### Random Sec 4 Differentiations

B6

$y = 3e^x + \frac{4}{e^x}$

$\frac{dy}{dx} = 3e^x - \frac{4}{e^x}$

$\frac{d^2y}{dx^2} = 3e^x + \frac{4}{e^x}$

let $\frac{dy}{dx} = 0$

$3e^x - \frac{4}{e^x} = 0$

$3e^{2x} = 4$

$2x = \mathrm{ln} \frac{4}{3}$

$x = \frac{1}{2} \mathrm{ln} \frac{4}{3}$

Sub $x = \frac{1}{2} \mathrm{ln} \frac{4}{3}$ to $\frac{d^2y}{dx^2}$

$\frac{d^2y}{dx^2} > 0$ Thus, it is a min point.

C7

$y = \mathrm{ln} \frac{5-4x}{3+2x}$

$y = \mathrm{ln} (5-4x) - \mathrm{ln} (3+2x)$

$\frac{dy}{dx} = \frac{-4}{5-4x} - \frac{2}{3+2x}$

let $\frac{dy}{dx} = 0$

$\frac{-4}{5-4x} - \frac{2}{3+2x} = 0$

$\frac{-4}{5-4x} = \frac{2}{3+2x}$

$-4(3+2x) = 2(5-4x)$

$-12 - 8x = 10 - 8x$

$-12 = 10$ (NA).

There are no stationary points for this curve.

C8

$x = \frac{1}{3}e^{y(2x+5)}$

$\mathrm{ln}(3x) = y(2x+5)$

$\frac{\mathrm{ln}(3x)}{2x+5} = y$

$y = \frac{\mathrm{ln}(3x)}{2x+5}$

$\frac{dy}{dx} = \frac{\frac{1}{x}(2x+5) - \mathrm{ln}(3x) \times 2}{(2x+5)^2}$

Let $x = e^2$

$\frac{dy}{dx} = \frac{\frac{1}{e^2}(2e^2+5) - \mathrm{ln}(3e^2) \times 2}{(2e^2+5)^2}$

Evaluate with a calculator…

### Reimagining mathematical notations

Maths notation is often overly complex. Let’s see how this can be better:

### Question of the Day #17

This is a pretty cool and interesting question from AMC.

There are four lifts in a building. Each makes three stops, which do not have to be on consecutive floors on include the ground floor. For any two floors, there is at least one lift which stops on both of them. What is the maximum number of floors that this building can have?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 12

### Question of the Day #16

The positive integers $x$ and $y$ satisfy

$3x^2 -8y^2 +3x^2 y^2 = 2008$

What is the value of $xy$?

### Arithmetic Problem #7

I came across this interesting algebra/ arithmetic problem the other day.

$(1 \times 2 \times 3) + 2 = 8 = 2^3$
$(2 \times 3 \times 4) + 3 = 27 = 3^3$
$(3 \times 4 \times 5) + 4 = 64 = 4^3$

Do you think this will always work?

Hint: We can use algebra to prove this cool sequence easily.

### O’levels Results 2016!

All of us wish the students receiving the O’levels Results 2016 the best! And do not let grades define you. 🙂

If you’re keen to meet up with us for the JC Talk, you may contact Newtonapple Learning Hub at +65 9222 3423 for more details.

This talk will be opened freely to the public and existing students. There will be discussion about Subject Combinations, discussion and introduction with different subjects. Come down to find out more 🙂

### 30 Free Awesome Resources for Students

New year, new you. Be ready to tackle on the new academic year with these awesome resources curated by Donut.sg

You’re welcome.

Did we miss anything out? Let us know by leaving a comment below.

### Arithmetic Problem #6

Try this question!

$\frac{999999 \times 999999}{1+2+3+4+5+6=7+8+9+8+7+6+5+4+3+2+1} = ?$
Hint: Simplify it first! 🙂

### Arithmetic Problem #5

Saw this question on the internet and thought its rather interesting. A-levels students should be all buried other their workload, amidst the mock papers and ten year series. Hopefully this might help.

$\frac{100001 + 100003 + 100005 + \ldots + 199999}{1 + 3 + 5 + 7 + 9 + 11 + 99999} = ?$

Hint: try and group the number together.

### Trick to squaring numbers

Many students go wow when I evaluate workings, without a GC. I’m not showing off, but it is because I don’t really carry a calculator with me. haha. So some students do ask me how I evaluate the square of numbers so quickly. I thought, in light of A-level’s coming, I should share something interesting.

Firstly, we trace back to a formula that we saw in primary school.

$(a+b)^2 = a^x + 2ab + b^2$

This formula is really going to be the core of us solving any square of numbers.

Next, we just need to split the number that you want to square effectively. So consider,

$63 = 60 +3$
$63^3 = (60+3)^2 = 60^2 + 2(60)(3) + 3^2$

Some students will ask why not use $(a-b)^2$ instead. This is plausible, but we usually are better with addition than multiplication hah.