differentiation

A-Math: Differentiation & Integration Application : Examples of Typical Kinematics Questions

I was looking through 2008 GCE O Level Additional Mathematics Exam Papers (Subject Code: 4038) and as expected, there was a Kinematics question (worth 6 marks) in Paper 1.

Kinematics is a application topic for Differentiation and Integration. To master this topic, you do not necessarily need to bring in your physics knowledge though it could be useful at times.

Instead, how I get my students to be a master in this topic is to be familiarize with a KINEMATICS VOCABULARY LIST.

Here's some of the vocabulary words that are useful and common:

Momentarily at rest, instantaneously at rest, changes direction of motion, stationary
Initial displacement, initial velocity, initial acceleration
Greatest displacement, greatest velocity, greatest acceleration
Distance travelled in the 4th second VS Distance travelled in the first 4 seconds
Maximum distance from Point O
Particle returns to Point O
Constant Velocity

I would say for Kinematics, it is one of the few topics in A-Math which uses extensive vocabulary. This is also the reason for you to decipher the meaning behind these words.

So do you know the meaning behind these words? I would love to hear about it in the comments section.

I have also taken a few questions from my A-Math TREQ book (Topical Real Exam Questions) to illustrate some common exam questions on Kinematics, further highlighting the importance of knowing your Kinematics well. (Click on the image for bigger view)

I would be sharing the step by step solutions for Question 8 in the next post. Subscribe to my blog to be updated again!

O level A-Math: Collection of Differentiation Tips

Usually at this period, many Additional Mathematics students would be 'sweating' over the rather foreign topic, Differentiation (part of calculus).

Today I have decided to gather all the tips I have written on Differentiation to be on this post so that they are easily accessible for you. Let's take a look what have been discussed so far:

Differentiation Basic Techniques
Stationary points (Includes a video on the step by step solution)
Differentiation involving exponential

So, how have you found Differentiation so far? Easy? Tough?I would love to hear from you on your learning experience so far.

In my coaching programme, I relate Differentiation to Freezing. Do you know why? Well, there is one common property both shared: Decrease.

A-Math: Find Coordinates and Nature of Stationery Point (Plus: Tips to note in Differentiation)

The following question is similar to June 2008 GCE O Level A-Math Paper question which is testing on the following concepts

Differentiation involving exponential (I have written a post on Differentiation of e; post on Basic Differentiation Techniques are available here too )

Finding coordinates and nature of stationary point (Application of Differentiation)

Watch the video for a few more important tips on handling this sort of question.

Integration Mixed With Differentiation

In Integration, unlike Differentiation, there isn't any product rule nor quotient rule. Having said this, examiners always like to present question in that form of either product or quotient. Students who aren't able to see through their plot to confuse your mind will fall straight into their trap.
So LOOK OUT!

The question most students will ask next will be what to do when the Integration is presented in the manner as if we can use product or quotient rule aka the given question is in the form of a fraction.

These are the few ways you can work around it

Simplifying either by breaking up the numerator
Simplification by applying Indices Law (very useful when you are working with exponential or bases-powers)
Apply partial fraction concept

Look at this question:
$\int (2x+1)\sqrt{8x+4} dx$

It is expressed as two product. How do we integrate?

Discuss your approach in the comment box below.

I will reveal the step by step approach by end of this week. Be sure to subscribe to www.singaporeolevelmaths.com/feed so that you will be notified of latest updates

Cheers!

Differentiation Basic Techniques

The big thing for Sec 4 A-Math ( Oh yar, since this blog has quite a number of visitors from other countries, I think it is important for me to mention that Secondary 3 - 4 is known as Grade 9 - 10 in other countries) is CALCULUS.

Well, you will hardly find this word in your A-Math textbook but you see a bulk of the chapters dedicated to Differentiation & Integration. These 2 topics are like freezing and melting processes.

Why? Because they are simply opposite of each other !

I am going to talk about the techniques of differentiation.

There are 3 main types for Sec 4 level ; you ought to learn these techniques real well and know when to apply each one of them as application problems follow after the basics.

Chain Rule
Product Rule
Quotient Rule reserved for fractions. * But some fractions can skip this rule

My personal favourite is No. 2 - Product Rule. Well, let's see the technique in action.

Differentiate with respect to x:

Now to do this, you can apply Product Rule - Differentiate Copy + Differentiate Copy

Now from this example, we notice some patterns, if you have a constant ( a fixed number) in front, you can simply focused on differentiating the portion with x involved. For example in this case, Focus on Differentiating so now, we don't even have to use product rule :

Leave the constant in front
Differentiate the portion with x involved by Power Front - Power Down by 1-Differentiate within also known as your Chain Rule. BINGO!

By realizing this pattern, it will save you some time and less pen ink as well.

Certainly hope it is useful.

:-)

alwaysLovely

Photo by just another paul

Application of Differentiation - Maximum/Minimum & Rate of Change

Qn2
A closed box with a square base of length x and height h, is to have a volume, F, of 150m^3. The material for the top and bottom of the box costs $2 per square metre, and the material for the sides of the box costs $1.50 per square metre. Fnd the value of x and h, correct to 3 decimal places, if the total cost of the materials, C is to be a minimum.
Qn3
A viscous liquid is poured onto a flat surface. It forms a circular patch which grows at a steady rate of 6cm^2/s. Find,
a) the radius r, in pie, of the patch 24 seconds after pouring has commenced.
b) the rate of increase of the radius at this instant, correct to 2 decimal places.

Done reading the qns? Do and see if your answers are correct!

Click on image for enlarged view.