# MEB 2010 Calculus

## Group (2010)

$\quad\;\,$$\, 1.Find \displaystyle\lim _{x \rightarrow 5} \dfrac{x^{2}+6 x-55}{x^{2}-2 x-15}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-10 x+25}{2 x^{2}-x-6}. \mbox{ (3 marks)} 2.Find \displaystyle\lim _{x \rightarrow-2} \dfrac{3 x^{2}+4 x-4}{x^{2}+3 x+2}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{4 x^{2}-10 x+15}{2 x^{2}-3 x-5}. \mbox{ (3 marks)} 3.Evaluate \displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-7 x+1}{x^{2}-2}, and \displaystyle\lim _{x \rightarrow 5} \dfrac{2 x^{2}-14 x+20}{x^{2}-25}. \mbox{ (3 marks)} 4.Calculate \displaystyle\lim _{x \rightarrow-5} \dfrac{x^{2}-25}{x^{3}+5 x^{2}} and \displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-25}{x^{3}+5 x^{2}}. \mbox{ (3 marks)} 5.Find \displaystyle\lim _{x \rightarrow 0}\left[\left(2 x+\dfrac{1}{x^{2}}\right)-\left(x+\dfrac{1}{x}\right)^{2}\right], \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{(3 x+2)(5 x-7)}{(2 x-3)^{2}}. \mbox{ (3 marks)} 6.Calculate \displaystyle\lim _{x \rightarrow-2} \dfrac{x^{3}+8}{x^{2}-4}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{\sqrt{2 x}-\sqrt{a}}{\sqrt{2 x}+\sqrt{a}}. \mbox{ (3 marks)} 7.Calculate \displaystyle\lim _{x \rightarrow-1} \dfrac{x^{3}+1}{x+1} and \displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-2 x+1}{(x+1)(2-x)}. \mbox{ (3 marks)} 8.Calculate \displaystyle\lim _{x \rightarrow 4} \dfrac{x^{3}-64}{x-4} and \displaystyle\lim _{x \rightarrow \infty} \dfrac{x^{2}-2 x+1}{(x+1)(2-x)}. \mbox{ (3 marks)} 9.Evaluate \displaystyle\lim _{x \rightarrow 2} \dfrac{x^{3}-8}{x^{2}+3 x-10}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{2 x^{2}-1}{x^{3}-1}. \mbox{ (3 marks)} 10.Evaluate \displaystyle\lim _{x \rightarrow 2} \dfrac{x-2}{\sqrt{x}-\sqrt{2}}, \quad \displaystyle\lim _{x \rightarrow \infty} \dfrac{(x-4)(x-5)}{(x+1)(x-7)}. \mbox{ (3 marks)} 11.Evaluate \displaystyle\lim _{x \rightarrow 6} \dfrac{3 x-18}{\sqrt{2 x-3}-\sqrt{x+3}} and \displaystyle\lim _{x \rightarrow \infty} \dfrac{(1-2 x)(3+x)}{(x-2)^{2}}. \mbox{ (3 marks)} 12.Given that y=(1+x) e^{3 x}, prove that \dfrac{d^{2} y}{d x^{2}}-6 \dfrac{d y}{d x}+9 y=0. \mbox{ (5 marks)} 13.Given that y=(1-x) e^{2 x}, prove that \dfrac{d^{2} y}{d x^{2}}=4 \dfrac{d y}{d x}-4 y. \mbox{ (5 marks)} 14.Given that y=x \sin x, prove that x \dfrac{d^{2} y}{d x^{2}}-2 \dfrac{d y}{d x}+x y+2 \sin x=0. \mbox{ (5 marks)} 15.If x \cos y=\sin x, prove that \dfrac{d y}{d x}=\dfrac{\cos y(\cos y-\cos x)}{\sin x \sin y}. \mbox{ (5 marks)} 16.If y=\sin ^{2} x, show that \dfrac{d^{2} y}{d x^{2}}+4 y-2=0. \mbox{ (5 marks)} 17.If y=\sin ^{2} x and \dfrac{d^{2} y}{d x^{2}}+\dfrac{d y}{d x}=a \cos 2 x+b \sin 2 x, where a and b are constants, find the value of a and of b. \mbox{ (5 marks)} 18.If y=\sin ^{2} 3 x, prove that \dfrac{d^{2} y}{d x^{2}}+36 y=18. By using this result show that, if z=\cos ^{2} 3 x, then \dfrac{d^{2} z}{d x^{2}}+36 z=18. \mbox{ (5 marks)} 19.If y=\cos ^{2} 3 x, prove that \dfrac{d^{2} y}{d x^{2}}+36 y=18. By using this result show that, if z=\sin ^{2} 3 x, then \dfrac{d^{2} z}{d x^{2}}+36 z=18. \mbox{ (5 marks)} 20.If y=\cos ^{2} 2 x, prove that \dfrac{d^{2} y}{d x^{2}}+16 y=8. By using this result show that, if z=\sin ^{2} 2 x, then \dfrac{d^{2} z}{d x^{2}}+16 z=8. \mbox{ (5 marks)} 21.Differentiate f(x)=\dfrac{1}{5 \cos x} and g(x)=e^{5 x} \ln (\sqrt{5 x-1}) with respect to x. \mbox{ (5 marks)} 22.If y=A \cos (\ln x)+B \sin (\ln x), where A and B are constants, show that x^{2} y^{\prime \prime}+x y^{\prime}+y=0. \mbox{ (5 marks)} 23.Find the equation of the normal line to the curve y=x^{2}-3 x+2 at the point where x=3. \mbox{ (5 marks)} 24.Find the equations of the normal lines to the curve y=x^{2}-2 x-8 at the points where this curve cuts the x-axis.\mbox{ (5 marks)} 25.Find the equations of the tangent and the normal to the curve y=2 e^{3 x} at the point where x=0. \mbox{ (5 marks)} 26.Find the equation of the tangent line to the curve x^{2}+x y+y=5 at the point where the curve cuts the line x=1. \mbox{ (5 marks)} 27.Find the equation of the tangent line to the curve 2 x^{2}+3 y^{2}=2 x y+23 at the point (2,3). \mbox{ (5 marks)} 28.Find the stationary points of the curve y=x^{3}-3 x^{2}-9 x+10 and determine their nature.\mbox{ (5 marks)} 29.Find the stationary points of the curve y=x^{2}(3-x) and determine their nature.\mbox{ (5 marks)} 30.What is the smallest perimeter possible for a rectangle of area 16 \mathrm{ft}^{2} ? \mbox{ (5 marks)} 31.What is the smallest perimeter possible for a rectangle of area 25 \mathrm{~m}^{2} ? 32.A rectangular box has a square base of side x \mathrm{~cm}. If the sum of one side of the square and the height is 12 \mathrm{~cm}, express the volume of the box in terms of x. Use this expression to determine the maximum volume of the box.\mbox{ (5 marks)} 33.Using y=\sqrt{x}, find the approximate value of \sqrt{26}. \mbox{ (5 marks)} #### Answer (2010) \quad\;\,$$\,$
1.$2, \dfrac{1}{2}$
2.2,8
3.$1, \dfrac{3}{5}$
4.$-\dfrac{2}{5}, 0 \quad$
5.$-2, \dfrac{15}{4}$
6.$-3,1$
7.$3,-1$
8.$48,-1$
9.$\dfrac{12}{7}, 0$
10.$2 \sqrt{2}, 1$
11.$18,-2$
12.Prove
13.Prove
14.Prove
15.Prove
16.Prove
17.$a=2, b=1 \quad$
18.Prove
19.Prove
20.Prove
21.$f^{\prime}(x)=\dfrac{1}{5} \sec x \tan x, \quad g^{\prime}(x)=\dfrac{5}{2} e^{5 x}\left(\dfrac{1}{5 x-1}+\ln (5 x-1)\right)$
22.Prove
23.$x+3 y=9$
24.$6 y-x=2 ; x+6 y=4$
25.$y-6 x=2 ; 6 y+x=12$
26.$2 x+y=4 \quad$
27.$x+7 y=23 \quad$
28.$(-1,15)$ maximum, $(3,-17)$ minimum
29.$(0,0)$ minimum, $(2,4)$ maximum
30.16 ft
31.20 m
32.$12 x^{2}-x^{3} ;256 \mathrm{~cm}^{3} \quad$
33.$5.1$