MEB 2010 Calculus



Group (2010)


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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.$5.1$

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