Calculus I PU Spring question paper

• a) Find the asymptotes of the curve: xtx?y—xy?-y3+x?-y?-2=0.
• b) i. Find the radius of curvature at any point of the curve 4+4 x =a(1+sint) y =a(a-—cost)
• ii) Trace the curve of y? = (x — 2)3

Attempt (Any three) of the following: 3x5 1

• a) Evaluate: f > dx
• b) Prove that = =2 —f ° Vx+Ja—-x Saag ©
• c) Evaluate: {¢ cos* 30 sin? 60 d@ (apply beta gamma function)
• d) Obtain the reduction formula for J sec™ x dx and hence evaluate f sec’ x dx.

• a) State and prove Euler’s theorem on homogeneous functions of two : variables of degree n. If sinu= = then show that du du xx +y ra tanu. i i
• b) Find the volume of the solid generated by revolving be oe between the parabola y = x? and the line y=1 about the line y~ -. OR ; Find the surface area and volume of the solid generated by revolving 2 the asteroid x + y3 = a3 about x-axis. 14 m 1 _ F

• a) Find the maximum value of x? + y? +2? when +5 +7 = Lusing Lagrange’s multipliers method.
• b) Define Bernoulli’s equation. Solve: 2 +xsin2y =x° cos? y. OR (ay The growth rate of a culture of bacteria is proportional to the number of bacteria present. After one day it is 1.5 times of original number. Find after how many days it will be
• i) double
• ii) triple.

• a) Find the general solution by using the method of variation of 2x parameters: y"—4y'+4y = - x
• b) i. Solve the initial value problem: (x?D? — 3xD + 4)y = 0 ;y(1) = 0,y'(1) = 3. 543
• ii) Find the particular solution of _y"—y' — 2y = 2e*

Attempt all the questions 4x25

• a) A function f(x) is defined as follows: 4= f= { x, x<2 x-2, x22 p Show that f (x) is not differentiable at x = 2. (my
• b) Find the area enclosed by x-axis and the curve y = 3x —5x?,
• c) State Riccati's equation. Also write the condition when it becomes first order linear differential equation.
• d) What is the equation of Newton's Law of Cooling? fran “tVieroEnotes