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Calculus : Jun 2013 - First Year Engineering (Semester 1) | Gujarat Technological University (GTU)

## 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) = x3- 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 ex.
(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=|x2-1|(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 = y2 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 - x2= 0 (4 marks) 6 (b) Find all local maxima, local minima and saddle point off(x,y) = x2+ y2+ 4x + 6y + 13.(5 marks) 6 (c) Suppose that the Celsius temperature at the point (x,y,z) on the sphere x2+y2+ z2=1 is T = 400xyz2. 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 x2 + y2≤ 4.(5 marks) 7 (c) A solid E lies within the cylinder x2+ y2= 1, below the plane z = 4 and above the paraboloid z = 1 - x2- y2. The density at any point is proportional to its distance from the axis of the cylinder. Find mass of E.(5 marks)