### Differentiation Question #1

Given that $y = \frac{8}{x^3} - \frac{6}{x^2} + \frac{5}{2x}$, find the approximate percentage change in $y$ when $x$ increases from 2 by 2%.

### Probability Question #4

A gambler bets on one of the integers from 1 to 6. Three fair dice are then rolled. If the gambler’s number appears $k$ times ($k = 1, 2, 3$), he wins $$k$. If his number fails to appear, he loses$1. Calculate the gambler’s expected winnings

### 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…

### June Revision Exercise

You can find the solutions of all ten sets of the June Revision Exercise we did in class.

Have fun!

June Revision Exercise 1
June Revision Exercise 2
June Revision Exercise 3
June Revision Exercise 4
June Revision Exercise 5
June Revision Exercise 6
June Revision Exercise 7
June Revision Exercise 8
June Revision Exercise 9
June Revision Exercise 10

### APGP, Sequence & Series related articles

Here is a compilation of all the APGP, Sequence & Series articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

### Statistics related articles

Here is a compilation of all the Statistics articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

### Complex Numbers related articles

Here is a compilation of all the Complex Numbers articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

### Integration related articles

Here is a compilation of all the Integration articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

### Vectors related articles

Here is a compilation of all the Vectors articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

### Combinatorics related articles

Here is a compilation of all the Combinatorics articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂