Tutorial Week 7

Arc Length

The formula to find the arc length of a function is \(\int_a^b \sqrt{a + (f'(x))^2} \; dx\).

Q1: Find the length of the curve \(y = -ln(sin(x))\) on \(x \in [\frac{\pi}{4}, \frac{3\pi}{4}]\).

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Complex Numbers

Complex numbers gives us a way of working with the roots of negative numbers.

The key concepts are:

  • \(i = \sqrt{-1}\)

  • Standard form is \(z = a + ib\), where a and b are real numbers

  • Polar form is represented as \(z = re^{i\theta}\), where r and \(\theta\) are real numbers

  • Euler’s Formula gives us \(e^{i\theta} = cos\theta + isin\theta\)

Q2: Write \((1 + \frac{2i}{1 - i})^{25}\) in standard form.

\[\begin{split}\begin{aligned} 1 + \frac{2i}{1-i} &= 1 + \frac{2i(i+1)}{(i-1)(i+1)} \\ &= 1 + \frac{2i^2 + 2i}{1 - i^2} \\ &= 1 + \frac{2i - 2}{1 - (-1)} \\ &= 1 + \frac{2i - 2}{2} \\ &= 1 + -1 + i \\ &= i \end{aligned}\end{split}\]
\[\begin{split}\begin{aligned} (i)^25 &= i(i^2)^23 \\ &= i(-1)^23 \\ &= i(-1) \\ &= -i \end{aligned}\end{split}\]

Q3: Write \(z = 2\sqrt{3}e^{\frac{\pi i}{3}}\) in standard form.

We’re given the polar form of a complex number, that is, in the form of \(z = re^{i\theta}\).

So \(r = 2\sqrt{3}\) and \(\theta = -\frac{\pi}{3}\).

Euler’s formula gives us \(e^{i\theta} = cos\theta + isin\theta\).

It follows that:

\[\begin{split}\begin{aligned} z &= re^{i\theta} \\ &= 2\sqrt{3}e^{i\frac{-\pi}{3}} \\ &= 2\sqrt{3}(cos(-\frac{\pi}{3}) + isin(-\frac{\pi}{3})) \\ &= 2\sqrt{3}(\frac{1}{2} + i(-\frac{\sqrt{3}}{2})) \\ &= \sqrt{3} - 3i \end{aligned}\end{split}\]