Master Entrance Exam, Advanced Calculus, NCKU Math, Oct. 26, 2012 Show all works

[£~it~]:

Part I.

~ ~41&~~

Master Entrance Exam, Advanced Calculus, NCKU Math, Oct. 26, 2012 Show all works

1. Prove or disprove the following statements.

(a) Iff is continuous at x

=

0, then there exists a 8 > 0 such that f is continuous on ( -0, o). [5%]

(b) If

f :

R -+ R is differentiable on R, then

f'

is continuous on R. [5%]

(c) Let

f : n c

^{JR3 -+} lR be a smooth function. Consider a smooth surface S

=

{(x, y, z) f(x, y, z) = k} for some constant k and a curve {a(t) ⁼ (x(t), y(t), z(t}); a(t)

c

S, t E I} for some.

interval I. Show that \1

f

is perpendicular to the surfaceS. [10%]

2. (a) State the Inverse FUnction Theorem. (Do not prove it.) [5%]

0, x=O.

(b) Let f(x) = { x + x

3

sin~'

^x

^=I

⁰· Using the Inverse FUnction Theorem to prove that /(x)

has an inverse on a small neighborhood contai~ing 0. {10%]

3. (a) Let an > 0 and

L

^an converge. Do the series

La!

^and

L ^Fn

converge? or diverge?

n n n

Prove it if it is correct. Disprove it if it is wrong by giving examples.

(b) State the definitions of a function being continuous, uniformly continuous, and absolutely continuous. Also gives examples of functions being uniformly continuous but not absolutely continuous,

[5%]

and continuous but not uniformly continuous. (10%]

(c) Give examples for function being Riemann integrable but not Lebesgue intetgrable and being

Lebesgue intetgrable but not Riemann integrable. [10%]

4. Let ¢ be continuous and bounded. Let u£(x, t) = ₃1

. ₃ { { { e

-J(.~~;i

^{2 ¢(x') dx' with}

(27rt)2(€

+

t)2 ./1JR3 .

E, t > 0, where i =

v'=I.

State the theorems you use in the following problems.

1

(a) Use the change of variables x- x' = (2(€² + 1}t) ²y to show that (W%]

... lim vAx, t) = (E

- 3 ^{i)~ f{{}

e-<£-i)IYI² dy¢(x).

t-to+ 1r2

}}}Ra

(b) Use the spherical coordinates and integration by parts to show that 2( ^{€ - 't} ')1 ²

100

^{. 2}

lim u£(x, t) = e-(t-t)r dr<fJ(x).

t--+O+ yl1r ⁰

(c) Use the facts

1

^00cosr2dr=

1

^{00sin r2dr=}

ji,

^{ei8 =} cos9+isin9, to show that lim lim u£(x, t) = ¢(x).

£--+0+ t--+0+

(d) Show that the integral

1

^{00 cosr2dr} converges.

1

1 1-i

and lim (E-i)2 = - -

t--+O+

.J'i

(10%]

[10%]

(10%]

[£-~tt~]:

Part II. /t·ti-1\tt

Notation

• AT: the transpose of the matrix A

• In: then x n identity matrix

• R: the field oheal numbers

1. (10%) Compute the dimension of the subspace

{ (a

+

2b

+

3c - d, a

+

c

+

d, a+ 2b

+

3c-d, 4a

+

5b

+

9c - d)

I

a, b, c, d E R}

2. Let V be the vector space of all polynomials of degree at most 3 with real coefficients.

Suppose T: V - t V is the map defined by

T(p(x)) = p(x)

+

p'(x), where p'(x) is the derivative of p(x).

(a) (4%) Show that Tis a linear operator on V.

(b) (8%) Find [T]p where {3 = {1,x,x²,x^{3 }.} (Here [T]p is the matrix representation ofT with respect to the ordered basis [3.)

3. (10%) Let

A =

(2f

₁

7 3:

₂

7 6:

₃

7) .

a4 as Q.()

Find suitable ai E R, ⁱ = 1, ... ,6, such that AT A= AAT = 1_{3 •}

4. (10%) Show that there are no 5 x 5 invertible matrices A, B over R satisfying AB = -BA.

5. Let A be an m x n matrix over R.

(a) (7%) Show that rank(BA) ~ rank(A) for every n x m matrix B over R.

(b) (7%) Show that rank(AT A) = rank(A).

· 6. Consider the following matrix

A=

(~1 ~ ~).

-1 1 2 (a) (5%) Find the eigenvalues of A.

(b) (12%) Find an invertible matrix Q such that Q-¹ AQ is a diagonal matrix.

(c) (5%) Find a matrix B such that B³ =A.

7. (a) (10%) Show that if A is a 102 x 102 matrix over R such that A¹⁰² = 102A, then the matrix A - I 102 is invertible.

(b) (12%) Find a 102 x 102 matrix B over R such that B¹⁰²

+

B¹⁰¹

+ ... +

B

+

^!102 = 0.

Justify your answer.