Winter 2008 - Cioaba's Class - Exam 1 (Version 1)

# Winter 2008 - Cioaba's Class - Exam 1 (Version 1) - Name:...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Name: PID: Midterm 1, Math 109 - Winter 2008 Duration: 50 minutes Please close your books, turn off your calculators and phones. To get full credit you should explain your answers. 1.a. (2 points) Show that P ⇒ Q is equivalent to ( P or Q ) ⇔ Q . Proof. Use truth tables. P Q P ⇒ Q (P or Q) (P or Q) ⇔ Q T T T T T T F F T F F T T T T F F T F T b.(3 points) Show that parenleftbigg a + 1 b parenrightbiggparenleftbigg b + 1 a parenrightbigg ≥ 4 for any real and positive numbers a and b . When does equality hold ? Proof. Solution 1 If x and y are real numbers, then ( x- y ) 2 ≥ 0. This implies that x 2 + y 2- 2 xy ≥ which means that x 2 + y 2 ≥ 2 xy (1) In (1), replace x by √ a and y by 1 √ b . We obtain a + 1 b ≥ 2 radicalbigg a b > (2) In (1), replace x by √ b and y by 1 √ a . We obtain b + 1 a ≥ 2 radicalbigg b a > (3) Multiplying (2) and (3), we obtain parenleftbigg a + 1 b parenrightbiggparenleftbigg b + 1 a parenrightbigg ≥ (2 radicalbigg a b )(2 radicalbigg...
View Full Document

## This note was uploaded on 04/30/2008 for the course MATH 109 taught by Professor Knutson during the Winter '06 term at UCSD.

### Page1 / 3

Winter 2008 - Cioaba's Class - Exam 1 (Version 1) - Name:...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online