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01000MA 101032103

Find the radius of convergence and interval of convergence of the series

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Module II
Answer any two questions. Each question carries 5 Marks
Find the local linear approximation of f(x,y) = /x? + y2 at (3,4) and compare
the error in approximation by L(3.04,3.98) with the distance between the points.
If = f(x —y,y - മമ — x), prove that _ ಕ ‏جس‎ 0.
Find the relative extrema of f(x,y) = xy — x? — 12
Module III
Answer any two questions. Each question carries 5 Marks
The position function of a particle is given by? = (ಟೆ — 2t)# + (ಟಿ - 4(۸ Find the
scalar tangential and normal components of acceleration. Also find the vector

tangential and normal components of acceleration at t = 1.
Prove that Vf(r) = LO; where 77 = ‏غير‎ + yf+ zk andr = ||.
Find an equation of the tangent plane to the ellipsoid x? + 4y? + 22 = 18 at the
point (1,2,1) and the parametric equation for the normal line to the surface at the
point.
Module IV
Answer any two questions. Each question carries 5 Marks
Evaluate She ydxdy where R is the region bounded by the parabola y? = 4x
and x? = 4y.
Change the order of integration and evaluate | त 1 /2 e* dxdy.
Find the volume of the solid within the cylinder x? + y? = 9 and between the
planes z= 1 andx+z=5.
Module V

Answer any three questions. Each question carries 5 Marks
Prove that 72/7) = 770) + 2700 where 7 = xf + yf+ 2.
Evaluate the line integral i —ydx + xdy along y? = 3x from (3,3) to (0,0)
Find the work done by the force field F = (y—x?)t+(z—y?)p+(x—z2)k

along the curve # = tt + {2 +- 30 > £ < 1 from (0,0,0) to (1,1,1)

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