Harvard-MIT Mathematics Tournament March 15. 2003 Individual Round: Calculus Subject Test a point is chosen randomly with uniform distribution in the interior of a circle of radius 1. What is its expected distance from the center of the circle? 2. a particle moves along the -axis in such a way that its velocity at position is given by the formula v(ar)=2+ sin r. What is its acceleration at a 3. What is the area of the region bounded by the curves y= x2003 and y=x/2003 and ing above the x-axis 4. The sequence of real numbers 21, 2, T3,. satisfies limn-oo(a2n+ 2n+1)= 315 and limn-oo(2n +I2n-1=2003. Evaluate limn-oo(a2n/ 2n+1) 5. Find the minimum distance from the point (0, 5/2)to the graph of y=x4/8 6. For n an integer, evaluate 3(√n2-02+√m2 7. For what value of a> 1 is 1 minimum? 8. A right circular cone with a height of 12 inches and a base radius of 3 inches is filled with water and held with its vertex pointing downward. Water flows out through a hole at the vertex at a rate in cubic inches per second numerically equal to the height of the water in the cone.(For example, when the height of the water in the cone is 4 inches, water Hows out at a rate of 4 cubic inches per second. Determine how many seconds it will take for all of the water to How out of the cone 9. Two differentiable real functions f(ar) and g(ar) satisfy f(x)-9(x) for all r, and f(0)=g(2003)= 1. Find the largest constant c such that f(2003)>c for all such functions f, g valuate d 1Harvard-MIT Mathematics Tournament March 15, 2003 Individual Round: Calculus Subject Test 1. A point is chosen randomly with uniform distribution in the interior of a circle of radius 1. What is its expected distance from the center of the circle? 2. A particle moves along the x-axis in such a way that its velocity at position x is given by the formula v(x) = 2 + sin x. What is its acceleration at x = π 6 ? 3. What is the area of the region bounded by the curves y = x 2003 and y = x 1/2003 and lying above the x-axis? 4. The sequence of real numbers x1, x2, x3, . . . satisfies limn→∞(x2n + x2n+1) = 315 and limn→∞(x2n + x2n−1) = 2003. Evaluate limn→∞(x2n/x2n+1). 5. Find the minimum distance from the point (0, 5/2) to the graph of y = x 4/8. 6. For n an integer, evaluate limn→∞ µ 1 √ n2 − 0 2 + 1 √ n2 − 1 2 + · · · + 1 q n2 − (n − 1)2 ¶ . 7. For what value of a > 1 is Z a 2 a 1 x log x − 1 32 dx minimum? 8. A right circular cone with a height of 12 inches and a base radius of 3 inches is filled with water and held with its vertex pointing downward. Water flows out through a hole at the vertex at a rate in cubic inches per second numerically equal to the height of the water in the cone. (For example, when the height of the water in the cone is 4 inches, water flows out at a rate of 4 cubic inches per second.) Determine how many seconds it will take for all of the water to flow out of the cone. 9. Two differentiable real functions f(x) and g(x) satisfy f 0 (x) g 0 (x) = e f(x)−g(x) for all x, and f(0) = g(2003) = 1. Find the largest constant c such that f(2003) > c for all such functions f, g. 10. Evaluate Z ∞ −∞ 1 − x 2 1 + x 4 dx. 1