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. [4]
(ii) Find the coordinates of the point where this tangent meets the curve again. [3]

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 $\D \frac{dy}{dx}.$ [2]
(ii) By using the substitution $\D u = e^{2x} ,$ find the value of $\D y$ when $\dydx = 3.$ [4]
(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]

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

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. [7]

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 $\D\dydx=-\frac{2x(2A+B)}{(x^2-2)^2}.$  [4]

It is given that $\D y = -3$ and $\D \dydx = -10$ when $\D x = 1.$
(ii) Find the value of $\D A$ and of $\D B.$ [3]
(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. [4]


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 $\D t = \frac{\pi}{6}
.$ [1]

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 $\D t = \frac{\pi}{6}.$ [3]


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]

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. [4]




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
\[T=\frac{200-x}{10}+\frac{\sqrt{(x^2+6400)}}{6}\] [2]
(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 $\D T.$ [6]

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. [4]
This tangent meets the curve again at the point B.
(ii) Find the coordinates of B. [4]
(iii) Find the equation of the perpendicular bisector of the line AB. [4]

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. [6]


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 $\D y = 8.$ [6]

Answers

$ 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

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