# Indian Institute of Information Technology 2007 B.E Information Technology Calculus - Question Paper

Wednesday, 23 January 2013 12:00Web

**1.**Let g

**(**be a continuous function such that 0ò1 g(t) dt =

**x)****2.**Let f

**(**= 1/2 0òx (x-t)2 g(t) dt then obtain f ’

**x)****(**and hence evaluate f "(x).

**x)****2.**obtain area of region bounded by the curve y=[sinx+cosx] ranging from x=0 to x=2p.

**3.**Solve: [3Öxy + 4y - 7Öy]dx + [4x - 7Öxy + 5Öx]dy = 0.

**4.**f(x+y) = f

**(**+ f(y) + 2xy - one " x,y. f is differentiable and f ‘(0) = cos

**x)****a.**Prove that f

**(**> 0 " x Î R.

**x)****5.**Show that 0òp q3 ln sin q dq = 3p/2 0òp q2 ln [Ö2 sin q] dq.

**6.**A function f : R® R satisfies f(x+y) = f

**(**+ f(y) for all x,y Î R and is continuous throughout the domain. If I1 + I2 + I3 + I4 + I5 = 450, where In = n.0òn f

**x)****(**d

**x)****x.**obtain f(x).

**7.**[Figure for ques. 7]Let T be an acute angled triangl

**e.**Inscribe rectangles R & S as shown. Let A

**(**denote the area of any polygon

**X)****X.**obtain the maximum value of [A(R)+A(S)]/A(T).

**8.**Evaluate 0òx [x] dx .

**9.**Let f

**(**be a real valued function not identically equal to zero such that f(x+yn)=f(x)+(f(y))n; y is real, n is odd and n >1 and f ‘(0) ³ 0. obtain out the value of f ‘

**x)****(**and f(5).

**10)****10.**Evaluate: 0ò11/{ (5+2x-2x2)(1+e(2-4x)) } dx

**1**If f

**1.****(**is a monotonically increasing function " x Î R, f "

**x)****(**> 0 and f -1

**x)****(**exists, then prove that å{f -1(xi)/3} < f -1({x1+x2+x3}/3), i=1,2,3

**x)****1**Let P(x)= Õ (x-ai), where i=1 to n. and all ai’s are real. Prove that the derivatives P ‘

**2.****(**and P "

**x)****(**satisfy the inequality P’(x)2 ³ P(x)P"

**x)****(**for all real numbers x.

**x)****1**Determine the value of 0ò1 xa-1.(ln x)n dx where a Î {2, 3, ...} and n Î N.

**3.****1**Evaluate ò sinx dx/[sin(x-p/6).sin(x+p/6)]

**4.****1**explain the applicability of Rolle’s theorem to f(x)=log[x2+ab/{(a+b)x}] in the interval [a,b].

**5.****1**Let the curve y=f

**6.****(**passes through (4,-

**x)****2)**and satisfies the differential equation: y(x+y3)dx = x(y3-x)dy and y=g(x)=[Integral 1/8 to sin sq. x]sin -1Öt dt + [Integral 1/8 to cos sq. x]cos -1 Öt dt, 0 £ x £p/

**2.**obtain the area of the region bounded by the curves y=f(x), y=g

**(**and x=0.

**x)****1**obtain the polynomial function f

**7.****(**of degree six which satisfies : Lim(x®0)[1 + f(x)/x3]1/x = e2 and has local maxima at x=1 and local minima at x=0,2.

**x)****1**obtain all the values of a (a¹ 0) for which: 0òx (t2-8t+

**8.****1**dt = x sin(a/x). obtain that solution.

**3)****1**[Figure for ques. 19]Three squares are shown in the diagram. The largest has side AB of length

**9.****1.**The others have side AC of length x and side DE of length y. As D moves along AB, obtain the values of x and y for which x2+y2 is a minimum. What is this minimum?

**20.**Suppose that the cubic polynomial h(x)=x3 - 3bx2 + 3cx + d has a local maximum A(x1,y

**1)**and a local minimum at B(x2,y2). Prove that the point of inflection of h is at the midpoint of the line segment AB.

**2**y = f

**1.****(**be a curve passing through (1,

**x)****1)**such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the 1st quadrant and has area

**2.**Form the differential formula and determine all such possible curves.

**2**Let f be a real-valued function described for all real nos. x such that, for a few positive constant a, the formula f(x+

**2.****a)**= 1/2 + Ö(f(x)-(f(x))

**2)**holds for all

**x.**

**(**Prove that the function f is periodi

**a)****c.**

**(**For a=1, provide an example of a non-constant function with the needed properties.

**b)****2**Suppose p, q, r, s are fixed real numbers such that a quadrilateral can be formed with side’s p, q, r, and s in clockwise order. Prove that the vertices of the quadrilateral of maximum area lie on a circle.

**3.****2**Evaluate 0òµ (x - x3/2 + x5/(2.

**4.****4)**- x7/(2.4.

**6)**+ ... )(1 + x2/22 + x4/(22

**4**+ x6/(2242

**2)****6**+ ... ) dx.

**2)****2**Let f be a twice-differentiable real-valued function satisfying f

**5.****(**+ f "e;

**x)****(**= -xg(x)f '(x), where g

**x)****(**³ 0 for all real

**x)****x.**Prove that |f(x)| is bounded.

**2**Let f be a real function on the real line with continuous third derivativ

**6.****e.**Prove that there exists a point such that f(a).f ‘(a).f "(a).f "’

**(**³ 0.

**a)****2**The area ranging from the two curves described by y=½x-(x)½ and y=ksin(px), k ³ 1, in the interval 0£ x £1, is obtained to be equal to two sq. units. obtain the constant k.Here

**7.****(**denotes an integer closest to

**x)****x.**e.g. (1.3)=1; (1.5)=2; (1.7)=2 and so on.

Earning: Approval pending. |