Reference no: EM131110622
Honors Exam in Real Analysis 2007
1. Suppose K is a subset of Rn. Prove that K is compact if and only if every continuous function f: K → R is bounded.
2. Suppose f: (0, 1] → R is differentiable and satisfies |f'(x)| < 1 there. Show that the sequence {f(1/n)} converges.
3. Suppose f: [0,∞) → R is of class C2, and f(x) → 0 as x → ∞.
(a) If f'(x) → b as x → ∞, show that b = 0.
(b) If f'' is bounded, show that f'(x) → 0 as x → ∞.
(c) Give an example of such an f for which f'(x) does not converge as x → ∞.
4. Suppose f: [0, 1] → R is upper semicontinuous: This means that for every x ∈ [0, 1] and every ε > 0, there exists δ > 0 such that |y -x| < δ implies f(y) < f(x) + ε. Prove that f is bounded above and achieves its maximum value at some x ∈ [0, 1].
5. Define a function f: [0, 1] → R by setting f(x) = 1/n if the first 0 after the decimal point in the decimal expansion of f occurs in the nth place after the decimal point, and f(x) = 0 if there are no 0's after the decimal point. Prove that f is Riemann-integrable. (To avoid ambiguity, choose decimal representations ending in 9's instead of 0's when both are possible. For example,
f(1/10) = f(0.0999 ... ) = 1,
f(1/2) = f(0.4999 ... ) = 0,
f(1) = f(0.9999 ... ) = 0.
Note that only digits after the decimal point are counted.)
6. Suppose f: Rn → Rk is continuous. Let λ be a positive real number, and assume that for every x ∈ Rn and a > 0, f(ax) = aλf(x).
(a) If λ > 1, show that f is differentiable at 0.
(b) If 0 <λ< 1, show that f is not differentiable at 0.
(c) If λ = 1, show that f is differentiable at 0 if and only if it is linear.
7. Let M2 be the set of 2 × 2 real matrices, identified with R4 in the obvious way, and define F: M2 → M2 by F(X) = X2 (i.e., the matrix X multiplied by itself). Does F have a local smooth inverse in a neighborhood of I (the 2 × 2 identity matrix)? Answer the same question when I is replaced by
Prove your answers correct.
8. Let T ⊂ R3 denote the doughnut-shaped surface obtained by revolving the circle (y - 2)2 + z2 = 1 around the z-axis. Give T the orientation determined by the outward unit normal.
(a) Compute the surface area of T.
(b) Compute the integral ∫Tω, where ω is the 2-form z dx ∧ dy.
9. Suppose M is a smooth, compact n-manifold with boundary in Rn. If f is a smooth real-valued function and X is a smooth vector field on Rn, use the general version of Stokes's theorem to prove the following "integration by parts formula":
∫M(grad f,X) dV = ∫∂Mf(X, N) dV - ∫Mf div X dV,
where (·, ·) denotes the Euclidean inner product or dot product, and N denotes the outward unit normal to ∂M. Explain what this has to do with integration by parts.
Read in the interest rate
: In C++ Using a while loop, code and run a program that will calculate how long it will take $10,000.00 to become $1,000,000 with 0.08 interest rate.
|
How high above the center of a circle of radius 10.0 in
: How high above the center of a circle of radius 10.0 in. should a light be placed so that illuminance at the circumference will be a maximum? See Fig. 27.51.
|
Data structures and algorithm analysis2d arrays in java
: Data Structures and Algorithm Analysis2d arrays in Java. This program will return the smallest number of coins. There are 3 different type of coins. We have 10cent coin, 6 cent coin, and 1 cent coins. For example if i wanted to give 12 cents, using t..
|
Organizational behaviour week discussion
: Have you ever witnessed escalation of commitment in your organization--or seen it take place with governments? Why do you think that escalation of commitment continued instead of stopping support of the original decision?
|
Compute the surface area of t
: Let T ⊂ R3 denote the doughnut-shaped surface obtained by revolving the circle (y - 2)2 + z2 = 1 around the z-axis. Give T the orientation determined by the outward unit normal. Compute the surface area of T
|
Compute net cash flow from operating activities
: 1. The net income for Letterman Company for 2010 was $320,000. During 2010, depreciation on plant assets was $124,000, amortization of patent was $40,000, and the company incurred a loss on sale of plant assets of $21,000. Compute net cash flow from ..
|
Write a program in c++ to accept a string
: Write a program in C++ to accept a String and print the total no of vowels in it. Also print the string in upper and lower case
|
The art of oratory discussion question
: After watching several films (or plays) over your lifetime, and now reading Syd Field's theory on the Three-Act-Paradigm, what do you notice about the commonalities of good presentations?
|
What shape will require the least amount of material
: A Y-shaped metal bracket is to be made such that its height is 10.0 cm and its width across the top is 6.00 cm. What shape will require the least amount of material? See Fig. 27.52.
|