math2015Term2

=Class of 2015 Mathematics HL Term 2 & Term 3=

And we continue with the course.

=== //more revision//

Notice, a big mistake in the solutions to practice exam Q7 of P1!!!

Here some more solutions to the packages handed out in class:
 * [[file:mHLrev_ExpLogs_ms.pdf]]
 * [[file:mHLrev_FactorRemThm_ms.pdf]]
 * [[file:mHLrev_Trig_ms.pdf]]
 * [[file:mHLrev_Vectors_ms.pdf]]
 * [[file:mHLrev_Fns2013_ms.pdf]]
 * [[file:mHLrev_Calculus2013_ms.pdf]]

For more practice on complex numbers, the book has some great end of chapter questions based on past examination papers.

Success & enjoy, Arno.

=== //week 19 May//

Revision has started. To see what questions you are required to be in a position to tackle, see all the unit tests as well as the prep package (P1 & P2) given out. The solutions to the latter are here:
 * P1: [[file:EoYP1Practice.pdf]]
 * P2: [[file:EoYP2PracticeTK.pdf]]

Success & enjoy, Arno.

=== //Mon 12 May//

Last week in vectors: asking ourselves what we can do with three planes.

So what can they do:
 * 1) Meet in a line, i.e., all three planes meet in a single line
 * 2) Meet in a point.
 * 3) Are all parallel.

To investigate this we use Gaussian elimination of the augmented matrix into a row reduced Echelon form (such a fantastic phrase!); notes:

Homework from the textbook: Exercise 14F p.462-3, Qs 1, 2, 5, 6, 9.

I'll show you on Thursday how to use your calculator to get the row reduced Echelon augmented matrix, but remember you must be able to do it without your calculator as well for Paper 1.

Success & enjoy, Arno.

=== //week 5 May//

Planes -the infinite kind.

We lay out how we describe a plane mathematically, in various forms. And by the end of the week we will have systematically gone through questions we can ask when working with the following mathematical objects:
 * 2 points
 * 1 point and a line
 * 1 point and a plane
 * 2 lines
 * 1 line and a plane
 * 2 planes

Pretty impressive stuff!

Some notes and assessment at the level of expected in this unit:
 * notes vector planes: [[file:Notes 43_01_Vector Planes.pdf]]
 * planes represented in their different forms: [[file:Notes 43_02_Planes and Their Different Forms.pdf]]
 * angles in 3D space: [[file:Notes_43_03_Angles_in_Space.pdf]]
 * problem set: [[file:20 point Assessment.pdf]]

Success & enjoy, Arno.

=== //Sat 3 May//

Finally created the screencast Hank asked about: Q8 of the complex number test. And here she is media type="custom" key="25805102"

Enjoy, Arno.

=== week 28 Apr

Vector multiplication week:

With assessment to check various properties of the scalar and vector product:

Also, be sure to work through the textbook problems, these product rules will be the cornerstone of the final two weeks of this course.

Success & enjoy, Arno.

=== //Thu 24 Apr//

We finished the problem set ... almost; solutions here:

This weekend you are working on this in preparation for next week's work:

We are on a trajectory now, do not fall off the line we are walking!

Success & enjoy, Arno.

=== //Wed 23 Apr//

Alas we lost Monday, fortunately we started to explore vector's way of looking at lines last week.

Here the notes:.

Here assessment problem set:.

Please have the problem set done by tomorrow morning's class.

Success & enjoy, Arno.

=== //Thu 17 Apr//

Here the solutions to the problem set:

Success & enjoy, Arno.

=== //Mon 14 & Wed 16 Apr//

Oh yes, a new topic ... vectors.

We will, in a couple of weeks be doing this kinds of calculations: media type="custom" key="25657464"

But for now, we start with definitions and first explorations, see notes for this week here:

On Wed I gave out this problem set, to be finished by Thursday's class:

Recall the exciting news, GeoGebra 5 will have 3D capabilities, beta version can be downloaded already to give it a spin: GeoGebra 5 beta

Success & enjoy, Arno.

=== //Thu 10 Apr//

Complex number test, finally time to show your stuff.

Success & enjoy, Arno.

=== //Wed 9 Apr//

Practice time for the test tomorrow, bring on the questions!

Success & enjoy, Arno.

=== //Tue 8 Apr//

Here the solutions to the problem set:.

Enjoy, Arno.

=== //Mon 7 Apr//

Getting ready for the test, via this Problem Set, self-assessed or assessed by me (solutions go out tomorrow, get it to me before if I am assessing it for you):.

Success & enjoy, Arno.

=== //Wed 2 Apr//

Recipe for n'th root problems with examples.

Homework from textbook p.516.

Tomorrow problem set in class, can be handed in for credit.

Unit test to be determined.

Success & enjoy, Arno.

=== //Mon 31 Mar//

Last week of Complex numbers.

Today we formalised De Moivre's Theorem (proof later, by induction):.

As did we the Fundamental Theorem of Algebra, Conjugate Root Theorem and consequences for solutions to polynomial equations, which **some** of you discovered in the process of the investigation.

For next class, please go over the following presentation on Nth Roots **carefully**, i.e., make notes:. An executive summary is this one:.

We will do some work on this on Wednesday with perhaps a finishing Test on Thursday.

Success & enjoy, Arno.

=== //Thu 27 Mar//

Oh the glory of GeoGebra!

All late comers should be up to date with the notes and homework by now.

All should have the Roots of Complex Numbers inquiry emailed to me in PDF by Monday, or else given to me in hardcopy.

Criteria are in yesterday's post.

Success & enjoy, Arno.

=== //Wed 26 Mar//

Finishing up our discussion of the various forms of complex numbers and how useful they are for different operations on complex numbers.

While some stragglers a catching up on missed work, others are working on an investigation:. Hand in of this investigation is Monday March 31st and will be graded according to the following rubric:.

Success & enjoy, Arno.

=== GeoGebra:

Here a few of GeoGebra files to play with:
 * complex conjugates on the Argand diagram [[file:complexcong.ggb]]
 * sum and differences of complex numbers and the effect on their modules and argument (great visualisation and helpful for when we will do vectors later): [[file:complex_sum_and_diff .ggb]]
 * what happens when you multiply a complex number by //i,// notice in particular the four-part cyclic nature of that process: [[file:multiplication_by_i_dynamic.ggb]]

Success & enjoy, Arno.

=== //Mon 24 Mar//

Revisiting:
 * Argand diagram, in particular where conjugate pairs live
 * various forms of complex numbers: rectangular, polar form and now also the Euler form
 * n solutions to degree n polynomial equations

And then we looked at some properties of the modulus and argument of complex numbers, these are very useful.

Some notes on this material:
 * polynomial equations: [[file:notes_05_22_02.pdf]]
 * polar form and properties of modulus and argument:[[file:Notes 05.22.02.pdf]]
 * Euler form and properties: [[file:22_03_notes.pdf]]

Homework from textbook pages 508-510, questions: 1; 2 a, c, f; 3, 4, 6, 7

Success & enjoy, Arno. === //Thu 6 Mar//

We went ahead to look at two different representations of complex numbers, namely
 * rectangular form: z = x + iy
 * polar form: z=r*cis(theta)

We identified these on the Argand diagram.

And then we started to explore using solutions over the complex plane to always get two solutions for quadratics, and realised that under certain circumstances, these are always conjugate pairs.

Notes for this class:

Practice: with solutions:

Success & enjoy, Arno.

=== //Wed 5 Mar//

Final theory for this week's work, though I may amp it up a little tomorrow with a preview or some other cool complex number stuff.

Last notes are:

At any rate, some more practice work:
 * quiz B: [[file:Diagnostic Quiz B.pdf]] with solutions: [[file:Diagnostic Quiz B Solutions.pdf]]
 * diagnostic quiz 2: [[file:Diagnostic Quiz 05.21.02.pdf]] with solutions: [[file:Diagnostic Quiz 05.21.02 solutions.pdf]]

And self-assessed Problem set:
 * questions: [[file:05_21_02 Problem Set.pdf]]
 * solutions: [[file:05_21_02 Problem Set solution.pdf]]
 * self -assessment criteria: [[file:Self AssessmentRubric.pdf]]

Success & enjoy, Arno.

=== //Mon 3 Mar//

For starters, solutions to last week's differential calculus test:

Then we started our 5th unit: Complex numbers. These are a treat, opens up ways to solve otherwise non-trivial problems like a breeze. Here is the podcast on some background on imaginary numbers and their uses, highly recommended: Imaginary Numbers

Today we introduced the idea of the imaginary number //i// and how we construct a complex number with that and some basic properties of complex numbers:

Solutions to the homework given out today:

Success & enjoy, Arno.

=== //Mon 24 Feb//

Snow day

But that didn't keep us from doing some final touches on our differentiation topic.

You were given the self-assed problem set (solutions attached).

Some last minute standard derivatives:

The solutions to some additional implicit differentiation work I handed out:

Remember, test on Thursday.

Success & enjoy, Arno.

=== //Mon 17 Feb//

After a disastrous time for mathematics due to all kinds of activities last week, back to work.

This week is great: implicit differentiation and rates of change -powerful stuff.

Today we looked at implicit differentiation, with the key to success being to consider d/dx as a operator! Notes here:.

Not much to the theory, but practice is crucial. Homework from Exercise 18D p.614 from the textbook.

This is the last new material for now on calculus (with more to come in two later installments!). Hence, next week a test.

Take care, Arno.

=== //Thu 13 Feb//

A very short week so we went over last week's quiz. The solutions to which are here:.

Enjoy, Arno.

=== //Wed 5 Feb//

Phenomenally, we have three rather useful rules to assist us in differentiating functions which are build up from the functions from which we know the standard derivatives: chain rule (to deal with composite functions), product rule, and quotient rule; these are summarized on the Notes shared on Mon 3 Feb below.

Homework from exercise 18A.

Tomorrow a wee quiz on what we have covered in differentiation so far.

Success & enjoy, Arno.

=== //Mon 3 Feb//

Another very, very poor submission rate of investigation ...

Today was a good day for differentiation, as we learned how to deal with the derivatives of the trig functions, natural exponentials (the beauty of this!) and logarithms.

Note that the rules for these are all in your Information Booklet:

Notes for this week are here:

Homework from the textbook 16E p. 548: 1, 2, 4, 5 16F p. 550: 1, 2, 3, 4 16G p.553: 3, 7 16H p.559: 1 b, c, 5

Success & enjoy, Arno.

=== //Thu 30 Jan//

Bo-ho, no snow = no skiing.

So we made good use of our 45 minutes by looking at the quartic investigation in particular how GeoGebra can help us.

Here are some screenshots from my work with GeoGebra and putting the results of various quartic functions in a spreadsheet to calculate the ratios of segments. Point is, software can assist you, in particular when you have repeat identical/similar analysis but for different input (in my set up, all I need to do is input a new quartic function and out rolls all the new results including the segments, I could even put them into GeoGebra spreadsheet so not to have to put that information into my OpenOffice spreadsheet).





Success & enjoy, Arno.

=== //Wed 29 Jan//

Oh beauty, the power of calculus is blinding!

Today we saw that, while from //f(x)// we can easily learn the axis-intercepts (//f(x)=//0 and //f(0//)//).// Now, from //f'(x)// we find **whether or not** //f(x)// has stationary points, namely the points which have //f'(x)=//0, these points are **necessarily stationary points**. We know on either side of these points the gradient has opposite sign.

Now when we were looking at a cubic function we discovered that
 * it may have either two, one or no points of inflection (remember how you can tell?!)
 * there are points where the curvature changes from positive to negative ... which means in between the curvature must be zero ... another interesting point ... which we call **points of inflection**!

Points of inflection are trickier than stationary points, because there are **two conditions which must be met in order for it to be a point of inflection**, each of these conditions individually are only necessary, not sufficient. The two conditions are:
 * on either side the gradient has the same sign (positive or negative)
 * on either side the curvature is opposite.

Homework:
 * textbook: 16H p. 559: Q 1-a, 2, 3, 9.
 * quartic investigation:[[file:Quartic_Investigation.pdf]] to be handed in on Monday and will be assessed according to these criteria: [[file:Rubric_Quartic_Investigation.pdf]].

Success & enjoy, Arno.

=== //Mon 27 Jan//

With the standard derivative as a tool and the discovered rule for how to differentiate a sum of functions, we can apply this knowledge and skill to the following:
 * 1) creating tangent and normal lines to curves (not surprising, after all, we got to differentiation from the concept of tangent line to a curve)
 * 2) finding the coordinates of local maxima/minima, namely places where the first derivative is zero (horizontal tangent line!) and getting a first method, using the first derivative, of categorising these as either a local maximum or a local minimum (the picture I have in mind is walking **up** to the summit and then **down** for a max, as oppose to walking **down** to a valley and then **up** from the valley floor).

Homework:
 * Ex 16G p.553 Q 1.a, 2, 5 & 8.

Remember, a short week with a look at some important applications of differentiation.

Success & enjoy, Arno.

=== //Thu 23 Jan//

We now have the standard derivative for power law function, yeah!

You should be able to show that using differentiation from first principles (you need not be able to show all standard derivatives for the functions we will consider, but this one is easy enough for you to tackle).

You are invited to let that knowledge loose on this extended problem, to be handed in on Monday:

Success & enjoy, Arno.

=== //Wed 22 Jan//

None of you bothered to calculate the 7 numbers representing the value of the gradient to our quadratic from first principles even though you had since Monday. Ouch!

Anyway, you spend the class in library doing that.

On top of that, you can do the work we were supposed to do in class today:
 * 1) plot the 7 points you found and find the equation of the curve
 * 2) Consider with careful consideration to the shapes of the functions and their derivatives, textbook pp. 528-534.
 * 3) Read these notes about the continuity of a function: [[file:ContinuityNotes.pdf]]
 * 4) Appreciate the parts of these Notes, including differentiation from first principles, and do the following problems from the textbook: Ex. 16B p. 538 Q 1.
 * 5) Read the standard derivative, namely for polynomial functions, textbook pp. 538-540, summarised here: [[file:Notes_Lesson_04_15.02.pdf]].
 * 6) Practice from textbook Ex 16c pp. 540-541 Qs 1 all, Q2 all, Q3 all.

Enjoy, Arno.

=== //Mon 20 Jan//

Introductory look at differentiation: gradients of lines and quadratics.

=== //Thu 16 Jan//

The wee little test this morning.

Next we are starting (drum rolling): //calculus//!

The core topic of calculus is divided up into two units, one on differentiation and one on integration. Next week we will start differentiation.

In preparation for that, you can have a look at the outline of the differentiation unit:

In addition, I ask you to listen to the following podcast for some historical background (45 minutes): Leibniz vs. Newton.

Enjoy, Arno.

=== //Wed 15 Jan//

Finished the last wee little bit trigonometry, which will serve us very well in complex numbers and calculus.

The solutions to the problem set are:

A little test tomorrow morning.

Success & enjoy, Arno.

=== //Mon 13 Jan//

Final touches on trigonometry with a look at inverse trig functions. Not that much to it, but just be careful that you ensure that the relations are well behaved, i.e., are functions.

Some notes on this week's work:

On Wednesday I will be handing out the following problem set after a little more work together:

A small test on trig on Thursday to round things off.

Success & enjoy, Arno.

=== //Mon - Fri 6 - 10 January// Feedback and corrections of semester test.