## Wednesday, December 5, 2018

### Application of Differentation

$\newcommand{\D}{\displaystyle} \def\dydx{\displaystyle\frac{dy}{dx}}$
1 (CIE 2012, s, paper 11, question 5)
(i) Find the equation of the tangent to the curve $\D y = x^3 + 2x^2 - 3x + 4$ at the point where the
curve crosses the y-axis. 
(ii) Find the coordinates of the point where this tangent meets the curve again. 

2 (CIE 2012, s, paper 11, question 10)
Variables $\D x$ and $\D y$ are such that $\D y = e^{2x} + e^{-2x} .$
(i) Find $\D \frac{dy}{dx}.$ 
(ii) By using the substitution $\D u = e^{2x} ,$ find the value of $\D y$ when $\dydx = 3.$ 
(iii) Given that $\D x$ is decreasing at the rate of 0.5 units $\D s^{-1},$ find the corresponding rate of changeof $\D y$ when $\D x = 1.$ 

3 (CIE 2012, s, paper 22, question 2)
(i) Given that $\D y =\sqrt{ (4x + 1)^3},$ find $\D \dydx.$ 
(ii) Hence find the approximate increase in $\D y$ as $\D x$ increases from 6 to $\D 6 + p,$ where $\D p$ is small. 

4 (CIE 2012, s, paper 22, question 8)
An open rectangular cardboard box with a square base is to have a volume of 256 cm$\D ^3.$ Find the dimensions of the box if the area of cardboard used is as small as possible. 

5 (CIE 2012, w, paper 12, question 11or)
A curve is such that $\D y = \frac{Ax^2+B}{x^2-2},$ where $\D A$ and $\D B$ are constants.
(i) Show that $\D\dydx=-\frac{2x(2A+B)}{(x^2-2)^2}.$  

It is given that $\D y = -3$ and $\D \dydx = -10$ when $\D x = 1.$
(ii) Find the value of $\D A$ and of $\D B.$ 
(iii) Using your values of $\D A$ and $\D B,$ find the coordinates of the stationary point on the curve, and determine the nature of this stationary point. 

6 (CIE 2012, w, paper 13, question 2)
The rate of change of a variable $\D x$ with respect to time $\D t$ is $\D 4\cos^2t.$
(i) Find the rate of change of $\D x$ with respect to $\D t$ when $\D t = \frac{\pi}{6} .$ 

The rate of change of a variable $\D y$ with respect to time $\D t$ is $\D 3\sin t.$
(ii) Using your result from part (i), find the rate of change of $\D y$ with respect to $\D x$ when $\D t = \frac{\pi}{6}.$ 

7 (CIE 2012, w, paper 21, question 6)
The normal to the curve $\D y = x^3 + 6x^2 - 34x + 44$ at the point $\D P (2, 8)$ cuts the x-axis at $\D A$ and the y-axis at $\D B.$ Show that the mid-point of the line AB lies on the line $\D 4y = x + 9.$ 

8 (CIE 2012, w, paper 22, question 2)
The total surface area, $\D A$ cm$\D^2,$ of a solid cylinder with radius $\D r$ cm and height 5 cm is given by $\D A = 2\pi r^2 + 10\pi r.$ Given that $\D r$ is increasing at a rate of $\D \frac{0.2}{\pi}$ cm s$\D ^{-1},$ find the rate of increase of $\D A$ when $\D r$ is 6. 

9 (CIE 2012, w, paper 23, question 10)
A track runs due east from $\D A$ to $\D B,$ a distance of 200 m. The point $\D C$ is 80 m due north of $\D B.$ A cyclist travels on the track from A to D, where D is $\D x$ m due west of B. The cyclist then travels in a straight line across rough ground from D to C. The cyclist travels at 10 m s$\D ^{-1}$ on the track and at 6 m $\D s^{-1}$ across rough ground.
(i) Show that the time taken, Ts, for the cyclist to travel from A to C is given by
$T=\frac{200-x}{10}+\frac{\sqrt{(x^2+6400)}}{6}$ 
(ii) Given that $\D x$ can vary, find the value of $\D x$ for which $\D T$ has a stationary value and the corresponding value of $\D T.$ 

10 (CIE 2013, s, paper 11, question 10)
The point $\D A,$ whose x-coordinate is 2, lies on the curve with equation $\D y= x^3- 4x^2+ x+ 1.$
(i) Find the equation of the tangent to the curve at A. 
This tangent meets the curve again at the point B.
(ii) Find the coordinates of B. 
(iii) Find the equation of the perpendicular bisector of the line AB. 

11 (CIE 2013, s, paper 12, question 6)
The normal to the curve $\D y + 2 = 3 \tan x,$ at the point on the curve where $\D x = \frac{3\pi}{4}$, cuts the y-axis at the point P. Find the coordinates of P. 

12 (CIE 2013, s, paper 21, question 3)
Variables $\D x$ and $\D y$ are related by the equation $\D y= 10- 4 \sin^2x$, where $\D 0\le x\le\frac{\pi}{2}.$
Given that $\D x$ is increasing at a rate of 0.2 radians per second, find the corresponding rate of change of $\D y$ when $\D y = 8.$ 

$1. (i)y = -3x + 4,$
(ii)(-2,10)
2. (i)$\D 2e^{2x} - 2e^{-2x}$
(ii)$\D y = \frac{5}{2} (iii)-7.25$
3. (i) $\D6\sqrt{4x+1} (ii)30p$
4. x = 8; h = 4
5. (ii)A = 2;B = 1
(iii)x = 0; y = -:5, max
6. (i)3,(ii)0.5
7. A(18; 0);B(0; 9);M(9; 4:5)
8. 6.8
9. (ii)x = 60; T = 92/3
10. (i)y = 1 - 3x
(ii)(0; 1)
(iii)y + 2 = (x - 1)/3
11. (0,-4.61)
12. -0:8