## CALCULUS : Question Paper First Semester B.tech Degree Examination, January 2016 (Free Download)

### PART A

Answer all questions, each question carries 3 marks

1. Show that the series  $\sum_{k=1}^{\infty&space;}\frac{\cos{k}}{k^{2}}$ is convergent.
2. Find  $\frac{\mathrm{d}&space;}{\mathrm{d}&space;x}(e^{x}sech^{-1}\sqrt{x}&space;)$
3. Identify the surfaces $5x^{2}-4y^2+20z^2=0$.
4. Equation of a surface in spherical coordinates is  $\rho&space;=sin\theta&space;sin\varphi$ Find the equation of this surface in rectangular coordinates.
5. Given $\inline&space;f=e^x\sin&space;y$ show that the function satisfies the Laplace equation $\inline&space;f_{xx}+f_{yy}=0$.
6.  Let $\inline&space;\mathbf{w=4x^2+4y^2+z^2}$ where $\inline&space;\mathbf{x=\rho&space;\sin&space;\phi&space;\cos\theta&space;},\mathbf{y=\rho&space;\sin&space;\phi&space;\sin\theta&space;},\mathbf{x=\rho&space;\cos\theta&space;}$ Find $\inline&space;\frac{\partial&space;w&space;}{\partial&space;\rho&space;}$ using chain rule.
7.  particle moves along a circular helix in 3-space so that its position vector at time t is r(t). (4 cos r t)i + (4 sin r t)j + tk Find the displacement of the particle during the interval $\inline&space;\mathbf{1\leq&space;t\leq&space;5}$.
8. Find the tangent to the curve r(t) = $\inline&space;(t^2-1)i&space;+&space;tj$ at t = 1.
9. Evaluate $\inline&space;\int_{1}^{a}\int_{1}^{b}\frac{dydx}{xy}$.
10. The line y = 2 - x and the parabola $\inline&space;y=x^2$ intersect at the points (-2, 4) and (1, 1),If R is the region enclosed by y=2-x and $\inline&space;y=x^2$ ,then find $\inline&space;\iint_{R}(y)dA$.
(10 x 3 = 30 Marks)

### PART-B

Answer any 2 complete questions each having 7 marks

1. Find the radius of convergence and interval of convergence of the series $\inline&space;\sum_{k=1}^{\infty&space;}\frac{(x-5)^k}{k^2}$.
2. Test the convergence of $\inline&space;\frac{x}{12}+\frac{x^2}{23}+\frac{x^3}{34}+&space;...$
3. Find the Taylors series of  $\inline&space;\frac{1}{x}$ about x = 1.
4.
Answer any 2 complete questions each having 7 marks

1. Find the domains of $\inline&space;(i)f(x,y)=&space;\sqrt{25-x^2-y^2-z^2}$  (ii) $f(x,y)=\ln&space;(x-y^2)$ and describe them in words.
2. Find the limit of $f(x,y)=\frac{-xy}{x^2+y^2}$  as $(x,y)\rightarrow&space;(0,0)$along (i) the X-axis , (ii) the Y- axis (iii) the line y = x.
3. Find the spherical and cylindrical coordinates of the point that has rectangular coordinates$\inline&space;(x,y,z)=(4,-4,4\sqrt{6})$

Answer any 2 complete questions each having 7 marks

1. Locate all relative maxima , relative minima and saddle point if any,of $\inline&space;f(x,y)=y^2+xy+4y+2x+3$ .
2. Let f be a differentiable function of 3 variables and suppose that $\inline&space;W=f(x-y,y-z,z-x)$  prove that $\inline&space;\frac{\partial&space;W}{\partial&space;x}+\frac{\partial&space;W}{\partial&space;y}+\frac{\partial&space;W}{\partial&space;z}=0$.
3. Find the local linear approximation L(x,y) to f(x,y) = $\inline&space;\frac{1}{\sqrt&space;{x^2+y^2}}$at the point P(4,3). Compare the error in approximating 'f' by L at the specified point Q (3.92, 3.01) with the distance between P and Q.

Answer any 2 complete questions each having 7 marks

1. Find y(t) where  $\inline&space;y^{''}(t)=12t^2&space;i-2tj,y(0)=2i-4j,y'(0)=0$ .
2. Find the arc length parametrization of the line x = 1 + t,y = 3 - 2t, z = 4 + 2t that has the same direction as the given line and has reference point (1, 3, 4).
3. Find the directional derivative of $\inline&space;f(x,y)=e^x&space;\sec&space;{y}$ at  $\inline&space;P(0,&space;\frac{\pi}{4})$  in the direction of PQ where Q is the origin.

Answer any 2 complete questions each having 7 marks

1. Find the area bounded by the x-axis, Y = 2x and x+y=1 using double integration.
2. Use a triple integral to find the volume of the solid ,within the cylinder $\inline&space;x^2+y^2=9$ and between the planes z = 1 and x + z = 5.
3. Sketch the region of integration and evaluate the integral $\inline&space;\int_{1}^{2}&space;\int_{y}^{y^2}dxdy$ by changing the order y of integration.

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