Download GTU BE/B.Tech 2019 Winter 1st And 2nd Sem New And Spfu MTH001 Calculus Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER- I & II (SPFU) EXAMINATION — WINTER 2019

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Subject Code: MTH0001 Date: 06/01/2020

Subject Name: Calculus

Time: 10:30 AM TO 01:00 PM Total Marks: 70

Instructions:

1. Attempt any five questions.

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

Q.1

(a) Test the convergence of S 1/(n=2)(n²+5) 07

(b) Test the convergence of 1/1e²e³ + 1/4e⁵e⁶ + 1/7e⁸e⁹ + ... 07

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Q.2

(a) Trace the curve r=a(1+cos?), a>0. 07

(b) Sketch the region of integration and evaluate ? e^(x2+y2) dydx 07

Q.3

(a) 1. If (cosx)^y =(siny)^x, find dy/dx 07

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2. If x^y +y^x =5xⁿ, find dy/dx

(b) Evaluate ? v(a² -r²)drd? over the upper half of the circle r =acos?. 07

Q.4

(a) Find the minimum and maximum value of f(x,y)=x³+y³ —3xy. 07

(b) Evaluate ? e^-y2 dxdy by changing order of integration. 07

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Q.5

(a) If u=tan^-1[ (x³+y³)/(x-y)], prove that x²u_xx + 2xyu_xy + y²u_yy =2cos²usinu. 07

(b) Evaluate ? xyzdzdydx (limits 0 to 1 for all integrals). 07

Q.6

(a) Find the equation of tangent plane and normal line at the point (-2,1,-3) to the ellipsoid x²/4 +y² +z²/9=3. 07

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(b) If u=f(r), where r² =x² +y², prove that u_xx + u_yy = f"(r)+(1/r)f’(r). 07

Q.7

(a) Test the convergence for S 2tan^-1(n)/(n² +1) 07

(b) Find the area of the region R enclosed by the parabola y = x² and the line y=x+2. 07

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