CHAPTER 8 Flashcards
BALL IS DROPPED 10 FT HIGH
Ball is dropped from a height of 10 feet. It rebounds one-half the distance on each bounce. What is the total distance it travels?
30 feet
GREATER e^π or π^e
Which is greater e^π or π^e
e^π > π^e
OBVIOUSLY THE SMALLER THE COMPUNDING PERIOD
Obviously the smaller the compounding period, the greater the interest. How much does one dollar amount to after one year at
100% per annum interest, compounded continuously, i.e., instantaneously?
e = 2.71..
SUM OF INFINITE SERIES
Find the sum of the infinite series 1 + 1/2 + 1/3 + 1/6 +1/8 + 1/9 + 1/12 + …, whose terms are the reciprocals of positive integers which are divisible by no prime > 3.
3
INFINITE PRODUCT
Evaluate the infinite product ( n=1 to ∞ ) ∏ (2^n + 1) / (2^n + 2) = 3/4 ⋅ 5/6 ⋅ 9/10
limit is 1/2 as n → ∞
RECTANGULAR OPEN BOX MADE FROM METAL SHEET
A rectangular box without a top is to be made from a sheet of metal in the manner familiar to all calculus students, i.e., by cutting out squares from the corners and bending up the sides. The finished product is to have maximum volume and its dimensions are to be all integers” How will these dimensions compare if the metal cutout amounts to 10% of the original sheet?
6 : 3 : 1
SERIES THAT IS FAMILIAR
A student studying series starts with the familiar 1 + 1/2 + 1/4 + 1/8 + … and inserts terms midway between these, obtaining 1 +
3/4 + 1/2 + 3/8 + 1/4 + … . He divides this by 2, since, as he explains it. “There are now twice as many terms as before.” He repeats the process, interpolating terms between those already placed, again dividing by 2. If he continues this indefinitely, what limit will the series approach?
limit is 3/2 as n → ∞
MR. X VEERS TO THE RIGHT WHEN HE WALKS
Mr. X veers to the right when he walks. The curvature of his path is proportional to his latitude. He starts walking North from point A on the equator, in the area of a large level plain, and finds he is proceeding East when he is one mile noith of the equator. He continues walking and arrives back at the equator at point B. What is the straight line distance from A to B?
Thus AB = 2x = a trifle less than 1.2 miles
ARCHIMEDES O’TOOLE FAVORABLE RESPONSE TO POETICIAN
Archimedes O’Toole was so overcome by the favorable response among “Poeticians” to his last mathematical limerick (# 1-41 ), that he composed another based on the above identity. Can you reconstruct the limerick?
( ∫ z^2dz, 1, 3 1/3)(cos(3π/9) = ln ∛e
Integral z squared dz
from one to the cube root of three
Multiplied by cosine of three pi over nine
is the log of the cube root of e.
WHICH CONTAINS MORE TERMS
Which contains more terms: the general polynomial of tenth degree in six variables or the general polynomial of sixth degree in ten variables?
In the specific case considered the number of terms is 8,008 in both instances
A sub(1+2) = 3A sub(a) + 2A sub(n+1)
Consider the sequence 0, 1, 2, 7, 20, 61,… in which A sub(1+2) = 3A sub(a) + 2A sub(n+1).
Assuming the ratio of successive terms approaches a limit r, compute r.
r = 3
DECIMAL 0.1 IN BASE TWO EQUALS 0.5
The decimal 0.1 in base two equals 0.5 in base ten. Likewise, 0.12 in base three and 0.123 in base four equal 0.556 and 0.422, respectively. Continuing in this manner, as the base increases, what is the limiting value of the decimal?
limiting value?
equals to?
B → ∞
- limiting value < 2 which equals to 2/B
thus, as B → ∞, 0.12345… → zero
A BOAT OWNER OUTING
A boat owner agrees to take a group on an outing at $4.50 apiece if the number of passengers is equal to or less than his break- even point. For each person above this he reduces the fare for all passengers 3 cents per person. If he has on board now the numberof passengers that maximizes the total collected, what is the boat owner’s profit?
0
Thus, the boat owner managed to achieve only his break-even point for a profit of zero.
A NOVICE IN CALCULUS REQUIRED TO DIFFERENTIATE AN EXPRESSION
A novice in calculus was required to differentiate an expression of the form A^x and evaluate at x=3. Naively using x A^(x-1) as the derivative, he nevertheless obtained the correct value. What was A?
A = 2.8564
PLATINUM TRAYS
The price per cubic inch for plantinum trays is the same as that per square inch for platinum sheets. A metal supply house has a square of platinum which will yield the same amount whether sold as a sheet, or fashioned into a tray of maximum volume with the four cut-out corners sold as sheets. How big is the square?
S = exactly 1 ft
