A+ Answers

Discrete Review

1. Prove or disprove

A. The difference of any two odd integers is odd.

B. An odd integer minus an even integer is odd.

2. Do a proof by mathematical induction using the four steps below that the following statement is true for every positive integer n.

A. Show that the base step is true:

B. What is the inductive hypothesis?

C. What do we have to show?

D. Proof:

3. Do a proof by mathematical induction using the four steps below that the following statement is true for every positive integer n.

A. Show that the base step is true:

B. What is the inductive hypothesis?

C. What do we have to show?

D. Proof:

4. Prove using mathematical induction that

2

3n

-1

is divisible by 7 for all

nλ

.

A. Show that the base step is true:

B. What is the inductive hypothesis?

C. What do we have to show?

D. Proof:

Calculus Review

1. Find the limit:

2. Find the limit:

lim

x®0

1-cos3x

x

2

3. Find the limit:

lim

x®¥

2x

2

-9

3x

2

+10x

4. Find the limit:

lim

x®¥

9x

2

+16

2+x

3

+1

5. Write the equation for each asymptote for:

2x

2

+x+5

x

6. Find the limit of

lim

x®0

1

x

-

1

sinx

æ

è

ç

ö

ø

÷