Math 90: Week 5

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 rest is after the fold –

The topic of the day was differentiation! The three questions of the day were –

  1. Differentiate the following functions:
    1. $latex e^x$
    2. $latex e^{e^x}$
    3. $latex e^{e^{e^x}}$
    4. $latex \sin x$
    5. $latex \sin (\sin x)$
    6. $latex \sin (\sin (\sin x))$
  2. A particle moves along a line with its position described by the function $latex s(t) = a_0t^2 + a_1t + a_2$. If we know that it’s acceleration is always $latex 20$ m/s/s, that its velocity at $latex t = 1$ is $latex -10$ m/s, and its position at $latex t = 2$ is $latex 20$ m. What are $latex a_0, a_1, a_2$?
  3. Given that $latex u(x) = (x^2 + x + 2$, what are the following:
    1. $latex \frac{d}{dx} (u(x))^2$
    2. $latex \frac{d}{dx} (u(x))^n$
    3. $latex \frac{d}{dx} (5 + x^3)^{-3}$
    4. $latex \frac{d}{dx} ((u(x))^n)^m$

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 $latex e^x$ is $latex e^x$. To compute the derivative of $latex e^{e^x}$, we might think of $latex u(x) = e^x$, so that we have $latex e^u$. The derivative of $latex e^u$ will be $latex e^u u’$, which gives us $latex e^{e^x}e^x$. Let’s look at the other way of understanding the chain rule to compute the derivative of $latex e^{e^{e^x}}$. The “outer function” is $latex e^{(\cdot)}$. It’s derivative is just itself. The first “inner function” is $latex e^{e^x}$. We have just computed its derivative above (it’s $latex e^{e^x} e^x$). So we multiply them together to get $latex e^{e^{e^x}}e^{e^x}e^x$.

Similarly, the derivative of $latex \sin x$ is $\cos x$. The derivative of $\sin \sin x$ requires the chain rule. On the one hand, the outer function is $latex \sin$, and the derivative of $latex \sin$ is $\cos$. So we know we will have a $latex \cos (\sin x)$ in the answer. The inner function is also $latex \sin x$, so we need to multiply by its derivative. The final answer will be $latex \cos (\sin x )\cos x$. To compute the derivative of $latex \sin \sin \sin x$, we again use the chain rule. I will again use helper functions, to illustrate their use. We might call $latex u(x) = \sin \sin x$, so that we are computing the derivative of $latex \sin (u)$. Then we get $latex \cos u u’$. We happen to have computed $latex u’$ just a moment ago, so the final answer is $latex \cos \sin \sin x \cos \sin x \sin x$.

Question 2

The key idea of this question is to remember that the function $latex s(t)$ gives position at time $latex t$. 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 $latex 2a_0t + a_1$, and the acceleration is $latex 2a_0$. Since we know that the acceleration is always $latex 20$, we know that $latex 2a_0 = 20$ so that $latex a_0 = 10$. The velocity at $latex t = 1$ is $latex -10$, so we know that $latex 2(10)(1) + a_1 = -10$, so that $latex a_1 = -30$. Finally, our position at time $latex t = 2$ is $latex 20$, so that $latex 4(10) + 2(-30) + a_2 = 20$, so that $latex a_2 = 40$. 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 $latex u(x)^2$. By the chain rule, this will be $latex 2u(x)u'(x)$. What is $latex u'(x)$?. It’s $latex 2x + 1$. So the derivative of $latex u(x)^2$ is $latex 2(x^2 + x + 2)(2x + 1)$. This is a single case of the slightly more general $latex u(x)^n$. Here, the power rule tells us that the derivative will be $latex nu(x)^{n-1}u'(x)$, which is $latex n (x^2 + x + 2)^{n-1}(2x + 1)$.

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 $latex (5 + x^3)^{-3}$ will be $latex -3(5 + x^3)^{-4}(3x^2)$. If we want to see the use of helper functions, call $latex v(x) = 5 + x^3$, so that we are computing the derivative of $latex v^{-3}$. The derivative will be $latex -3v^{-4}v’$, which is exactly what we have above.

I look forward to seeing some of you on Monday, and happy studying!

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