1. Logic and Set Theory

(a) Let A = {x e R I - 1 < x < 3}, B = {2, 3, 4}, and C = {n e Z I - 2 < n < 3}. Find the

following sets:

(i) A n C                                                                                                                [1 mark]

(ii)  B u C                                                                                                                [1 mark]

(iii)  C \ B                                                                                                                [1 mark]

(iv) AR(c)                                                                                                                                                                                    [1 mark]

(b)     (i) What is the negation of the statement: ”There is no rational number less than -,3”? [2 marks]

(ii) What is the negation of the statement: ”(3q e Q)(Vn e N)(nq  Z)”?        [2 marks]

(iii) Write the contraposition to the implication concerning x e R:

If x is rational then x2  is rational. (Note: proof is not required)                  [2 marks]

(c) Prove that the following statement is false:

For all subsets A, B, and C of R we have A \ (B n C) = (A \ B) n (A \ C).   [4 marks]

(d) Prove that the infimum of the set S = {2 +  I n e N} is 2.    [6 marks]

2. Functions

(a) Find the natural domain of the function f : D → R defined by [2 marks]

(b) Find the range of the function g : R → R defined by

g(x) = cos(5x) + 3

[2 marks]

(c) Let f (x) = tan(x) and g(x) = sin(x2 ).  Give a formula for the composition function f 。g and state its natural domain.                                                                                  [3 marks]

1 - x

5 - x

[8 marks]

3. Limit of a function

(a) Use the limit laws to evaluate the following limits (show all the steps):

(i)

5x(x2 - 25)

x→0   5x2 - 25x

(ii)

5x+1 + 7x一1 

x→& 5x+2 + 3(7x )

(b) Use the definition of a limit to prove that lim(10x - 6) = 4.

x→ 1

(c) Use the Squeeze Theorem to prove that

lim                         = 0

.                                                                                                                                [5 marks]

4. Differentiation

(a) Find a function f : R → R with infinitely many inflection points. Justify your claim.

[5 marks]

cos(πxy)

at the point (1, 2) is 0.                                                                                             [5 marks]

5. Integration

(a)  Consider the integral     x ln(x)dx. Explain (in 1-2 sentences) why integration by substitu-

tion is ηot likely to work on this integral. Then, pick a valid method for integration, and use it to find the antiderivative.

[6 marks]

(b) When you were revising with your friend for the exam, you came across the following

problem:

x - 25      

Your friend solved this problem, and says that the answer here is 0 .  ．乞t在out calculating the integral or referring to its exact answer, explain why your friend is wrong.

[4 marks]

6.  Sequences and Calculus

Consider the following graph of y = (x + 1)一1.5 .

y

y=(x+1)-1.5

x

1        2        3        4        5        6

(a) By referring to the drawing above, for any natural number N , which of the following is

N                                           N

0                                             n=2  n1.5

[5 marks]