MEB 2011 Calculus



Group (2011)


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1.Evaluate $\displaystyle\lim _{x \rightarrow 4} \frac{x^{2}-x-12}{x^{2}-11 x+28}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{3 x^{2}-x+2}{x^{2}+1}$. $\mbox{ (3 marks)}$
See Answer 1
2.Calculate $\displaystyle\lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{4}-3 x^{3}}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{x^{2}-9}{x^{4}-3 x^{3}}$. $\mbox{ (3 marks)}$

3.Find $\displaystyle\lim _{x \rightarrow 2} \frac{2 x-x^{2}}{x^{2}-3 x+2}, \displaystyle\lim _{x \rightarrow \infty} \frac{2 x^{2}-3 x+1}{x^{2}-x+2}$. $\mbox{ (3 marks)}$

4.Find $\displaystyle\lim _{x \rightarrow 2} \frac{x^{3}-8}{x^{2}-x-2}, \displaystyle\lim _{x \rightarrow \infty} \frac{x^{2}-x-2}{x^{3}-8}$. $\mbox{ (3 marks)}$

5.Calculate $\displaystyle\lim _{x \rightarrow 5} \frac{x^{3}-125}{5-x}$, and $\displaystyle\lim _{x \rightarrow \infty} \frac{9 x^{2}-1}{x^{2}-x}$. $\mbox{ (3 marks)}$

6.Calculate $\displaystyle\lim _{x \rightarrow 1} \frac{x^{3}-2 x^{2}}{x}$, and $\displaystyle\lim _{x \rightarrow \infty} \frac{(x+1)(x+2)}{x^{2}-4}$. $\mbox{ (3 marks)}$

7.Find the limits $\displaystyle\lim _{x \rightarrow \infty} \frac{(x-3)(2 x+4)}{3 x^{2}-2}$, and $\displaystyle\lim _{x \rightarrow 0} \frac{x(1-x)}{x^{3}+3 x}$. $\mbox{ (3 marks)}$

8.Find the limits $\displaystyle\lim _{x \rightarrow \infty} \frac{2 x^{2}-5}{(3 x-2)(x+4)}$, and $\displaystyle\lim _{x \rightarrow 0} \frac{x(1-x)}{x^{3}+3 x}$. $\mbox{ (3 marks)}$

9.Calculate $\displaystyle\lim _{x \rightarrow 5} \frac{2 x^{2}-14 x+20}{x^{2}-25}$, and $\displaystyle\lim _{x \rightarrow \infty} \frac{2 x^{-2}-14 x^{-1}+20}{x^{-2}-25}$. $\mbox{ (3 marks)}$

10.Calculate $\displaystyle\lim _{x \rightarrow 3} \frac{\sqrt{x}-\sqrt{3}}{x^{2}-3^{2}}, \displaystyle\lim _{x \rightarrow \infty} \frac{x^{3}-2}{2 x^{3}+3 x^{2}-1}$. $\mbox{ (3 marks)}$

11.Calculate $\displaystyle\lim _{x \rightarrow 2} \frac{\sqrt{x+2}-2}{x-2}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{5-x^{2}}{x^{2}-x}$. $\mbox{ (3 marks)}$

12.Differentiate $y=2 \pi x+2 \cos \pi x$ and $y=\frac{x^{2}+\tan 3 x}{e^{x}}$ with respect to $x$. $\mbox{ (5 marks)}$

13.If $x+\sin y=\cos (x y)$, find $\frac{d y}{d x}$. If $z=\sqrt{\frac{x^{2}+1}{x^{2}-1}}$, find $\frac{d z}{d x}$. $\mbox{ (5 marks)}$

14.Find $\frac{d y}{d x}$ if $y=\frac{3 x+8}{2 x^{2}+5}$. Find also $\frac{d z}{d x}$ if $x+\cos z=\tan (x z)$. $\mbox{ (5 marks)}$

15.Find the approximate change in the volume of a sphere when its radius increases from $2 \mathrm{~cm}$ to $2.05 \mathrm{~cm}$. $\mbox{ (5 marks)}$

16.Using the derivative of a suitable function, find an approximate value of $\sqrt{143.5}$. $\mbox{ (5 marks)}$

17.Given that $y=e^{3 x} \sin 2 x$, prove that $\frac{d^{2} y}{d x^{2}}-6 \frac{d y}{d x}+13 y=0$. $\mbox{ (5 marks)}$

18.If $y=3 x \sin 3 x+\cos 3 x$, then prove that $x \frac{d^{2} y}{d x^{2}}+9 x y=2 \frac{d y}{d x}$. $\mbox{ (5 marks)}$

19.Find $\frac{d y}{d x}$ if $y=(5+3 x) e^{-2 x}.$ Find also $\frac{d y}{d x}$ if $y=\frac{\sin 3 x}{\sqrt{x^{2}+1}}$. $\mbox{ (5 marks)}$

20.Given that $x y=\sin x$, prove that $\frac{d^{2} y}{d x^{2}}+\frac{2}{x} \frac{d y}{d x}+y=0$. $\mbox{ (5 marks)}$

21.Given that $x^{2}-y^{2}=5$, show that $y^{2} y^{\prime \prime}+x y^{\prime}=y$. $\mbox{ (5 marks)}$

22.Given that the gradient of the curve $y=a x^{2}-b x+3$ at the point $(2,7)$ is 8. Find the values of $a$ and $b$. $\mbox{ (5 marks)}$

23.Find the stationary points of the curve $y=x^{3}-3 x+2$ and determine their nature. $\mbox{ (5 marks)}$

24.Find the stationary points of the curve $y=x^{3}(4-x)+5$ and determine their nature. $\mbox{ (5 marks)}$

25.Given that $x+y=5$, calculate the maximum value of $2 x^{2}+x y-3 y^{2}$. $\mbox{ (5 marks)}$

26.Find the minimum value of the sum of a positive number and its reciprocal. $\mbox{ (5 marks)}$

27.Find the $x$-coordinate, for $0 < x< \frac{\pi}{2}$, of the stationary point on the curve $y=e^{\sqrt{3} x} \cos x$. $\mbox{ (5 marks)}$

28.Given that $y=\frac{3 x^{2}+2}{x}$, prove that $x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-y=0$. $\mbox{ (5 marks)}$

29.Find the equations of the tangent and normal lines to the curve $y=x^{2}-5 x+6$ at the point $(1,2)$. $\mbox{ (5 marks)}$

30.
Find the equation of the normal line to the curve $y=x^{2}-3 x+2$ given that the gradient of the normal is $\frac{1}{2}$. $\mbox{ (5 marks)}$

31.Show that the equation of the tangent to the curve $x^{2}+x y+y=0$ at the point $(a, b)$ is $x(2 a+b)+y(a+1)+b=0$. $\mbox{ (5 marks)}$



Answer (2011)


$\quad\;\,$$\,$
1.$-\frac{7}{3}, 3$Question 1
2.$\frac{2}{9}, 0$
3.$-2,2$
4.4,0
5.$-75,9$
6.$-1,1$
7.$\frac{2}{3}, \frac{1}{3}$
8.$\frac{2}{3}, \frac{1}{3}$
9.$\frac{3}{5},-\frac{4}{5}$
10.$\frac{\sqrt{3}}{36}, \frac{1}{2}$
11.$\frac 14,-1$
12.$2 \pi(1-\sin (\pi x)) ; \frac{2 x+3 \sec ^{2} 3 x-x^{2}-\tan 3 x}{e^{x}}$
13.$\frac{-y \sin (x y)-1}{\cos y+x \sin (x y)} ; \frac{-2 x \sqrt{x^{2}-1}}{\left(x^{2}-1\right)^{2} \sqrt{x^{2}+1}}$
14.$\frac{-6 x^{2}-32 x+15}{\left(2 x^{2}+5\right)^{2}} ; \frac{1-z \sec ^{2}(x z)}{\sin z+x \sec ^{2}(x z)}$
15.$0.8 \pi \mathrm{cm}^{3}$
16.$11.979$
17.Prove
18.Prove
19.$(-7-6 x) e^{-2 x} ; \frac{\left(3 x^{2}+3\right) \cos 3 x-x \sin 3 x}{\left(x^{2}+1\right) \sqrt{x^{2}+1}}$
20.Prove
21.Prove
22.$a=3 ; b=4$
23.$(1,0)$ minimum; $(-1,4)$ maximum
24.$(0,5)$ inflexion; $(3,32)$ maximum
25.$\frac{625}{8}$
26.2
27.$\frac{\pi}{3}$
28.Prove
29.$y+3 x-5=0 ; 3 y-x-5=0$
30.$2 y-x-1=0$
31.Prove

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