

A1. Prove that (a  x)^{6}  3a(a  x)^{5} + 5/2 a^{2}(a  x)^{4}  1/2 a^{4}(a  x)^{2} < 0 for 0 < x < a.


A2. Define f(x) = ∫_{0}^{x} ∑_{i=0}^{n1} (x  t)^{i} / i! dt. Find the nth derivative f^{ (n)}(x).


A3. A circle radius a rolls in the plane along the xaxis the envelope of a diameter is the curve C. Show that we can find a point on the circumference of a circle radius a/2, also rolling along the xaxis, which traces out the curve C.


A4. The real polynomial x^{3} + px^{2} + qx + r has real roots a ≤ b ≤ c. Prove that f ' has a root in the interval [b/2 + c/2, b/3 + 2c/3]. What can we say about f if the root is at one of the endpoints?


A5. The line L is parallel to the plane y = z and meets the parabola y^{2} = 2x, z = 0 and the parabola 3x = z^{2}, y = 0. Prove that if L moves freely subject to these constraints then it generates the surface x = (y  z)(y/2  z/3).


A6. f is defined for the nonnegative reals and takes positive real values. The centroid of the area lying under the curve y = f(x) between x = 0 and x = a has xcoordinate g(a). Prove that for some positive constant k, f(x) = k g'(x)/(x  g(x))^{2} e^{ ∫ 1/(t  g(t)) dt}.


A7. Do either (1) or (2):
(1) Let A be the 3 x 3 matrix
1+x^{2}y^{2}z^{2} 2(xy+z) 2(zxy)
2(xyz) 1+y^{2}z^{2}x^{2} 2(yz+x)
2(zx+y) 2(yzx) 1+z^{2}x^{2}y^{2}
Show that det A = (1 + x^{2} + y^{2} + z^{2})^{3}.
(2) A solid is formed by rotating about the xaxis the first quadrant of the ellipse x^{2}/a^{2} + y^{2}b^{2} = 1. Prove that this solid can rest in stable equilibrium on its vertex (corresponding to x = a, y = 0 on the ellipse) iff a/b ≤ √(8/5).


B1. A particle moves in the plane so that its angular velocity about the point (1, 0) equals minus its angular velocity about the point (1, 0). Show that its trajectory satisfies the differential equation y' x(x^{2} + y^{2}  1) = y(x^{2} + y^{2} + 1). Verify that this has as solutions the rectangular hyperbolae with center at the origin and passing through (±1, 0).


B2. Find:
(1) lim_{n→∞} ∑_{1≤i≤n} 1/√(n^{2} + i^{2});
(2) lim_{n→∞} ∑_{1≤i≤n} 1/√(n^{2} + i);
(3) lim_{n→∞} ∑_{1≤i≤n2} 1/√(n^{2} + i);


B3. Let y_{1} and y_{2} be any two linearly independent solutions of the differential equation y'' + p(x) y' + q(x) y = 0. Let z = y_{1}y_{2}. Find the differential equation satisfied by z.


B4. Given an ellipse center O, take two perpendicular diameters AOB and COD. Take the diameter A'OB' parallel to the tangents to the ellipse at A and B (this is said to be conjugate to the diameter AOB). Similarly, take C'OD' conjugate to COD. Prove that the rectangular hyperbola through A'B'C'D' passes through the foci of the ellipse.


B5. A wheel radius r is traveling along a road without slipping with angular velocity ω > √(g/r). A particle is thrown off the rim of the wheel. Show that it can reach a maximum height above the road of (rω + g/ω)^{2}/(2g). [Ignore air resistance.]


B6. f is a real valued function on [0, 1], continuous on (0, 1). Prove that ∫_{x=0}^{x=1} ∫_{y=x}^{y=1} ∫_{z=x}^{z=y} f(x) f(y) f(z) dz dy dx = 1/6 ( ∫_{x=0}^{x=1} f(x) dx )^{3}.


B7. Do either (1) or (2):
(1) f is a realvalued function defined on the reals with a continuous second derivative and satisfies f(x + y) f(x  y) = f(x)^{2} + f(y)^{2}  1 for all x, y. Show that for some constant k we have f ''(x) = ± k^{2} f(x). Deduce that f(x) is one of ±cos kx, ±cosh kx.
(2) a_{i} and b_{i} are constants. Let A be the (n+1) x (n+1) matrix A_{ij}, defined as follows: A_{i1} = 1; A_{1j} = x^{j1} for j ≤ n; A_{1 (n+1)} = p(x); A_{ij} = a_{i1}^{j1} for i > 1, j ≤ n; A_{i (n+1)} = b_{i1} for i > 1. We use the identity det A = 0 to define the polynomial p(x). Now given any polynomial f(x), replace b_{i} by f(b_{i}) and p(x) by q(x), so that det A = 0 now defines a polynomial q(x). Prove that f( p(x) ) is a multiple of ∏ (x  a_{i}) plus q(x).

