A few administrative notes before we review the day's material: I will not be holding office hours this Wednesday. And there are no classes next Monday, when my usual set of office hours are. But I've decided to do a sort of experiment: I don't plan on reviewing for the exam specifically next week, but a large portion of the class has said that they would come to office hours on Monday if I were to have them. So I'm going to hold them to that - I'll be in Kassar House 105 (the MRC room) from 7-8:30 (or so, later perhaps if there are a lot of questions), and this will dually function as my office hours and a sort of review session.
But this comes with a few strings attached: firstly, I'll be willing to answer any question, but I'm not going to prepare a review; secondly, if there is poor turnout, then this won't happen again. Alrighty!
The topic of the day was differentiation! The three questions of the day were -
- Differentiate the following functions:
- $e^x$
- $e^{e^x}$
- $e^{e^{e^x}}$
- $\sin x$
- $\sin (\sin x)$
- $\sin (\sin (\sin x))$
- A particle moves along a line with its position described by the function $s(t) = a_0t^2 + a_1t + a_2$. If we know that it's acceleration is always $20$ m/s/s, that its velocity at $t = 1$ is $-10$ m/s, and its position at $t = 2$ is $20$ m. What are $a_0, a_1, a_2$?
- Given that $u(x) = (x^2 + x + 2$, what are the following:
- $\frac{d}{dx} (u(x))^2$
- $\frac{d}{dx} (u(x))^n$
- $\frac{d}{dx} (5 + x^3)^{-3}$
- $\frac{d}{dx} ((u(x))^n)^m$
Question 1
This is all about the chain rule. Please note that this is a big deal, so if you have any trouble at all with the chain rule, seek extra help. The derivative of $e^x$ is $e^x$. To compute the derivative of $e^{e^x}$, we might think of $u(x) = e^x$, so that we have $e^u$. The derivative of $e^u$ will be $e^u u'$, which gives us $e^{e^x}e^x$. Let's look at the other way of understanding the chain rule to compute the derivative of $e^{e^{e^x}}$. The "outer function" is $e^{(\cdot)}$. It's derivative is just itself. The first "inner function" is $e^{e^x}$. We have just computed its derivative above (it's $e^{e^x} e^x$). So we multiply them together to get $e^{e^{e^x}}e^{e^x}e^x$.
Similarly, the derivative of $\sin x$ is $\cos x$. The derivative of $\sin \sin x$ requires the chain rule. On the one hand, the outer function is $\sin$, and the derivative of $\sin$ is $\cos$. So we know we will have a $\cos (\sin x)$ in the answer. The inner function is also $\sin x$, so we need to multiply by its derivative. The final answer will be $\cos (\sin x )\cos x$. To compute the derivative of $\sin \sin \sin x$, we again use the chain rule. I will again use helper functions, to illustrate their use. We might call $u(x) = \sin \sin x$, so that we are computing the derivative of $\sin (u)$. Then we get $\cos u u'$. We happen to have computed $u'$ just a moment ago, so the final answer is $\cos \sin \sin x \cos \sin x \sin x$.
Question 2
The key idea of this question is to remember that the function $s(t)$ gives position at time $t$. So its derivative gives a result in terms of position per time, the velocity. And the derivative of velocity will give a result in terms of position per time per time, or acceleration. So the velocity of our particle is $2a_0t + a_1$, and the acceleration is $2a_0$. Since we know that the acceleration is always $20$, we know that $2a_0 = 20$ so that $a_0 = 10$. The velocity at $t = 1$ is $-10$, so we know that $2(10)(1) + a_1 = -10$, so that $a_1 = -30$. Finally, our position at time $t = 2$ is $20$, so that $4(10) + 2(-30) + a_2 = 20$, so that $a_2 = 40$. I used different numbers between the two classes, so don't pay too much attention if the exact details are different between one class and the other.
Question 3
This is more about the chain-rule! This is sort of an explicit example of helper functions. We first want to compute the derivative of $u(x)^2$. By the chain rule, this will be $2u(x)u'(x)$. What is $u'(x)$?. It's $2x + 1$. So the derivative of $u(x)^2$ is $2(x^2 + x + 2)(2x + 1)$. This is a single case of the slightly more general $u(x)^n$. Here, the power rule tells us that the derivative will be $nu(x)^{n-1}u'(x)$, which is $n (x^2 + x + 2)^{n-1}(2x + 1)$.
The idea behind the third question is to see if we can work out the same sort of idea, but without starting with a helper function. (It's perfectly fine to always use helper functions to use the chain rule - that's not a problem at all). The derivative of $(5 + x^3)^{-3}$ will be $-3(5 + x^3)^{-4}(3x^2)$. If we want to see the use of helper functions, call $v(x) = 5 + x^3$, so that we are computing the derivative of $v^{-3}$. The derivative will be $-3v^{-4}v'$, which is exactly what we have above.
I look forward to seeing some of you on Monday, and happy studying!
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Comments (2)
2012-10-09 confused
Firstly, why is it that if we calculate the points of continuity of $\dfrac{xtan x}{x^2 + 1}$, we only end up worrying about when $tan x$ is continuous?
Secondly, the book has a question about finding the discontinuities of $csc (2x)$, but I disagree with the book's answer.
2012-10-09 davidlowryduda
Let's find the points of continuity of $\csc(2x)$. We know that $\csc(2x) = \dfrac{1}{\sin 2x}$, so we are wondering about when $\sin (2x) = 0$
To do this, we first need to remember when $\sin x = 0$. The zeroes of $\sin x$ are $x = 0, \pm \pi, \pm 2 \pi, ...$ or $x = n \pi$ for any integer $n$ (this includes $\sin \pi = 0$, which you mentioned in your email). But we now have $2x$ instead, so we might ask ourselves when $2x = n \pi$ for any integer $n$. This happens when $x = \dfrac{n \pi}{2}$.
Let's check ourselves - $\sin (2 \frac{n \pi}{2}) = \sin (n \pi) = 0$. So this seems about right.
comment 3: On 2012-10-09 davidlowryduda
I was asked: why do we only care about when $\tan x$ is continuous when we consider $\dfrac{x \tan x}{x^2 + 1}$? To answer this, we need to remember something from the book:
If $f(x)$ and $g(x)$ are both continuous at a point $x = c$, then their product $f(x)g(x)$ is also continuous at the point $x = c$.
For our problem, we might realize that $\dfrac{x}{x^2 + 1}$ is continuous everywhere, but $\tan x$ is only continuous whenever $x \neq \frac{\pi}{2} + \pi n$ for any integer $n$. This means that their product $\dfrac{x \tan x}{x^2 + 1}$ is continuous at least when both $\tan x$ and $\dfrac{x}{x^2 + 1}$ are continuous. So our function is continuous at least when $x \neq \frac{\pi}{2} + \pi n$.
But what about when $x = \frac{\pi}{2} + \pi n$? Let's look at one in particular: say $x = \pi/2$. $\tan \pi/2$ is not defined (the left limit is $+ \infty$, the right limit is $- \infty$, but $\tan x$ has a vertical asymptote there). In fact, $\tan x$ isn't defined at any of these points $x = \frac{\pi}{2} + \pi n$, so these points are not in the domain of our function. So the function can't be continuous there.
So the discontinuities of $\dfrac{x \tan x}{x^2 + 1}$ are exactly the places where $\tan x$ is discontinuous.
During my office hours, I mentioned that this not because of anything particularly special about $\dfrac{x}{x^2 + 1}$, but simply because $\dfrac{x}{x^2 + 1}$ is continuous. In a sense, multiplying by a continuous function never reduces the points of continuity.