Part I: Constructing proofs.

You must write down all proofs in acceptable mathematical language: make sure you mark the beginning and end of the proof, define all variables, use complete, grammatically correct sentences, and give a justification for each assertion (e.g., by definition of…).

Definitions:

• An integer ? is even if and only if there exists an integer ? such that ? = 2?.

• An integer ? is odd if and only if there exists an integer ? such that ? = 2? + 1.

• Two integers have the same parity when they are both even or when they are both odd. Two integers have opposite parity when one is even and the other one is odd.

• An integer ? is divisible by an integer ? with ? ≠ 0, denoted ? | ?, if and only if there exists an integer ? such that ? = ??.

• A real number ? is rational if and only if there exist integers ? and ? with ? ≠ 0 such that ? = ?/?.

• For any real number ?, the absolute value of ?, denoted |?|, is defined as follows: |?| = { ? if ? ≥ 0; −? if ? < 0 4.

Prove each of the following statements using a direct proof, a proof by contrapositive, a proof by contradiction, or a proof by cases. For each statement, indicate which proof method you used, as well as the assumptions (what you suppose) and the conclusions (what you need to show) of the proof.

a. If ? is divisible by ? and ? is divisible by ?, where ?, ?, and ? are positive integers, then ? +? is divisible by ?.

b. The difference of any rational number and any irrational number is irrational.

c. There is no integer that is both even and odd.

d. Any two consecutive integers have opposite parity.

e. For all real numbers ? and ?, ???(?, ?) = ?+?−|?−?| ? and ???(?, ?) = ?+?+|?−?| ? .

A farmer has 80 meters of fence to make an enclosure for his cows.
The enclosure is divided into 2 sections by a fence parallel to one of the sides, as in
the drawing below. One section is shaped like a half-hoop and the other is shaped
rectangular. Find the dimensions that maximize the total area of ​​the enclosure.

A window has the shape of a rectangle surmounted by a semicircle similar to older buildings. The diameter of the semicircle is equal to the width of the rectangle. If the perimeter of the window is 30ft, find the dimensions of the window so that the greatest possible amount of light is admitted.

1. A principal amount P0 is deposited in a bank account earning 3% interest compounded monthly. After 20 years there is $1000 in the account. What was the principal P0?

2. Suppose you want to buy a house for $300, 000, but you only have $100,000. You are able to make an investment that pays 7.3% annual interest. If the interest is compounded continuously, how long will it take before you can buy your new house?

3. You are made an offer. You can have $1 million in cash, or you can get a penny today. But then the next day you will get two pennies, the day after that 4 pennies, and so on, the number of pennies doubling each day, for one month.

   (a) Explain why the number of pennies on the nth day is 2n.
   (b) How long will it take for the quantity of pennies to exceed $1 million worth?

   (c) Should you take the million or the penny?

There are four apples in a fruit basket which contains 10 fruits. What fraction of the fruits are apples as it is (do not simplify)?

There are four apples in a fruit basket which contains 10 fruits. What fraction of the fruits, in simplest form, are apples?

There are four apples in a fruit basket which contains 10 fruits. What percentage of the fruits are apples?

There are four apples in a fruit basket which contains 10 fruits. What proportion of the fruits are apples? Write your answer as a decimal.

There are three apricots in a fruit basket which contains 14 fruits. What proportion of the fruits are apricots? Round up your answer to the nearest hundredth.

In a fruit basket, there are three apples, four apricots, two peaches, and one watermelon. What is the probability of choosing an apple?

Find a matrix representation of transformation T(x)= 2x1w1+x2w2-3x3w3 from R3 to a vector space W, where w1,w2, and w3 ∈ W. Clearly state how this matrix is representing the transformation.

The table below gives the net worth of the first 13 U.S. Presidents as measured in millions of 2010 dollars (that just means the numbers have been adjusted for inflation). President Washington Adams Jefferson Madison Monroe Adams Jackson Van Buren Harrison Tyler Polk Taylor Fillmore Net Worth $525 $19 $212 $101 $27 $21 $119 $26 $5 $51 $10 $6 $4 Determine the following for this data set. Round to one decimal place, as necessary. Mean. Median. Mode Midrange Range Standard Deviation

Let R be the region bounded by the following graphs. Find the quantities below.y= (x−1)^2+ 1,y= 1,x= 0

a. Volume of the solid formed by rotating R about the x-axis, using the shell method.

b. Volume of the solid formed by rotating R about they-axis, using the disc method.

Solve the equation

(? − 2)2 + 2?2 − 13? = 10

by the method requested. Show the work in details. Do not skip any step. If steps are skipped you will receive zero credit.

a) Factoring

2. b) Completing the square and square root property

3. c) Quadratic Equation

Consider the function on the interval (0, 2π).

f(x) = sin(x) cos(x) + 2

(a) Find the open interval(s) on which the function is increasing or decreasing. (Enter your answers using interval notation.)

(b) Apply the First Derivative Test to identify all relative extrema.

indicate which are the following options is the subsidiary equation of the problem with initial conditions y(0) = 0 and y '(0) = 0

15 y + 8 y ′ + y ″ = U₄(t) + U₈ (t)

Suppose we know that the radius of a circle is increasing at a rate of 10 feet each second, and we want to know how fast the area of the circle is increasing when the radius is 5 feet. What can we do?

Find the reduced row echelon form of the following matrices. Interpret your result by giving the solutions of the systems whose augmented matrix is the one given.

[ 0 0 3 -1 5

1 0 0 4 2

4 1 3 0 -8

1 2 7 9 0 ]