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Calculus
Math 294A: Problem Solving Seminar – Vera Furst
November 18, 2006
Hard calculus problems are common on the Putnam Exam. The problems below require (more or less) the techniques of Calculus I–III and not those of a real analysis course. That said, the key tools at your disposal are:
- The fundamental theorem of calculus (1 and 2).
- The intermediate value theorem.
- The mean value theorem (differential and integral).
- L’Hopital’s Rule.
Example 1. Suppose f is a twice continuously differentiable function such that |f (x)| ≤ 1 for all x and (f (0))^2 + (f ′(0))^2 = 4. Prove that there exists a real number x 0 such that f (x 0 ) + f ′′(x 0 ) = 0.
Example 2. Show that the improper integral ∫ (^) ∞
0
sin x sin x^2 dx
converges.
Example 3. Let A be a positive real number. What are the possible values for
∑^ ∞
i=
x^2 i , given that
{ xi
i=0 is a sequence of positive numbers that satisfy
∑^ ∞
i=
xi = A?
Example 4. Find the maximum value of the function
F (y) =
∫ (^) y
0
x^4 + (y − y^2 )^2 dx.
Problem 1. Find all possible functions f : R → R with continuous derivative f ′^ such that
(f (x))^2 =
∫ (^) x
0
[
(f (t))^2 + (f ′(t))^2
]
dt + 1990.
Problem 2. Evaluate the integral
∫ (^4)
2
ln(9 − x) √ ln(9 − x) +
ln(x + 3)
dx.
Problem 3. A clock’s minute hand has length 4, and its hour hand has length 3. What is the distance between the tips at the moment when this distance is increasing most rapidly?
∗These problems (or some version of them) all appear either on previous Putnam exams or in the book Problem-Solving Through Problems by L.C. Larson.
Problem 4. For what positive real numbers α, β does the integral ∫ (^) ∞
β
x + α −
x −
x −
x − β dx
converge?
Problem 5. Find
lim x→∞ x
∫ (^) x
0
et
(^2) −x 2 dt.
Problem 6. Evaluate
lim n→∞
n^4
∏^2 n
i=
(n^2 + i^2 )^1 /n.
Problem 7.
(a) Suppose that f : [a, b] → R and g : [a, b] → R are continuous and that g(x) ≥ 0 for all x in [a, b]. Prove that there exists a number c in [a, b] such that ∫ (^) b
a
f (x)g(x) dx = f (c)
∫ (^) b
a
g(x) dx.
(b) Suppose that f : [a, b] → R is increasing (and therefore integrable), and g : [a, b] → R is integrable and satisfies g(x) ≥ 0 for all x in [a, b]. Prove that there exists a number c in [a, b] such that ∫ (^) b
a
f (x)g(x) dx = f (a)
∫ (^) c
a
g(x) dx + f (b)
∫ (^) b
c
g(x) dx.
Problem 8. Let C(α) denote the coefficient of x^1992 in the power series for (1 − x)α. Find
∫ (^1)
0
C(−y − 1)
k=
y + k
dy.
Problem 9. Suppose that f is differentiable, and that f ′(x) is strictly increasing for x ≥ 0. If f (0) = 0, prove that f (x)/x is strictly increasing for x > 0.
Problem 10. Let f be a differentiable function on [0.1] with f (0) = 0 and f (1) = 1.
(a) For each positive integer n, show that there exist distinct points x 1 , x 2 ,... , xn in [0, 1] such that ∑^ n
i=
f ′(xi)
= n.
(b) For each positive integer n and arbitrary positive numbers k 1 , k 2 ,... , kn, show that there exist distinct points x 1 , x 2 ,... , xn such that ∑^ n
i=
ki f ′(xi)
∑^ n
i=
ki.
Problem 11. In the (x, y)-plane, let R be the set of points inside and on a convex polygon, and let D(x, y) be the distance from (x, y) to the nearest point of R. Show that there exist constants a, b, and c, independent of R, such that ∫ (^) ∞
−∞
−∞
e−D(x,y)^ dx dy = a + bP + cA,
where P and A are the perimeter and area of R, respectively. Find the values of a, b, and c.
Problem 12. Sum the series 1 − 13 + 15 − 17 + 19 − · · ·.
