Calculus I PU Fall question paper

8 ay State Leibnitz theorem. If y= (c +V1l+x? y , Show that i, (14+x?)y, +29, —mPy =0 (4x? yg (20 Dag +07 -m?)y, =0 b)/ State Rolle’s Theorem and wriie its geometrical meaning. Verify it for f(x) = (x —@)™(x — b)", xe[a, b] where m and n are positive integers. . CR

• b) Assuming the validity of the expansion, expand f(x) = tanx using Maclaurin’s series and hence the expansion of sec? x.

oa Find the asymptotes of the curve (x—yPx? a(x? +7 = 0.

• b) Find the radius of curvatwe at any point of the parametric curve x =a(t+sint),y = a(1—cost). +

Integrate Any Three of the following: 3x5 vice 2=3sin 2x pt/2_Neotx =

• b) Prove that: (0° joe aan X= 00 x. A Evaluate: rerecy iii
• d) By using gamma function evaluate Jsin*3xcos? 6xdx 0 4, a) Find the volume of the solid in the region in the first quadrant bounded by the curve y = x? , and the lines y = 0,x = 1 about the line x = —1. ae eae a w/ State and prove Euler's theorem of homogenous function of two variables x and y of degree n, fu = sin“? (222 ou 4 yu Ifu = sin ( Te *) then show that x mrty iar tanu S. ay Use Lagrange's multipliers Method, Find the minimum value of f = x2+y? 427 such thatx + y + 2 = 3a? by Define Bernoulli’s equation. Solve: 2 —ytanx = —y? secx.

OR

• b) A tank initially contains 40 kg of salt dissolved into 200 liter of water, A solution of 2 kg of salt per liter is allowed to enter the tank at the rate of 5 liters per minute and uniform solution is drained from the tank at the same rate. Find the amount of salt at any time t. Also determine the salt in tank in 15 minutes. : 6/2) Define Euler -Cauchy equation. Solve the initial value problem: 4x2y" + 24xy! + 25y = 0,y(1) = 2,y'(1) = -6 et py Solve by method of variation of paiameter y” — 2y' + y => q x 7. Attempt all Questions. 4 4x25 2—-x forl<% <2 2 weoson 0) =| x-= forx>2 Examine continuity of a function at x=-2.
• b) Check line of symmetry of the curve y’(a-x) = x7(a-x).
• c) Find the arc length of the curve x?+y"=a? .
• d) Solve: (1+ x)ydx+(i+y)x dy=0. Farr iriaiee weap) n