level:

Remember that the function $f(x) = |x|$ is defined as:

\eqalign{ y &= x \ {\rm for } \ x\geq 0 \cr &= -x \ {\rm for } \ x < 0 .}

Split up the domain of the function.