# MEB 2011 Calculus

## Group (2011)

$\quad\;\,$$\, 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