I've neglected to post homework answers for a while; the purpose of this post is to repair this. I'll try to finish it soon. Let me know if I've missed anything.

• page 125:
• page 129:
• page 149: (a) (0,-7); (b) (2,1); (c) (-2,-1); (d) (3,-9).
• page 154: (-1,0,9)
• page 160: (a) ; (b) 7.
• page 163: .
• page 184: 8.
• page 185: 6, 0, and -1.
• page 188:
• page 210: and .
• page 214: and (-1/2, -1/2, 0).
• page 216: (1/2, -5/2, 3).
• page 227: .
• page 229: .
• page 240: intersect the paths and to get the points (2, -3) and (7, 12).

The advice I give below is based on personal experience taking exams, and on the assumption that your objective is to maximise your exam mark. If your objective is different (e.g. to put in only as much work as is absolutely necessary to pass; or to write the most hilarious solutions to the exam problems), you might have tweak things a bit.

It is a common fear that, the day of the exam, you would find yourself in the following situation:

Rest assured that this will not happen in this course. Overall, the exam is of reasonable length and difficulty, and the problems are representative of the material covered in the course. The word overall is emphasised to indicate a statement of average: some problems are easier and others harder, but overall the exam is reasonable.

How to prepare for the exam

You are "responsible" for the amount of material covered in:

• the notes (theory and examples),
• the blue problem book,
• the tutorial sheets,
• the homework assignments.

Other aids that might come in handy include:

• the answers to the notes' "homework" questions (at the bottom of this page, click on the "calculus" tag to get the list of all posts),
• the past exams you can get from the LMS,
• various consultation hours next week (see the LMS),
• problems from the textbook.

Some people are really good at taking exams, more or less independently of the exam's subject matter. Here is some inside information I got from such people: you have to think of the preparation as a video game. Maybe you are playing a game of Tetris with the admirable goal of improving your visio-spatial perception (as opposed to say, killing time).
Even so, getting a high score on the game is generally not achieved by first getting a PhD in cognitive science -- but rather by playing the game so much that you can do it automatically, without much involvement from your conscious self. It has to be in your fingers rather than in your head, because the fingers are much closer to the keys. If you have to think about it, you are about to lose the game.

The exam situation is very similar, except that you don't really need such a short reaction time. But you will have to solve problems, so you have to practice solving a lot of them. The exam preparation and the actual exam are no time to seek a deeper understanding of integrals and trigonometric functions and complex numbers -- it is too late for that, you need to focus on solving problems rather than understanding big concepts now. (It is likely that by solving a lot of problems you will gain a better understanding of the concepts; that is awesome, you should think of it as a bonus, but it is not your main concern at this point in time.)

This may sound somewhat cynical, but I think it's true: the markers who give you points for the problems look at whether you solved the problem correctly, and they base this decision on what you wrote on the paper. They do not have a crystal ball or an MRI of your brain to check whether you really understood the material.

How to write the exam

Remember that the exam as a whole should be treated as an optimisation problem: you are trying to maximise your final mark.

• there is a 15-minute reading period prior to the start of the exam; this means that the reading period starts at 9:15am and the exam at 9:30am; for good measure, I recommend making it to the venue by 9:00am, especially if it is the first time you go there;
• during the reading period: identify the problems that you think are easy and you can do quickly; if you see a problem that seems ambiguous to you, ask for a clarification;
• some people like to do the following at the start: take the formula sheet and add to it whatever things you learned by heart and are not there already; I never felt the need to do this in an exam, but if you are worried you will forget things later on, this might help;
• start with the easy problems, but do not rush through them; you want to make sure you get all the points on these;
• if you get to an answer that looks like it will be a pain to simplify, leave that for later and move on to another problem;
• try to write down at least the beginning of the solution for every single problem you think you know how to start;
• if you have spent a while on a problem and you are not getting anywhere, move on;
• leaving things for later in the hope of coming back to them seems to assume that you will have time to review your answers; this might or might not happen -- however, if you have to choose between making a desperate attempt to write "anything" in a question you know absolutely nothing about, and going back to simplify/check answers to questions you are happy with, I recommend going with the latter.

DON'T PANIC!

Homework answers

• page 542: clockwise rotation by about the origin.
• page 556: .
• page 558: .
• page 559: circle of radius 4 centred at the origin.

Today we continued our foray into the realm of complex numbers. The lecture was basically about the exponential (polar) form of a complex number; for this we defined the modulus and the argument, and did enough examples to last us a couple of lifetimes. We covered pages 531-541 and 561-564.

Homework answers

• page 536: , .
• page 562: .
• page 563: .

Today we started the last chapter: complex numbers. We defined them and learned how to do arithmetic with them.

Homework answers

• page 520:
• page 528: (i) ; (ii) .

Back to the normal set of notes.

We continued our short exploration of applications of differential equations with an example of Newton's law of cooling (pages 502-506).

Then we went back to page 485 to talk about separable differential equations. I repeat that this is the only method of solution that you need to know for this class (the "type 1 and type 2" methods from last time are just special cases).

This concludes the chapter on differential equations, and next time we'll start our last chapter: complex numbers.

Homework answers

None!

You received a handout with the material for this lecture; it is mostly a rearrangement of some of the differential equations topics. If you lost your handout or never got it (I ran out of them in class), here is an electronic copy.

We continued our introduction to differential equations by talking about the difference between general and particular solutions. We then discussed the simplest method of solving differential equations, by "direct antidifferentiation". Note that doing this is mostly of a pedagogical nature. In the next lecture we will see the technique of separation of variables, and "direct antidifferentiation" is just a special case. The moral of the story is that any differential equation that you will see in this class is separable and can be solved as such.

We then started looking at applications of differential equations and talked about modelling population growth. I stopped on page 478, having only started the example (which will be finished next time).

Homework answers

These are on pages 481-483 of the handout.

1. Check it.
2. Check it.
3. a=0, b=-1, c=1.
4. n=-2, 5.
5. order 2, .
6. Check it.
7. Check it.
8. .
9. (a) ; (b) 2828 bacteria; (c) about 10 hours.

We discussed the second application of integrals, this time to the computation of volumes of solids of revolution. This was the last topic in the integration chapter.

We started very briefly the chapter on differential equations, simply introducing them and talking about what a solution is and how to check it. We stopped on page 464 of the notes (this is page 463 in the revised notes that would be handed out on Wednesday).

Homework answers

None!

We reviewed a couple of important concepts: the definite integral (defined as an area under a curve, or by limits of Riemann sums), and the fundamental theorem of calculus linking definite integral and derivative.

We also talked about integration by substitution in the context of definite integrals, and we saw our first application of integrals, to the computation of areas between curves.

We stopped on page 444.

Homework answers

None!

Today we talked about integration by partial fractions. This is a good technique to master, since breaking up into partial fractions comes up in other situations as well (for instance, for Laplace transforms). Keep in mind that repeated factors in the denominator complicate things slightly. Also, if your factorisation of the denominator has quadratic factors that cannot be factored any further, you will have to complete the square and use arctan's for these.

We stopped on page 422.

Homework answers

• page 415:
• page 416: (a) ; (b) ; (c) ; (d) .
• page 421: (a) ; (b) .

Ah yes, the joy of trigonometric integrals. Enough said.

We stopped on page 406.

Homework answers

• page 402: (a) ; (b) ; (c) .
• page 405: .

We moved into Topic 4 of this course: integral calculus. This started with a revision of antiderivatives of familiar functions, and continued with the most important integration technique: substitution. It's not particularly hard, but you have to work through a bunch of examples yourselves until you get the hang of it.

We stopped on page 392.

Homework answers

• page 377: just do it
• exercise on page 387: substitute , finally get .
• homework on page 387: substitute , finally get .
• page 390: substitute , finally get .

We finished the section on graphing functions with the long example on page 343 (I gave all the answers in class but left out some of the details, make sure you go through it carefully).

We then looked at a typical application of derivatives: optimisation problems. We skipped the first example which is quite straightforward (do it as homework! see the answers below), and worked in detail through the examples on pages 356 and 359. One thing you have to keep in mind is that after you find the stationary points of the function, you must also check the value of the function at the endpoints of the interval on which it is defined, since these endpoints could very well be global maxima or minima (see the graphs on page 353).

We skipped the kangaroo example, which you can look at as homework if you wish (see answers below).

Homework answers

• example on page 354: maximal area is 1250 m^2, attained when the dimensions are 25 m and 50 m
• example on page 362:
  • (a)
  • (b) K(0)=200; 2 years later
  • (c) -110/pi
  • (d) at t=0
  • (e) after 6 years

We finished the long example on page 327, and got an awesome graph to show for it. Then we moved into the last graph-related topic: asymptotes. You should already be familiar with vertical asymptotes, which come up when a denominator becomes 0. We discussed horizontal and oblique asymptotes for rational functions (quotients of polynomials). To find these, use polynomial long division.

We stopped before the long example on page 343.

Homework answers

None!

The next couple of lectures are dedicated to increasing the number of graph-sketching tools at your disposal. Here we are of course talking about sketching graphs by hand (since you won't have access to a graphical calculator or a computer at the final exam).

You should already be aware of finding x-intercepts and y-intercepts. Today's class was about using the first and second derivatives of the function you have to graph, in order to figure out where the graph is increasing/decreasing, where it is concave up/concave down, and where it transitions from one type of behaviour to another (giving local maxima/minima and inflection points).

We stopped in the middle of the long example on page 327, which we will finish on Friday.

Homework answers

None!

Continuing with applications of derivatives, we spent some time looking at problems involving related rates. The notes emphasise approaching these problems via the chain rule, however I prefer using implicit differentiation because

• I find it more intuitive
• there are problems that cannot be done via the chain rule, but can be done by implicit differentiation (see page 311 in the notes).

Any related-rates questions on the final exam (if there are any) can be done using either method, so just go with what you find easiest.

We finished the section around page 309, and then we spent a bit of time talking about the SSLC survey results. If I get around to it, I might write another post about that.

Homework answers

• page 307:

This was one class dedicated to one of the important applications of implicit differentiation: computing the derivative of an inverse function. We focused on inverse trigonometric functions, but the same method works very well in general. For the inverse trigonometric functions, I gave a pictorial method for figuring out how to rewrite the expression for the derivative (which might have y's in it) purely in terms of x. We finished on page 296.

Homework answers

• page 285:
• page 290:
• page 294:

Today we discussed implicit differentiation, or how to take the derivative of y with respect to x in a situation where you cannot obtain an explicit expression for y as a function of x. We ended on page 280.

Homework answers

• page 274: .
• page 278: or .

Back from the Easter break, we started the chapter on differential calculus. Today was mostly a revision of various derivative rules, together with the introduction of the idea that one can take as many derivatives of a (nice) function as one feels like. We stopped on page 264.

It's very easy to get carried away and think of differentiation as just another bunch of rules that you apply blindly and get an answer. I suggest that you have a quick look at the first couple of pages of the wikipedia article on derivatives to refresh your memory on the definition and geometric interpretation of the derivative. This is the starting point of most of the actual applications of derivatives.

Homework answers

• page 247: .
• page 249: .
• page 251: .
• page 253: .
• example on page 254: .
• example on page 255: .
• homework on page 255: .

I see a lot of sad faces and hear the despair in many voices as soon as the topic of implied domain and range of a function comes up. Therefore I decided to give you a very detailed example, illustrating my own minimise-the-confusion way of thinking about these problems.

I will use the following pictorial way of representing a function h: I draw two white boxes, each representing the real numbers , and I draw a blue arrow labelled h going from one box to the other. I want you to think of a white box simply as a box that contains all the real numbers in some random unimportant order (as opposed to the usual picture of the real line, where the real numbers are placed in their usual order).

Next I want to represent the domain and range of h. I will look into the left white box, and shove all the real numbers which are in the domain of h into one corner, so I can keep an eye on them. While I'm at it, I will colour all of them blue, hence I end up with a white box that has a blue corner representing the domain of h. Similarly, I will look into the right white box, and push all the real numbers which are in the range of h into one corner, and colour them blue. We end up with a picture like this:

It might be good to have a concrete example, so let us have a look at the function . We know that its domain is all of , so we colour the entire left box blue. The range is , so we cram this into a corner of the right box and colour it blue as well:

Similarly, if we take the function , we can represent it by the following picture (I'm using red for g to avoid confusing it with h):

So far, nothing much has happened. Let us consider the following fairly typical problem: Find the implied domain and range of the function .

This f is a composition of functions, more precisely with the functions g and h considered above. So we will use the information that we already know about g and h and put it together in a new picture:

Everything should have been fairly automatic to this point, without much thinking required. Now we have to start thinking about the implied domain. What happens when we try to evaluate the function f?

We start with some x in the blue area of the leftmost box (in our example, this happens to be the entire box). Then the blue arrow h takes x and turns it into a real number h(x) in the blue area of the middle box. At this point, we would like to continue and follow the red arrow g; but we can only do this if the blue real number h(x) also happens to be in the red area of the middle box! This gives us a condition that needs to be satisfied: , which is the same as , which is the same as , which is the same as . We need to solve these two inequalities for x. The inequality is satified for all x, since the absolute value is always nonnegative. But the second inequality has as solution the region .

Conclusion: the implied domain of f is the interval [-1/2, 1/2]. As a sanity check, you should convince yourself that it is possible to evaluate f for any x in that interval, and impossible to evaluate f for any x outside of that interval. (Just pick a few values inside the interval and a few outside and see what happens.)

It remains to figure out the range of f. For this we are allowed to feed any number in the implied domain [-1/2, 1/2] into f, and we have to keep track of all the real numbers that we obtain in this process. In the spirit of this example, we will think of this pictorially, by taking the last picture and adding a little bit to it:

What did I do? I drew three new regions in green: on the left, I put the implied domain of f as we just figured it out; in the middle, I put the numbers obtained by plugging the left green region into h (it's no coincidence that it is precisely the intersection of the red and blue regions in the middle -- think about it!); finally, on the right I put the numbers obtained by plugging the middle green region into g. The green region on the right is precisely what we are after: the range of f.

Now we know what is going on: the range of f is obtained by plugging the middle green region [0, 1] into g, i.e. into arcsin. By looking at the graph of arcsin, we realise that if we give it all the real numbers between 0 and 1, we will obtain all the real numbers in . So the range of f is the interval .

Did this make sense? If yes, great. Go and do some more examples on your own. If not, try to go over this example slowly and carefully and pinpoint exactly which part does not make sense to you. Ask a friend, or come to consultation hours and ask about that specific part.

I will go out on a limb and guess that most of you just want to be able to apply this process to a problem on the assignment/final exam, and hopefully get the right answer. That is a reasonable goal, but I hope that you will also make an effort to really understand the process and why it works. It will improve your understanding of functions as mathematical objects, and that is a Really Good Thing.

We spent some time talking about circles and ellipses, and only barely mentioned hyperbolas, stopping on page 126 of the notes.

I emphasised the fact that all of these curves have intrinsic definitions that are independent of any particular way of placing them in a Cartesian plane (e.g. centred at the origin, etc.) I find these definitions much more satisfying than the alternative given in the notes: "a circle is a curve defined by the equation ..., unless you move it to (h, k), in which case it is defined by this other equation ..., but if you instead move it to (-c, -d), then it is defined by this third equation ..."

In the context of this course, you will not have to know the nice geometric definitions of these curves. Most of the work will consist in looking at an equation and being able to recognise which type of curve it represents and what the important points and dimensions are.

I went through the very easy connection between the geometric definition of the circle and its equation. I did not do the same for the ellipse, but I encourage whoever is curious to try it out. It involves a little bit of algebraic manipulation -- the interesting bit is to see how the positions of the foci depend on the parameters a and b.

I went at a leisurely pace through the section on implied domain and range, ignoring a lot of the theory prose and trying to replace it by pictorial representations. I made it to the end of the section (page 115), but I skipped the Example on page 109 (worked out below) and I didn't do the range in the Example on page 111 (to be finished in Monday's class).

I want to work through the Example on page 109 here, as an illustration of the pictorial method that I used throughout the lecture.
Our job is to find the implied domain and range of the function . Let's write for the outer function and for the inner one, so that .

We start by representing the domain and range of h (in green) and g (in red) on the following diagram:

Now we think about the implied domain. We can start with any x in the green region of the leftmost box, which is actually all of . Then we get sent as h(x) into the green region of the middle box; but in order to continue and follow the arrow g, we actually need to be in the green+red region of the middle box, that is in the interval [-1,1]. So the question we need to answer is: For which x in do we have ? Playing with these two inequalities we eventually get , which means that the implied domain is the interval [1,2].

How about the range? If we feed the whole implied domain [1,2] into h, we get the interval [-1,1]. This now gets fed into g, but it is the whole domain of g, so in the end we obtain the whole range of g, which is [0,π].

Today we covered the three main inverse trigonometric functions: arcsine, arccosine, and arctangent, stopping at the end of page 96. (Note that the link above is to the corresponding Wikipedia article, which, as is often the case, tells you more than you ever wanted to know.)

Coming up on Friday: implied domain and range of a function. It's not a bad idea to try to read/think about this a bit before class.

Homework answers

• page 95: (a) , (b) , (c) , (d) .

We continued to talk about trigonometric formulae, including double angles.

I finished by asking some questions about inverse functions, algebraically and graphically. We'll focus on inverses of trigonometric functions on Wednesday.

I made it to the end of page 75, more or less.

I also handed out the questions for the first 5 assignments; this includes the first assignment, due at 10am on Monday 15 March. If you missed class, I have a vague recollection that the questions will be put on LMS.

Homework answers

• page 68: .
• page 71: .

We talked a bit about trigonometric identities, and I finished the class at the end of page 63.

I'll record here what happened during the last 10 minutes of class, since it's not in the notes. If it doesn't make sense to you, don't panic! It is an optional topic, you won't be assessed on it, and you will have plenty of opportunity to see these concepts again (more leisurely) in calculus 2 and linear algebra.

Here goes: we want to figure out formulae for the sine and cosine of a sum of two angles α and θ. One way of doing this is by looking at a clever geometric construction involving a bunch of triangles and a rectangle. See pages 60--63 in the lecture notes for this; if that makes you happy, great!

It doesn't make me particularly happy. If I am going to learn tricks, I prefer ones that apply to a wide range of circumstances, so that I can reuse them without too much effort. The one given in the notes doesn't really apply to anything else. However, there is a way of solving the same problem using two-by-two matrices, and it is made of intermediate steps all of which can be used in a great variety of ways. As I said in class, think of this as a lightning preview of some important mathematical concepts that you will encounter again later in your studies.

Alright, that is enough propaganda, let's see how this works.

If I am given an angle α, I can rotate everything in the plane by that angle, keeping the origin fixed:

(In the picture, I am keeping track of the unit circle -- which looks the same after the rotation -- and of the new position of the coordinate axes.)

This pictorial representation of the rotation is very clear, but somewhat cumbersome to calculate with. Luckily, there is a short algebraic representation that captures all the information and is amenable to computation: this is done using matrices. As you will see, in order to completely determine a "nice" (the actual technical term is linear) transformation of the plane such as our rotation, it is sufficient to record what happens to the two "standard" points (1,0) and (0,1). From the picture we drew above, we see that (1,0) turns into ; after a moment's thought, we realise that (0,1) becomes . We will use a two-by-two matrix to organise this information in a compact way by putting the destination of (1,0) into the first column, and the destination of (0,1) into the second column):

So this is a way of representing the rotation by the angle α. Similarly, the rotation by θ can be written:

and the rotation by the angle (α+θ) can be written:

At this point we should notice that it is possible to perform a rotation by the angle (α+θ) in two steps: first rotate by θ and then rotate the result by α. In terms of the representing matrices, composing two transformations corresponds to multiplying the two matrices:

Ah, but how does one multiply matrices? The quick answer is this (for two-by-two matrices):

In our case, the product of the two matrices we are interested in is

Not pretty, is it? However, here is the good news: this is supposed to be a matrix representing the rotation by the angle (α+θ), so it is equal to the matrix

We now compare each entry of the two matrices and conclude that

These are the two formulae for sine and cosine of a sum of two angles.

Homework answers

• page 50: , , , , , , .

Not much to report: I talked about graphs of trigonometric functions and solving trigonometric equations, and about reciprocal trigonometric functions (secant, cosecant, cotangent). I stopped on page 45 (just before the "harder example", which I will talk about briefly at the beginning of class on Friday).

While I was preparing today's class, it occurred to me that I knew nothing about the etymology of the names of trigonometric functions. So I did a bit of reading on the origins of these names, and they don't seem so arbitrary anymore (except for sine, which was apparently mistranslated from Sanskrit to Arabic to Latin). In particular, cosine is actually much more recent and simply stands for complementi sinus, or the sine of the complementary angle (and the same is true of the other co- functions).

An aside: while reading about these things, I came across Dave's short course in trigonometry. If you're unsure of your trigonometry skillz, give it a try.

Homework answers

• page 25: (a) , (b) , (c) , (d) .
• page 26:

• page 35:

• page 40:

• page 45:

Yet another first day of classes. I ran into some technical trouble with the equipment in Rivett, since the computer was booted into Windows and I had my notes on a hfsplus-formatted USB stick (d'oh!). Moreover, my plan B (get the notes from my web page) also failed miserably since apparently these computers block Internet access -- I guess lecturers are not to be trusted. This is actually rather annoying, because I wanted to do some interactive Sage stuff throughout the course, and I'd rather not have to carry my laptop around...

Anyway, I had to go old-school and use the blackboard (which I like) while projecting my own copy of the notes with the document camera (which I do not like). It was mostly administrative beginning-of-the-semester stuff, with a bit of trigonometry revision (for those who are hoping to get something useful out of this post: I made it to page 19-ish in the notes).

Plan for Wednesday:

1. win the war against technological terror;
2. remember to talk about SSLC representatives and about academic dishonesty;
3. do some more trigonometric function goodness;
4. (if time permits) world domination.

Homework answers

• page 16: , , , , , , , , .
• page 19: (a) , ; (b) , ; (c) , .
• page 22 (drawing these takes forever, so I'll just give you the relevant quadrants -- remember that they are numbered starting with one for the "x and y both positive" one, and going counterclockwise around the circle): (a) second, (b) third and fourth, (c) second and third, (d) fourth, (e) third, (f) second and fourth.
• page 28: (a) , ; (b) ; (c) , ; (d) , ; (e) , , .

---

PS: I hesitated to mention this, but here it is. For a second year in a row, we had a representative of the textbook's publisher come and present the (recommended, not required) textbook and the online features to the students during the first class. This year's instalment was somewhat less advertisement-like than the last's, but I'm still quite uneasy about it. The students pay for their right to sit in lectures -- if they are international students, then they pay a whole lot. I don't think it is appropriate for us to take time out of the lectures in order to sell them a product, no matter how beneficial that product may be to their education. I really have nothing against the publisher's representative; I'm actually grateful for her help in distributing the course materials today. And I have nothing against the publisher trying to connect with interested students and sell the textbook. I just think that an actual University lecture is the wrong venue for this type of activity. Even doing this at the orientation-week information sessions instead seems somewhat more appropriate.

If you're a student in this semester's calculus 1 and you have an opinion (whether positive or negative) about this, I encourage you to give us feedback. There are several ways you can do this: the SSLC survey in about three weeks, or the quality of teaching survey at the end of the semester, or, if you think you'll forget about it by then, directly to Deb, the course coordinator.