GTU Calculus - June 2013 Exam Question Paper

GTU First Year Engineering (Semester 1)
Calculus
June 2013

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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 M

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 M

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 M

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 M

2 (b) Find the Taylor polynomial of order 3 generated by

$f(x)=\sqrt{x} \ at a=4$

5 M

2 (c) Sketch the curve y=|x²-1|

5 M

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 M

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 M

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 M

4 (a)

$Evaluate \ \int^{3}_0\dfrac {dx}{x-1} \ if \ possible. \\Evaluate \ \int^{\infty}_{-\infty}\dfrac {dx}{1+x^2}$

4 M

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 M

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 M

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 M

5 (b) Determine whether

$u(x,y)=ln \sqrt{x^2+y^2}$ is a solution of Laplace's equation.

5 M

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 M

6 (a) Find equation for the tangent plane and normal line at point (2,0,2) on the surface 2z - x²= 0

4 M

6 (b) Find all local maxima, local minima and saddle point off(x,y) = x²+ y²+ 4x + 6y + 13.

5 M

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 M

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 M

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 M

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 M

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