### FIRST SEMESTER B.TECH DEGREE EXAMINATION, JANUARY 2016

### Course Code: MA101

### Course Name: CALCULUS

#### Max. Marks: 100 Duration: 3 Hours

###
__PART A __

__PART A__

**Answer all questions, each question carries 3 marks**

**Show that the series is convergent.****Find****Identify the surfaces .****Equation of a surface in spherical coordinates is Find the equation of this surface in rectangular coordinates.****Given show that the function satisfies the Laplace equation .****Let where Find using chain rule.****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 .****Find the tangent to the curve r(t) = at t = 1.****Evaluate .****The line**y = 2 - x**and the parabola intersect at the points (-2, 4) and (1, 1),If R is the region enclosed by y=2-x and****,then find .**

**(10 x 3 = 30 Marks)**

###
__PART-B__** **

__PART-B__

**Answer any 2 complete questions each having 7 marks**

**Find the radius of convergence and interval of convergence of the series .****Test the convergence of****Find the Taylors series of about x = 1.**

**Answer any 2 complete questions each having 7 marks**

**Find the domains of (ii) and describe them in words.****Find the limit of as along (i) the X-axis , (ii) the Y- axis (iii) the line y = x.****Find the spherical and cylindrical coordinates of the point that has rectangular coordinates**

**Answer any 2 complete questions each having 7 marks**

**Locate all relative maxima , relative minima and saddle point if any,of .****Let f be a differentiable function of 3 variables and suppose that prove that .****Find the local linear approximation L(x,y) to f(x,y) = 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**

**Find y(t) where .****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).****Find the directional derivative of at in the direction of PQ where Q is the origin.**

**Answer any 2 complete questions each having 7 marks**

**Find the area bounded by the x-axis, Y = 2x and x+y=1 using double integration.****Use a triple integral to find the volume of the solid ,within the cylinder and between the planes z = 1 and x + z = 5.****Sketch the region of integration and evaluate the integral by changing the order y of integration.**

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**eduktu@gmail.com**.
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