How do we prove the sum of two angles formulae?
This was a question O.B. asked in office hours. We drew lots of triangles and got lots of relations, but didn't get to the triangles we needed: which are here, for example.
My own usual way of seeing the answer doesn't come from geometry as much as from linear algebra. Namely: in MAT1341 you learn about vector spaces, and matrices. One interpretation of a matrix (among the many you will see) is that if you multiply a matrix \(A\) by a (column) vector \(v\) then the answer \(Av\) is another vector; thus multiplication by \(A\) is a function from one vector space to another. It is called a linear transformation because for any constants \(c,d\) and vectors \(v,w\) you have \(A(cv+dw)= c(Av)+d(Aw)\) -- a really strong property that basically means if you know what your linear transformation does to the vectors in a basis, you know the entire function.
So what does this have to do with angle formulae? Rotate the plane \(\mathbb{R}^2\) counterclockwise through angle \(y\) around the origin. This is a linear transformation, therefore it is given by a matrix, and to find that matrix, I just have to find out where this transformation sends the vector of my favourite basis, say \(\left\{\left[\begin{matrix}1\\0\end{matrix}\right], \left[\begin{matrix}0\\1\end{matrix}\right]\right\}\). You work that out (geometry required a bit here), and put those two vectors (in order) as the columns of a matrix, which gives: \[ A = \left[\begin{matrix}\cos(y) & -\sin(y)\\ \sin(y) & \cos(y) \end{matrix}\right]. \] (Check, if you know matrix multiplication, that this is correct.) So now: suppose I start at a point \((\cos(x),\sin(x))\) on the unit circle, and I rotate by angle \(y\); then I will end up at the point with coordinates \((\cos(x+y),\sin(x+y))\). Alternately, I can use matrix multiplication by \(A\) to work out the same thing: \[ A\left[\begin{matrix}\cos(x)\\ \sin(x)\end{matrix}\right] = \left[\begin{matrix}\cos(y) & -\sin(y)\\ \sin(y) & \cos(y) \end{matrix}\right]\left[\begin{matrix} \cos(x)\\ \sin(x)\end{matrix}\right] = \left[\begin{matrix}\cos(y)\cos(x)-\sin(y)\sin(x)\\ \sin(y)\cos(x)+\cos(y)\sin(x)\end{matrix}\right]. \] As these two are equal, we get our formulae.
Linear algebra is amazing, once you get past the torture of twisting your brain to understand what vector spaces, linear independence of vectors, and spanning sets actually mean.
What was that squeeze theorem with \(\displaystyle \lim_{h\to 0}\frac{\sin(h)}{h}\) all about?
So when we wrote down what we needed to know to compute (using the definition) the derivative of \( y=\sin(x)\), it came down to understanding \(\displaystyle \lim_{h\to 0}\frac{\sin(h)}{h}\) and \(\displaystyle \lim_{h\to 0}\frac{\cos(h)-1}{h}\), both of which are indeterminate forms of type \( \displaystyle \frac{0}{0}\), meaning we do not know their value by inspection.
We then used geometry to prove that \(\cos(h) \leq \frac{\sin(h)}{h} \leq 1\) if \(h\) is close to \(0\). (We did the proof for the case that \(h > 0\) but because \(\frac{\sin(h)}{h}\) is even, it is the same for \(h < 0\)). Here is a graph (sorry, the axis should be labeled \(x\) not \(t\)):
About the midterm:
1) Will we get an equation sheet of any kind? (I assume no, we must have all the derivatives memorized)
No equation sheets and yes, you need to know your derivative formulas; but the only fancy trig I expect you to memorize is \(\sin^2x +\cos^2x=1\) and the double angle formulas (any other trig formulas needed I would include).
2) Will we have to prove any derivative rule on an exam (like the quotient rule) like we did today in class? I wasn't clear on the answer.
No. I might ask you to compute a derivative using the definition, like we did last week, but not to prove the product rule (for example). (But wasn't it nice to see where the magic in that formula came from?)
3) Will we have to use the squeeze theorem for anything on an exam?
If I did, I would set things up to give you the two functions for squeezing, since that part takes a lot of creativity (which is nice for homework or in real life, but less so on an exam).
4) Do you have any old exams/midterms to practice on in preparation for the upcoming assessment?
I will describe the number of questions and format in class on Wednesday, but won't be circulating old exams. Each of the questions is really very strongly correlated to the homework or suggested exercises --- so the best practice is definitely to do the suggested exercises.
Why doesn't MapleTA accept \(\cos(\frac{\pi}{2}-y)\) as an answer in homework 1?
It is true that \(\sin(y)=\cos(\frac{\pi}{2}-y)\), and this is a good identity, but in this question, you want to use an identity that relates \(\sin(y)\) to \(\cos(y)\) (and not to \(\cos(\pi/2-y)\)).
How do I find the functions \(g(x)\) and \(h(x)\) to use the squeeze theorem?
There is no set rule; this is one technique among many that you might use in a given situation to figure out the limit. The important part is the conceptualization: can you see how this theorem logically makes sense, and how it is only true if ALL the hypotheses hold? (For example, if I have \(g(x)\leq f(x) \leq h(x)\) but \(\lim_{x\to a}g(x) \neq \lim_{x\to a}h(x)\), then you can't pretend you know what \(\lim_{x\to a}f(x)\) is --- there wasn't enough information.)
On a test: I would ask you to use the squeeze theorem and set things up so you knew which functions to take. The prototypical example is one like \(f(x) = x^2\sin(s(x))\), for any weird function \(s(x)\), where you know that \( -1 \leq \sin(\text{anything}) \leq 1\) so can squeeze \(f(x)\) between \(-x^2\) and \(x^2\).
In reality: this is one step in how you use approximations to help make sense of functions; Calculus will give you a whole toolbox of these.
Where can I get the textbook? Do I need the online content if I buy a used book?
You can buy the book from the campus bookstore in the University centre or from Agora, the SFUO's bookstore (off campus). The one for sale is the 8th edition because they are no longer printing the 7th. Last year, MAT1320 and MAT1322 used the 7th edition, and it is fine to use that for this year as well. You don't need the online content that comes with the book.
What do I need to bring to class? Textbook? Laptop?
You won't need to bring the book, or your laptop to class. I post my course notes on Blackboard Learn, which are approximately what I will do in class; if copying notes from the board during lecture is causing problems for you, you might try bringing the course notes for the day and just taking the notes for things that are particularly interesting or different. Taking notes is a really good idea; it is the first step in practicing math.
What was that weird function notation about?
In class, I defined a function as a rule that associates to each element in the domain an element of the range. We give our functions names; my favourite function name is \(f\) but \(\ln\), \(\sin\) and \(T\) are also great names.
A compact way to communicate all this information about the function (its name, its domain, its range and its rule, as well as specifying what letter I would like to use for the dependent variable) is the following: \[ \begin{align*} f \quad \colon\quad \mathbb{R} &\to \mathbb{R} \\ x &\mapsto e^x+3 \end{align*}\] When I think the domain or the range are obvious enough, I could just write \(f(x) = e^x+3\). When I don't need to give a name to the function, I could just write \(y=e^x+3\).
So for a more complex example: In MAT1322 I could be interested in the function \[ \begin{align*} g \quad \colon\quad \mathbb{R}^2 &\to \mathbb{R}^3 \\ (x,y) &\mapsto (x+1, xy, y\sin(x)+2) \end{align*}\] which in shorthand would be \(f(x,y) = (x+1,xy, y\sin(x)+2)\). The notation just helps us with any potential ambiguity or confusion. A more down-to-earth example could be: Let \(\mathbb{R}_{>0}\) denote the set of all strictly positive numbers. Then we can define \[ \begin{align*} h \quad \colon\quad \mathbb{R}_{>0} &\to \mathbb{R}_{>0} \\ x &\mapsto (\ln(x))^2+1 \end{align*}\] where I have modified the domain (because \(x > 0\) is necessary for \(\ln(x)\) to make sense) and the range (to give you a bit of information, that you could have deduced anyway here, that the output is always positive.
What is the difference between the range and the image of a function?
The image of a function is exactly the \(y\) values that you can get as an output of the function. The image of \(\sin(x)\) is the interval \([-1,1]\); the image of \(h\) (just above here) is the interval \( [1,\infty)\); the image of \(g\) (just above here) is a really complicated surface in \(\mathbb{R}^3\); the image of \(f\) above is the interval \( (3,\infty)\).
The range is a more generic term, that says where the \(y\)-values live. So in our course, I am always allowed to say that the range is \(\mathbb{R}\), because we only care about functions whose output is a real number. In other courses, your output might be a vector or a complex number, so the range would be the set of all vectors (\(\mathbb{R}^2\), for example, or \(\mathbb{C}\) ).