Integration By Parts

Integration By Parts

JC Mathematics

Today, I’ll share a little something about Integration by Parts. I want to share this because I observe that several students are over-reliant on the LIATE to perform Integration by Parts. This caused them to overlook/ appreciate its use.

Lets start by reviewing the formula for Integration by Parts

\int v \frac{du}{dx} ~dx = uv - \int u \frac{dv}{dx} ~dx

I like to share with students that Integration by Parts have two interesting facts.

  1. It allows us to integrate expression that we cannot integrate. Eg. \text{ln}x or inverse trigonometric functions. This is also closely related to how LIATE is established.

  2. This is closely related to point 1. That is, instead of trying to integrate the expression, we are differentiating it instead. And that itself, is a very important aspect.

Next, I like to point out to students that LIATE is a general rule of thumb.
GENERAL – because it does not work 100% of the times. And today, I’ll use an example to illustrate how LIATE actually fails.

\int \frac{te^t}{(t+1)^2} ~ dt

Here we observe two terms \frac{t}{(t+1)^2} and e^t. Going by LIATE, we should be differentiating \frac{t}{(t+1)^2} and integrating e^t.

\int t^2 e^{t} ~ dt
= \frac{t}{(t+1)^2} e^{t} - \int \frac{(1)(t+1)^2 - t(2)(t+1)}{(t+1)^4} e^t ~dt

Now observe what happened here, after applying integration by parts, it got “worst”. We are stuck to integrating \frac{(1)(t+1)^2 - t(2)(t+1)}{(t+1)^4} with an exponential… This should raise some question marks. But we did follow LIATE.

I’ll leave students to try out this on their own. And feel free to ask questions here. Have fun!

Random Questions from 2017 Prelims #1

Random Questions from 2017 Prelims #1

JC Mathematics

Last year, I shared a handful of random interesting questions from the 2016 Prelims. Students feedback that they were quite helpful and gave them good exposure. I thought I share some that I’ve seen this year. I know, its a bit early for Prelims. But ACJC just had their paper 1. 🙂

This is from ACJC 2017 Prelims Paper 1 Question 7. And it is on complex numbers.

7
(a) Given that 2z + 1 = |w| and 2w-z = 4+8i, solve for w and z.

(b) Find the exact values of x and y, where x, y \in \mathbb{R} such that 2e^{-(\frac{3+x+iy}{i})} = 1 -i

I’ll put the solutions up if I’m free.

But for students stuck, consider checking this link here for (a) and this link here for (b). These links hopefully enlightens students.

Just FYI, you cannot \text{ln} complex numbers as they are not real…

H2 Mathematics (9740) 2016 Prelim Papers

H2 Mathematics (9740) 2016 Prelim Papers

JC Mathematics

So many students have been asking for more practice. I’ll put up all the Prelim Papers for 2016 here. Do note that the syllabus is 9740 so students should practice discretion and skip questions that are out of syllabus. 🙂

Here are the Prelim Paper 2016. Have fun!

Here is the MF26.


2016 ACJC H2 Math Prelim P1 QP
2016 ACJC H2 Math Prelim P2 QP


2016 CJC H2 Math Prelim P1 QP
2016 CJC H2 Math Prelim P2 QP


2016 HCI JC2 Prelim H2 Mathematics Paper 1
2016 HCI JC2 Prelim H2 Mathematics Paper 2


2016 IJC H2 Math Prelim P1 QP
2016 IJC H2 Math Prelim P2 QP


2016 JJC H2 Math Prelim P1 QP
2016 JJC H2 Math Prelim P2 QP


2016 MI H2 Math Prelim P1 QP
2016 MI H2 Math Prelim P2 QP


2016 MJC H2 Math Prelim P1 QP
2016 MJC H2 Math Prelim P2 QP


2016 NJC H2 Math Prelim P1 QP
2016 NJC H2 Math Prelim P2 QP


2016 NYJC H2 Math Prelim P1 QP
2016 NYJC H2 Math Prelim P2 QP


2016 PJC H2 Math Prelim P1 QP
2016 PJC H2 Math Prelim P2 QP


2016 SAJC H2 Math Prelim P1 QP
2016 SAJC H2 Math Prelim P2 QP


2016 SRJC H2 Math Prelim P1 QP
2016 SRJC H2 Math Prelim P2 QP


2016 TJC H2 Math Prelim P1 QP
2016 TJC H2 Math Prelim P2 QP


2016 TPJC H2 Math Prelim P1 QP
2016 TPJC H2 Math Prelim P2 QP


2016 VJC H2 Math Prelim P1 QP
2016 VJC H2 Math Prelim P2 QP


2016 YJC H2 Math Prelim P1 QP
2016 YJC H2 Math Prelim P2 QP


Students, test your Vectors!

Students, test your Vectors!

JC Mathematics

As the prelims examinations draw really close, many students were asking me to give questions to test their concepts for several topics. In class, I had the opportunity to explore several applications questions too. We saw several physics concepts mixed. We also have some conceptual questions that need students to be able to use the entire topic to solve it.

So I’ll share one here. This involves several concepts put together. I’ll put the solution up once I find the time. Concepts that will be involved, will be

  1. Vector Product
  2. Equations of Plane
  3. Finding foot of perpendicular of point

The question in one a reflection of a plane in another plane. I think such questions will come out in a few guided steps in exams. But should a student be able to solve it independently, it shows that he has good understanding.

The plane p has equation x + y + z = 9 and the plane p_1 contains the lines passing through (0, 2, 3) and are parallel to (1, -1, 0) and (0, 1, 1) respectively. Find, in scalar product form, the equation of the plane which is the reflection of p_1 in p.

Thinking Math@TheCulture #9

Thinking [email protected] #9

JC Mathematics, Mathematics

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here


This is a standard summation question. I’m interested in the last part only.

Summation Question

The answer to (ii) is written there by the student. I’ll only do the solution to (iv).

Quick Summary (Probability)

Quick Summary (Probability)

JC Mathematics, Mathematics, University Mathematics

University is starting for some students who took A’levels in 2016. And, one of my ex-students told me to share/ summarise the things to know for probability at University level. Hopefully this helps. H2 Further Mathematics Students will find some of these helpful.


Random Variables

Suppose X is a random variable which can takes values x \in \chi.

X is a discrete r.v. is \chi is countable.
\Rightarrow p(x) is the probability of a value of x and is called the probability mass function.

X is a continuous r.v. is \chi is uncountable.
\Rightarrow f(x) is the probability density function and can be thought of as the probability of a value x.

Probability Mass Function

For a discrete r.v. the probability mass function (PMF) is

p(a) = P(X=a), where a \in \mathbb{R}.

Probability Density Function

If B = (a, b)

P(X \in B) = P(a \le X \le b) = \int_a^b f(x) ~dx.

And strictly speaking,

P(X = a) = \int_a^a f(x) ~dx = 0.

Intuitively,

f(a) =  P(X = a).

Properties of Distributions

For discrete r.v.
p(x) \ge 0 \forall x \in \chi.
\sum_{x \in \chi} p(x) = 1.

For continuous r.v.
f(x) \ge 0 \forall x \in \chi.
\int_{x \in \chi} f(x) ~dx = 1.

Cumulative Distribution Function

For discrete r.v., the Cumulative Distribution Function (CDF) is
F(a) = P(X \le a) = \sum_{x \le a} p(x).

For continuous r.v., the CDF is
F(a) = P(X \le a ) = \int_{- \infty}^a f(x) ~dx.

Expected Value

For a discrete r.v. X, the expected value is
\mathbb{E} (X) = \sum_{x \in \chi} x p(x).

For a continuous r.v. X, the expected value is
\mathbb{E} (X) = \int_{x \in \chi} x f(x) ~dx.

If Y = g(X), then     For a discrete r.v. X,   latex \mathbb{E} (Y) = \mathbb{E} [g(X)] = \sum_{x \in \chi} g(x) p(x)$.

For a continuous r.v. X,
\mathbb{E} (Y) = \mathbb{E} [g(X)] = \int_{x \in \chi} g(x) f(x) ~dx.

Properties of Expectation

For random variables X and Y and constants a, b, \in \mathbb{R}, the expected value has the following properties (applicable to both discrete and continuous r.v.s)

\mathbb{E}(aX + b) = a \mathbb{E}(X) + b

\mathbb{E}(X + Y) = \mathbb{E}(X) + \mathbb{E}(Y)

Realisations of X, denoted by x, may be larger or smaller than \mathbb{E}(X),

If you observed many realisations of X, \mathbb{E}(X) is roughly an average of the values you would observe.

\mathbb{E} (aX + b)
= \int_{- \infty}^{\infty} (ax+b)f(x) ~dx
= \int_{- \infty}^{\infty} axf(x) ~dx + \int_{- \infty}^{\infty} bf(x) ~dx
= a \int_{- \infty}^{\infty} xf(x) ~dx + b \int_{- \infty}^{\infty} f(x) ~dx
= a \mathbb{E} (X) + b

Variance

Generally speaking, variance is defined as

Var(X) = \mathbb{E}[(X- \mathbb{E}(X)^2] = \mathbb{E}[X^2] - \mathbb{E}[X]^2

If X is discrete:

Var(X) = \sum_{x \in \chi} ( x - \mathbb{E}[X])^2 p(x)

If X is continuous:

Var(X) = \int_{x \in \chi} ( x - \mathbb{E}[X])^2 f(x) ~dx

Using the properties of expectations, we can show Var(X) = \mathbb{E}(X^2) - \mathbb{E}(X)^2.

Var(X)
= \mathbb{E} [(X - \mathbb{E}[X])^2]
= \mathbb{E} [(X^2 - 2X \mathbb{E}[X]) + \mathbb{E}[X]^2]
= \mathbb{E}[X^2] - 2\mathbb{E}[X]\mathbb{E}[X] + \mathbb{E}[X]^2
= \mathbb{E}[X^2] - \mathbb{E}[X]^2

Standard Deviation

The standard deviation is defined as

std(X) = \sqrt{Var(X)}

Covariance

For two random variables X and Y, the covariance is generally defined as

Cov(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]

Note that Cov(X, X) = Var(X)

Cov(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[y]

Properties of Variance

Given random variables X and Y, and constants a, b, c \in \mathbb{R},

Var(aX \pm bY \pm b ) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X, Y)

This proof for the above can be done using definitions of expectations and variance.

Properties of Covariance

Given random variables W, X, Y and Z and constants a, b, \in \mathbb{R}

Cov(X, a) = 0

Cov(aX, bY) = ab Cov(X, Y)

Cov(W+X, Y+Z) = Cov(W, Y) + Cov(W, Z) + Cov(X, Y) + Cov(X, Z)

Correlation

Correlation is defined as

Corr(X, Y) = \dfrac{Cov(X, Y)}{Std(X) Std(Y)}

It is clear the -1 \le Corr(X, Y) \le 1.

The properties of correlations of sums of random variables follow from those of covariance and standard deviations above.

Thinking Math@TheCulture #8

Thinking [email protected] #8

JC Mathematics

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here


This is a interesting Complex Number Question.

The complex number z satisfies z + |z| = 2 + 8i. What is |z|^2

Thinking Math@TheCulture #7

Thinking [email protected] #7

JC Mathematics, Mathematics

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here


This is an application question for hypothesis testing from the 9758 H2 Mathematics Specimen Paper 2 Question 10.

The average time required for the manufacture of a certain type of electronic control panel is 17 hours. An alternative manufacturing process is trialled, and the time taken, t hours, for the manufacture of each of 50 randomly chosen panels using the alternative process, in hours, is recorded. The results are summarized as follows

n = 50
\sum t = 835.7
\sum t^2 = 14067.17

The Production Manager wishes to test whether the average time taken for the manufacture of a control panel is different using the alternative process, by carrying out a hypothesis test.
(i) Explain whether the Production Manager should use a 1-tail or a 2-tail test.
(ii) Explain why the Production Manager is able to carry out a hypothesis test without knowing anything about the distribution of the times taken to manufacture the control panels.
(iii) Find unbiased estimates of the population mean and variance, and carry out the test at the 10% level of significance for the Production Manager.
(iv) Suggest a reason why the Production Manager might be prepared to use an alternative process that takes a longer average time than the original process.
The Finance Manager wishes to test whether the average time taken for the manufacture of a control panel is shorter using the alternative process. The Finance Manger finds that the average time taken for the manufacture of each of the 40 randomly chosen control panels, using the alternative process, is 16.7 hours. He carries out a hypothesis test at 10% level of significance.
(v) Explain, with justification, how the population variance of the times will affect the conclusion made by the Finance Manager.

Thinking Math@TheCulture #6

Thinking [email protected] #6

JC Mathematics, Mathematics

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here


This is a very interesting vectors question from a recent JC BT

Shown in the diagram is a methane molecule consisting of a carbon atom, G, with four hydrogen atoms, A, B, C, and D, symmetrically placed around it in three dimensions, such that the four hydrogen atoms form the vertices of a regular tetrahedron.

Methane

Treat A, B, C, D, and G as points. The coordinates of A, B, C, and D are given by (5, -2, 5), (5, 4, -1), (-1, -2, -1) and (-1, 4, 5) respectively, By considering the line DG and the symmetrical properties of methane, find the bond angle of methane, that is, \angle DGA.

Thinking Physics@TheCulture #5

Thinking [email protected] #5

JC Physics

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by Aaron. More of him can be found here.


Given a chance to counter Sir Isaac Newton’s famous quote of “What goes up must come down”, do you think it is true in all scenarios?