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

1. Let g(x) be a continuous function such that 0ò1 g(t) dt = 2. Let f(x) = 1/2 0òx (x-t)2 g(t) dt then obtain f ’(x) and hence evaluate f "(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(x) + f(y) + 2xy - one " x,y. f is differentiable and f ‘(0) = cos a. Prove that f(x) > 0 " x Î R.
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(x) + 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) dx. obtain f(x).
7. [Figure for ques. 7]Let T be an acute angled triangle. Inscribe rectangles R & S as shown. Let A(X) denote the area of any polygon X. obtain the maximum value of [A(R)+A(S)]/A(T).
8. Evaluate 0òx [x] dx .
9. Let f(x) 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 ‘(10) and f(5).
10. Evaluate: 0ò11/{ (5+2x-2x2)(1+e(2-4x)) } dx
11. If f(x) 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
12. Let P(x)= Õ (x-ai), where i=1 to n. and all ai’s are real. Prove that the derivatives P ‘(x) and P "(x) satisfy the inequality P’(x)2 ³ P(x)P"(x) for all real numbers x.
13. Determine the value of 0ò1 xa-1.(ln x)n dx where a Î {2, 3, ...} and n Î N.
14. Evaluate ò sinx dx/[sin(x-p/6).sin(x+p/6)]
15. explain the applicability of Rolle’s theorem to f(x)=log[x2+ab/{(a+b)x}] in the interval [a,b].
16. Let the curve y=f(x) passes through (4,-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(x) and x=0.
17. obtain the polynomial function f(x) 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.
18. obtain all the values of a (a¹ 0) for which: 0òx (t2-8t+13) dt = x sin(a/x). obtain that solution.
19. [Figure for ques. 19]Three squares are shown in the diagram. The largest has side AB of length 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,y1) 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.
21. y = f(x) be a curve passing through (1,1) 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.
22. 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+a) = 1/2 + Ö(f(x)-(f(x))2) holds for all x. (a) Prove that the function f is periodic. (b) For a=1, provide an example of a non-constant function with the needed properties.
23. 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.
24. Evaluate 0òµ (x - x3/2 + x5/(2.4) - x7/(2.4.6) + ... )(1 + x2/22 + x4/(2242) + x6/(224262) + ... ) dx.
25. Let f be a twice-differentiable real-valued function satisfying f(x) + f "e;(x) = -xg(x)f '(x), where g(x) ³ 0 for all real x. Prove that |f(x)| is bounded.
26. Let f be a real function on the real line with continuous third derivative. Prove that there exists a point such that f(a).f ‘(a).f "(a).f "’(a) ³ 0.
27. 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 (x) denotes an integer closest to x. e.g. (1.3)=1; (1.5)=2; (1.7)=2 and so on.

Earning:   Approval pending.