Dong Jun


3. Module 4 Calculus Part 2

2021 Test Prep 1: Integration and applications, P&C


Vectors worksheet


Test Prep


Comments on school work

Functions assignment 4
  • 1(i) looks good (the bell shaped curve). But must label! The axes, the origin, the eqn of the graph.
  • 1(i) Range: does the graph touch 0? Answer should be $R_f = (0, 4]$ instead.
  • 1(ii): correct
  • 1(iii): Will highlight the error in whatsapp
  • 1(iv), (v) I'd explain more in depth in class.
  • Q2: try to review your notes on composite functions. Recall that, for $gf$ to exist, we must have $R_f \subseteq D_g$ so we need to find $R_f$ and $D_g$. Then to find $gf(x)$, we sub $f$ into $g$.
    Finally, the range of $gf$ is the tough 2 step approach: first find $R_f$. Then use $R_f$ "as if" it is the domain of $g$ to get $R_{gf}$. If you're not sure we can discuss a simpler example in class.
Transformation tutorial
  • Q1: general transformations, graphs and points look good. Remember to label! (special points, axes, curve and the origin).
    For Q1(iii), point $B$, check your last step. The scale factor is 3.
  • We'd also use Q1(iii) to discuss about the importance of order in class
  • Q2: what's the problem? We can discuss together in class
  • Q3: we'd discuss in class. The key idea in transformations (especially for $x$-transformations is to do replacements so we have to be careful there
  • Q4a: remember for $x$ axis we do the "opposite". We need to replace $x$ with $x-1$ so it's translation of 1 unit in the positive $x$-axis direction.
    Q4b: missed out the $y$-transformation step.
  • We'd try Q5 again after we tackle Q3
  • Q6a: Take note the difference between $y=f(|x|)$ and $y=|f(x)|$. We'd discuss that in class.
    Q6b $y=f(|x|)$ is good: but need to label!
  • We'd tackle Q7(i) after discussing orders, and Q7(ii) after we review $y=\frac{1}{f(x)}$ in Q6.

Extra questions on graphing techniques 2:

Q2(i), 3(ai) 1. Module 2 Graphing Techniques
Explanation type questions: Transformations June 2020
Q1,2,3,4,5,7,8,10,11 2_Graphs_2019

Graphs of Inequalities

Q1, 3bi, iii, 6, 7 1. Module 2 Graphing Techniques

Vectors test prep

Vectors June 2020 Lines and Planes 2
Vectors June 2020 Lines and Planes

21st June: Comments on vectors worksheet

  • Q1b is on the right track. How come didn't continue?
  • Q1c: Our foot of perpendicular approach: $\overrightarrow{AF} = (\overrightarrow{AB}\cdot \hat{\mathbf{d}})\hat{\mathbf{d}}$.
  • Q1d: We'd discuss after we finished 1c.
  • Q2: The trick is to translate the story into our language of lines and planes
    • Q2a: the path along which the skydiver lands is a line. Are you able to find the direction vector of it? Meanwhile, the field is a plane with equation $x+2y-2z=0$. Able to recall how to find the angle between a line and a plane?
    • Q2b: Point $A$ has coordinates $(6.5,6,4)$. The field is a plen with equation $x+2y-2z=0$. Able to recall how to find the perpendicular distance between a point and a plane?
    • Q2c is much tougher. We'd discuss it once we're done with 2a and 2b. We'd also discuss the weirder Q2d after that.
  • Q3b: careless on the line "$-6-2\lambda = -1-5\mu$. Check the eqn in part (a) again: should be positive 6, not negative 6.
  • Q3c: This can be done without part b. Know how to?
  • Q3d: To practice for potential "hence or otherwise" questions, try out this using the otherwise method. Then I'd show you the hence method.
  • Q4a: concept is correct. But careless mistake in your cross product. The normal vector should be $\begin{pmatrix}1\\4\\-2\end{pmatrix}$. You had an extra negative for the first component. Check it again. Also remind me to discuss exam techniques with you here.
  • Q4b: Remember how to find intersection between line and plane?
  • Q4d: The foot of the perpendicular concept for planes. See if you're able to draw out a picture of $A$ and the plane and see that $\overrightarrow{FA} = (\overrightarrow{CA}\cdot \hat{\mathbf{n}})\hat{\mathbf{n}}$. I'd guide you if you get stuck.
  • Q5 is the nastiest question in this set. We'd cover it last.
  • Q6b is about the intersection between two lines. (similar to what we did for Q3b where your concept was correct). Able to try it out on your own?

14th June: Functions discussion

Summary in PDF form: Functions 14th may
  • Repeating/periodic functions $f(x+a) = f(x)$ or $f(x-b)=f(b)$ (periods are $a,b$ respectively).
  • Piecewise functions: concepts and use of GC
  • School notes example 17: combination of the 2
  • Tutorial question Q9 (see below)
  • Extra: advanced function manipulation
  • Tutorial question Q7(ii), 8(ii) (see below)
  • Extra: "self-inverse" functions
  • TYS questions to try: Functions TYS Collection
  • Odd and even functions:
    • Even function: $f(x) = f(-x)$. Symmetrical about the $y$-axis.
    • Odd function $f(-x) = -f(x)$. Symmetrical about the origin.
    • Uses: algebra and roots/$x$-intercepts. In integration and area.

Tutorial questions 7(ii), Q8(ii), Q9

Sketching of Q9 can be done from our discussion on piecewise functions. To find $f^{-1}$ for piecewise functions, we repeat our steps for finding inverse (the rule as well as the domain) for the individual pieces.

For Q7(ii), the longer, systematic way to approach will be to continue working with composite functions. $f^3 (x) = fff(x) = f(f^2(x)) = f(\frac{x-1}{x}) = \frac{1}{1-\frac{x-1}{x}} \ldots$ After quite a bit of algebraic simplification we can get the answer $f^3 (x) = x$.

But turns out there's a faster method! We can make use of part (i). $f^3 (x) = f f^2 (x) = f f^{-1} (x) = x$.

For Q8(ii), there's not much we can do except play around with algebra/substitution. We have $hf(x) = 4x^2-4x-7$ which means that $h(2x-1) = 4x^2-4x-7$. Our aim is to find $h(x)$. How can we get rid of the 2? How about the $-1$?

A modification of Q8(ii) will involve the question telling us that $fh(x) = 4x^2-4x-7$. and ask us to find $h(x)$. We can do a substitution approach as well, but there's an easier way!

Functions past prelims questions

Plan for 30th May

  • Continue with earlier topics: inequalities, graphs, vectors
  • Add-on to function discussion: describing sets
  • Functions Worksheet Q3-10 Functions Basic Questions

Selected questions

Comments on Topic 2 worksheet:

In general, most of your asymptotes and curves are drawn outright. Missed most of the $y$-intercepts and the labelling of the $x$ and $y$ axis. Legend: *: should look at them on your own first. **: I plan to do a more in depth discussion with you.
    1. Looks good.
    2. * The trick for conics is to force the standard form out. We always make the RHS 1: $$ \frac{(5x)^2}{13^2} + \frac{(12y)^2}{13^2} = 1$$ Then see if you understand this next step: $$ \frac{x^2}{\left (\frac{13}{5}\right)^2} + \frac{y^2}{\left (\frac{13}{12}\right)^2} = 1 $$ Are you able to draw the curve now?
  1. ** I'd discuss with you in person.
  2. * Must write the vertical asymptote $x=0$ beside the $y$-axis.
    ** I'd discuss the presentation of asymptote answers for Q3,4 with you.
    1. Looks good (with exception of presentation comments above).
    2. * Miss the $y$-intercept.
    3. Looks good
    1. * Careless in long division. Try it again and see what you get.
    2. ** We'd discuss this together
    1. ** Looks good. I'd explain the standard method to reinforce.
    2. * The algebraic method: using the discriminant. Can remember the method?
    3. * Looks good but missing $y$-intercept.
      1. * Careless in long division. Try again! ** I'd discuss ways to avoid careless mistakes costing us too much marks.
      2. ** We'd try this out after discussions from earlier questions
      3. ** We'd discuss later
    1. Looks good
    1. Looks good
    2. * Must do differentiation. Still remember how?
    3. ** We'd discuss together
    4. ** We'd try after discussing earlier questions

Comments on Topic 3 worksheet:

Legend: *: should look at them on your own first. **: I plan to do a more in depth discussion with you.
  1. * Need to comment that $x^2 + 7$ is always positive. For second part, need to reject $\mathrm{e}^x < - 11$.
  2. ** Check to make sure we know that the $-1$ is because of the asymptote of the $\ln(x+1)$ graph.
    ** For second part, must use hence method
  3. Looks good.
  4. * Careless with equation (2): should be $(-2)^2$, not $(-2)$. Similar mistake with equation 3.
  5. ** We'd discuss (iii) together
  6. * For the hence part, let's draw graph of $\sin$ rather than trying to take sine inverse (inequalities don't really work with trigo functions)
  7. * Hint: the curve is a circle/conics so we should probably complete the square first (in terms of $A$ and $B$. Able to do that? We'd discuss how to use that later
    Also, for the part about $y=|x|$, let $x=-2$. Then $y=4$. So we sub these values into our $x^2+y^2+Ax+By+C=0$ curve to find one equation. Try to do the same for $x=-8$.
  8. ** We'd discuss the hence together

Plan for 10th May:

HW for 3rd-10th May, Plan for 10th May:

Plan for 3rd May:

  • Graph school assignment:
    • Answer for part (i): $x=-\lambda$, $y=x-\lambda$.
    • Try parts (iii)-(v) first
    • Part (ii)
  • School work on inequalities
    • Q6 hence discussion (answer should be $-1 < x <1$ )
    • Q7 discussion (answer should be $\ln \frac{1}{12} < x < \ln 2$)
    • Q9 discussion (answer should be $x=15, y=25, z=10$
  • Past prelim questions on inequalities: Q3,5,6,7,9,11 Inequalities and Equations Worksheet
  • Functions discussion Functions Notes and Qns
  • 1 page summary for all topics: Notes - Elements of H2 A Level Math

Plan for 26th April:

Plan for 19th April:

Plan for 12th April:

Plan for 5th April:

  • Review any questions from school/tutorial
  • Discuss the new topic: curves.
    • Use of the GC. Graph, Zoom, Window, Calculate, Table
    • Asymptotes
    • Rational functions and long division
    • Common curve sketching mistakes
    • Practice on these concepts: Curve Sketching Companion Questions
    • Try out a past (modified) TYS question (ignore the transformations part for now): Randomly Generated Curves (TYS)
    • (Next week's plan, but discuss if got time/school went fast) Conics, Parametric curves, use of discriminant.
  • Work on test prep for vectors once we know the details.
  • Past prelim questions on curves and transformations: Curves and Transformations Worksheet. Relevant questions for this week: Q6, 7a,c,d.

Google Drive I will be storing files in for this period of time:

29th March-5th April:

  • Finish up school tutorial: planes

Plan for 29th March:

  • Check on what went on in school the past week + your questions
  • Continue drill work on lines and planes: Vectors 2: Lines and Planes (I believe I have already passed you this before).
  • Start work on a new topic?
Answers to functions assignment: Q1) $f^{-1}(x) = -\sqrt{\frac{4}{x}-1}$. $h(x) = -\sqrt{\frac{4}{x^2}-1}$, $D_h = (0, 2]$. Q2) $gf:x \to = \frac{1}{x^2-1}, x \geq 2$. $R_{gf} = (0, \frac{1}{2}]$. $x=4$.