1. Let \(a > 0\). Sketch the graph of

    \[y=\dfrac{a+x}{a-x} \quad \text{for} \quad -a < x < a.\]

Are there any noticeable features of this function over this interval?

  1. Let \(0 < \theta < \dfrac{\pi}{2}\). In the diagram below is the half-disc given by \(x^2 + y^2 \leq 1\) and \(y \geq 0\).

    The shaded region \(A\) consists of those points with \(-\cos \theta \leq x \leq \sin \theta\). The region \(B\) is the remainder of the half-disc.

    Find the area of \(A\).

The half-disc with shaded area A and unshaded area B

Can we divide the area \(A\) into regions where we can easily calculate the areas?

  1. Assuming only that \(\sin^2\theta + \cos^2\theta = 1\), show that \(\sin\theta\cos\theta \leq \dfrac{1}{2}\).

Would it help to consider \((\sin\theta-\cos\theta)^2\)?

  1. What is the largest \(\dfrac{\text{area of } A}{\text{area of } B}\) can be, as \(\theta\) varies?

How does our work for (i) help us here?