## Calculus - Jun 2013

### First Year Engineering (Semester 1)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1 (a)** (i) Find the intervals on which the function f(x) = x^{3}- 27x is increasing and decreasing.

(ii) Graph the set of points whose polar coordinates satisfy the conditions $$ 1 \le r \le 2 \ and \ 0 \le \theta \le \dfrac {\pi}{2} $$(4 marks)
**1 (b)** Which of following series converge and which diverge? $$ (i) \ \sum^{\infty}_{n=1}\dfrac {2n+1}{n^2+2n+1} \\ (ii) \ \sum^{\infty}_{n=1}e^{-n}$$(5 marks)
**1 (c) ** For what values of x do the power series $$ \sum^\infty_{n-1}\dfrac {(-1)^{n-1}x^{2n-1}}{2n-1} \ converge? $$(5 marks)
**2 (a)** $$ (i) \ Evaluate \ \lim_{x\rightarrow \frac{\pi}{2}}\dfrac {2x-\pi}{\cos x} $$

(ii) Find the Maclaurin's series for e^{x}.(4 marks)
**2 (b)** Find the Taylor polynomial of order 3 generated by $$ f(x)=\sqrt{x} \ at a=4 $$(5 marks)
**2 (c) ** Sketch the curve y=|x^{2}-1|(5 marks)
**3 (a)** (i) Obtain reduction formula that expresses the integral ∫(ln x)^{x} dx in terms of an integral of a lower power of (ln x).

$$ (ii) \ Find \ \dfrac {dy}{dx} \ if \ y=\int^5_x 3t\sin tdt $$(4 marks)
**3 (b)** State Leibniz's Rule. Use Leibniz's Rule to find the derivative of $$ g(y)=\int^{2\sqrt{y}}_{\sqrt{y}}\sin^2 tdt. $$(5 marks)
**3 (c) ** $$ Evaluate \\ (i) \int^{\frac{\pi}{4}}_{0}\sin^7 2\theta d\theta \$$ii) \ \int^{1}_{0}\dfrac {x^7}{\sqrt{1-x^4}}dx $$(5 marks)
**4 (a)** $$ Evaluate \ \int^{3}_0\dfrac {dx}{x-1} \ if \ possible. \\Evaluate \ \int^{\infty}_{-\infty}\dfrac {dx}{1+x^2} $$(4 marks)
**4 (b) ** Find the volume of the solid generated by revolving the region between the parabola x = y^{2} and the line x = 1 about line x = 1. (5 marks)
**4 (c) ** Use the shell method to find volume of solid generated by revolving the region bounded by y = x , x = 0 and y =1 about x-axis. (5 marks)
**5 (a)** $$ (i) \ Show \ that \ \lim_{(x,y)\rightarrow (0,0)}\dfrac {xy}{x^2+y^2} \ does \ not \ exist. $$

(ii) Determine set of all points at which the function $$ f(x,y)= \dfrac {x^2+Y^2}{x-y} \ is \ continuous. $$(4 marks)
**5 (b) ** Determine whether $$ u(x,y)=ln \sqrt{x^2+y^2} $$ is a solution of Laplace's equation.(5 marks)
**5 (c) ** $$ if \ f(x,y)=\dfrac {x^{\frac {1}{4}}+y^{\frac {1}{4}}}{x^{\frac {1}{5}}+y^{\frac {1}{5}}} \ then \ find \ x\dfrac {\partial f}{\partial x}+y\dfrac {\partial f}{\partial y} \ and \\x^2\dfrac {\partial^2f}{\partial x^2}+2xy\dfrac {\partial^2f}{\partial x \partial y}+ y^2 \dfrac {\partial^2f}{\partial y^2} $$(5 marks)
**6 (a) ** Find equation for the tangent plane and normal line at point (2,0,2) on the surface 2z - x^{2}= 0 (4 marks)
**6 (b)** Find all local maxima, local minima and saddle point off(x,y) = x^{2}+ y^{2}+ 4x + 6y + 13.(5 marks)
**6 (c) ** Suppose that the Celsius temperature at the point (x,y,z) on the sphere x^{2}+y^{2}+ z^{2}=1 is T = 400xyz^{2}. Locate the highest and the lowest temperature on the sphere. (5 marks)
**7 (a) ** $$ (i) \ Evaluate \ \int^2_1\int^1_0 (1+3xy)dxdy \$$ii) \ \int^1_{-1}\int^{2}_{0}\int^{1}_{0}xz-y^3 \ dzdydx $$(4 marks)
**7 (b)** Find the volume of the solid under the cone $$ z=\sqrt{x^2+y^2} $$ and above the disk x^{2} + y^{2}≤ 4.(5 marks)
**7 (c) ** A solid E lies within the cylinder x^{2}+ y^{2}= 1, below the plane z = 4 and above the paraboloid z = 1 - x^{2}- y^{2}. The density at any point is proportional to its distance from the axis of the cylinder. Find mass of E.(5 marks)