Tutor profile: Ryan F.

Explain how the asymptotes and x-intercepts of the graph $$ y = tan\theta$$ are related to $$cos\theta$$ and $$sin\theta$$

By definition, $$tan\theta = \frac{sin\theta}{cos\theta} $$ If we look at the numerator above, $$sin\theta$$, we can recognize that $$tan\theta = 0 $$ if $$sin\theta = 0 $$. From the unit circle and the properties of $$sin\theta$$ we know that $$sin\theta = 0$$ when $$\theta$$ = $$n\pi$$ where $$n$$ is a whole number (0, 1, 2, 3...). Hence, the x-intercepts of $$tan\theta$$ are when $$\theta$$ = $$n\pi$$ where $$n$$ is a whole number (0, 1, 2, 3...). If we look at the denominator above, $$cos\theta$$, we can recognize that if $$cos\theta = 0 $$ then $$tan\theta = \frac{sin\theta}{0} $$ which is undefined as we are dividing by zero. From the unit circle and the properties of $$cos\theta$$ we know that $$cos\theta = 0$$ when $$\theta$$ = $$\frac{n\pi}{2}$$ where $$n$$ is an integer (0, $$\pm$$1, $$\pm$$2, $$\pm$$3...). Hence, the asymptotes of $$tan\theta$$ are when $$\theta$$ = $$\frac{n\pi}{2}$$ where $$n$$ is an integer (0, $$\pm$$1, $$\pm$$2, $$\pm$$3...)

In your own words, describe the idea of an electron cloud and how it is related to the position of an electron inside an atom.

A poweful analogy that can be used to think of this is students inside a classroom. In your class there is no seating plan, the students (electrons) can sit wherever they like. Your teacher is late to class and can not see where any one student is seated. Now, even though the teacher is not in the room, they can have a good idea of where a student might be located. For example, the student is likely sitting at a desk. They might also be likely to be sitting at the desk they commonly sit at. However, there is no guarantee that they are at that desk. The studednt may not be sitting in a chair at all, but could be anywhere in the room (the atom). The only thing the teacher would able to say with guarnatee is that the student (electron) is somewhere in the class (atom). In short, the electron cloud models states that electrons do not orbit the nucleus in a fixed orbit and hence we can not know exactly where the electron is. We can only make an educated guess as to where the electron is likely to be.

Derive the Quadratic Formula $$ x = \frac{-b \pm \sqrt{b^{2} -4ac}}{2a} $$ starting from $$ ax^{2} + bx + c = 0 $$

We will make use of the fact that we know our end point the quadratic formula, and work backwards to get $$ ax^{2} + bx + c = 0 $$ . Starting with the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^{2} -4ac}}{2a} $$ Now, multiply both sides by 2a $$ 2ax = {-b \pm \sqrt{b^{2} -4ac}} $$ Now, add $$b$$ to both sides $$ 2ax + b = \pm \sqrt{b^{2} -4ac}$$ Now, square both sides $$ (2ax + b)^2 = b^{2} -4ac $$ Now, expand the Left hand side $$ (4a^2x^2 + 4abx + b^2) = b^{2} -4ac $$ Now, subtract $$b^2$$ and add $$4ac$$ to both sides $$ 4a^2x^2 + 4abx + 4ac = 0 $$ Now, factor out the common term, $$4a$$ $$ 4a(ax^2 + bx + c) = 0 $$ Now, divide both sides by $$4a$$. Note that $$ \frac{0}{4a} = 0$$ $$ ax^{2} + bx + c = 0 $$ These steps can be performed in reverse to show how one can derive the quadratic equation starting from $$ ax^{2} + bx + c = 0 $$