Download GTU BE/B.Tech 2019 Summer 1st And 2nd Sem (New And SPFU) MTH001 Calculus Question Paper

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Subject Code: MTH0001

GUJARAT TECHNOLOGICAL UNIVERSITY

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Subject Name: Calculus

Time: 10:30 AM TO 01:00 PM

Instructions:

1. Attempt any five questions.

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2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q1 (a) (i) Discuss the convergence of the series S (2 (n+1)" x") / n, n=1 to infinity. 04
(ii) Test the convergence of the series S ((-1)^(n-1) n) / (n^2 -1), n=1 to infinity. 03

(b) If u=f(r) and r^2 =x^2 +y^2 +z^2, prove that ?^2u/?x^2 + ?^2u/?y^2 + ?^2u/?z^2 = f''(r) + (2/r)f'(r) 07

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Q2 (a) (i) Show that X +y is continuous at the origin. 04
f(x,y) = (x^2 y) / (x^4 + y^2), (x,y) != (0,0)
f(x,y) = 0, (x,y) =(0,0)
(ii) If x=rcos?, y=rsin?, show that (?r/?x)^2 + (?r/?y)^2 =1. 03

(b) Determine absolute or conditional convergence of the series S ((-1)^n / (n + (1/n))), n=1 to infinity. 07

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Q3 (a) (i) Evaluate ? x / (v(x^2 + y^2)) dydx, limits 0 to 1, 0 to x. 04
(ii) Evaluate ? xydxdy over the region enclosed by the x-axis, the line x=2a and the parabola x^2 =4ay. 03

(b) If u=sin^-1 ((x-y) / (x+y)), prove that x(?u/?x) + y(?u/?y) = tanu 07

Q4 (a) (i) Find the equation of the tangent plane and normal line to the surface z=2x^2+y^2 at the point (1, 1,3). 04
(ii) If u=y^2 -4ax, x=at^2, y=2at find du/dt. 03

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(b) Use triple integral to find the volume of the solid within the cylinder x^2 +y^2 =9 between the planes z=1 and x+z=1. 07

Q.5 (a) (i) Investigate the convergence of the series S (n^2 / 2^n), n=1 to infinity. 04
(ii) If u=f(r) and r^2 =x^2 +y^2 +z^2, prove that ?^2u/?x^2 + ?^2u/?y^2 + ?^2u/?z^2 = f''(r) + (2/r)f'(r) 03

(b) If u=r^m, r² =x² +y² +z² show that (?²u/?x²) + (?²u/?y²) + (?²u/?z²) =m(m+1)r^m-2 07

Q.6 (a) (i) Discuss the maxima and minima of the function 3x^2 - 3x + x^3 04

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(b) Evaluate ? xyz dxdydz over the positive octant of the sphere x^2 +y^2 +z^2 =4. 07

Q.7 (a) Evaluate ? e^(-y^2) dx dy by changing the order of integration. Limits 0 to infinity, 0 to y. 07

(b) (i) Find the minimum value of x^2+ y^2, subject to the condition ax+by =c. 04
(ii) Expand e^xy in power of (x—1) and (y+1) up to first degree terms. 03

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Total Marks: 70

Date: 07/06/2019