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Any with an interest in science and technology interested in re-igniting an interest in Mathematics here is a starter copy and paste from my post on another forum with further free starter informtion & material.
Came across this actual problem in a book I chanced to glance through.
I provide solution calculations (I have proof checked - however, accommodate any typos) - some basic calculus.
Solutions verified against book answers - the book does not give calculations and the answeres can be derived by alternative methods.
The Question - 5.2 Contextual Q:3:
On a particular day, the Financial Times 100 Share Index (FTSE 100) opens in London at 4,000.
During the rest of the day, its value t hours at 9 a.m. is given as F = 4,000 - 16t^2 + 8t^3 - 3/4t^4.
A broker is instructed to sell the client’s position (in pre-agreed increments and criteria – not relevant to Question and Solutions) only if the value of the FTSE is falling.
Comment: 16t^2 = 16 x t to the exponent 2; 8t^3 = 8 x t to the exponent 3; etc. (in first example - t is the base and 2 is the exponent and t^2 is the power)
Questions:
A: What is the value of the FTSE at noon.
B: Calculate the highest value of the index during the day and at what time to the nearest minute did this occur.
C: If trading finishes at 4.30 p.m., by how much has the index risen or fallen during the day.
D: During which times of the day could the broker have sold off the clients position ?
Answers:
A: 4011.3 points (note: indexes are designated in points (professionally) not pips)
B: 4183.9 points at 3.19 p.m.
C: Rise of 102.0 points at 4.30 p.m.
D: Between 9 a.m. and 10.41 a.m. and between 3.19 p.m. and 4.30 p.m.
SOLUTIONS:
Solution A:
12.00 (noon) - 9.00 = 3. Therefore: t = 3. Note: 9.00 a.m. ~ t = 0; 10 a.m ~ t = 1; 11.00 a.m ~ t =2; 12.00 p.m. ~ t = 3.
At 9.00 a.m ~ t = 0
F = 4,000 - 16t^2 + 8t^3 - 3/4t^4
F= 4,000 - 16(0^2) + 8(0^3) - 3/4(0^4)
F= 4,000 - 16(0) + 8(0) - 3/4(0)
F= 4,000 - 0 + 0 - 0
The index = 4,000 points at 9.00 a.m. or when time t = 0.
At 12.00 ~ t = 3
F = 4,000 - 16t^2 + 8t^3 - 3/4t^4
F= 4,000 - 16(3^2) + 8(3^3) - 0.75(3^4)
F = 4,000 - 16(9) + 8(27) - 0.75(81)
F = 4,000 - 144 + 216 - 60.75
F= 4,011.25
F= 4,011.3
Answer A: The index vaue is 4,011.3 points at noon (12.00 p.m.)
Solution B:
See YouTube for basic differentiation
Differentiate F(t) to determine local and absolute maximum/minimum values; on a graph t(time) would be the horizontal axis and F (points) would be the vertical axis: normally the horizontal axis is designated the x-axis with x-values and the vertical axis is designated y-axis with y-values. In this instance t(time) is equivalent to the x-axis and index point values equivalent to the y-axis. ---> this may help:
Inserted Video
F = 4,000 - 16t^2 + 8t^3 - 3/4t^4
F'(t) - differentiated = -32t + 24t^2 -3t^3 (note: F'(t) is differentiated to the first order; F"(t) is defferentiated to the second order, F"(t) = -32 + 48t - 9t^2; third order, F'"(t) = 48 -18t)
Note: F = 4,000 - 16t^2 + 8t^3 - 3/4t^4 can also be arranged as (pay attention to the - and + signs): F = -3/4t^4 + 8t^3 - 16t^2 + 4,000
and F'(t) = -32t + 24t^2 -3t^3 can also be arrange: F'(t) = -3t^3 + 24t^2 - 32t
The latter is the normal way - presenting expressions/equations in decending order of powers - but it is not necessary.
It also become tedious to be pedantic about exact/strict/rigorous mathematical notation when knocking off a few quick solutions.
However, whether decending or assending - incremental order of the powers should be maintained.
Apply quadratic formula to -32 + 24t - 3t^2 or cubic formula to -32t + 24t^2 -3t^3 (Sharp EL-W505T, may be suitable, cheap and considered superior features to comparable Casio – and bonus one key press ease of access re Pi - (not referal links) EL-W506T - Sharp calculators
Applying quadratic formular to -32 + 24t - 3t^2 (note: calculator inputs will be in order, -3,+24,-32)
Then:
t2 = 1.690 (derived from factorisation -> quadratic formular calculation) [calculator or pen/paper if hardcore]
t3 = 6.309 (derived from factorisation -> quadratic formular calculation)
and t1 = 0 (derived from factorisation - refer a: )
Applying qubic formular to -32t + 24t^2 - 3t^3 (note: calculator inputs will be in order, -3,+24,-32,0)
Then:
t1 = 0 (derived from cubic formula calculation) [calculator or pen/paper if hardcore]. Note: The 'solve any cubic-synthetic division method' as touted by clickbate YouTuber's is not univeral and only solve for the YouTuber's cherry-picked equations. For a fuller understanding of cubic formular* see - Further Information - link at end of post
t2 = 1.690 (derived from cubic formula calculation)
t3 = 6.309 (derived from cubic formula calculation)
Process t1
F1 = 4,000 - 16t1^2 + 8t1^3 - 3/4t1^4
F1 = 4,000 - 16(0^2) + 8(0^3) - 4/4(0^4)
F1 = 4,000 - 0 + 0 - 0
F1 = 4,000 (point value at t = 0 or 9.00 a.m.)
Answer C: The index has risen by 102 points at 4.30 p.m.
Solution D:
At 9.00 a.m. point value = 4,000.0
At 10:41 a.m. point value = 3,986.8 (this was the min value- determined by 1st order calculus differentiation)
At 3.19 p.m. point value = 4,183.9 (this was the max value-determined by 1st order calculus defferentiation)
At 4.30 p.m point value = 4,102.0
The index was falling between 9.00 a.m and 10.41 a.m. and 3.19 p.m. and 4.30 p.m.
Answer D: The broker could have incrementally sold off the client's position between 9.00 a.m - 10.41 a.m and 3.19 p.m. and 4.30 p.m.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Further Information - FREE Mathematical Refresher Material:
-------------------------------------------------------------------------------------------------------------------------
I included this in the original post
Good joke – the nonsense should be fairly obvious. However, I wonder how obvious....
There is nothing special about today, this year, any other year or every 1,000 years.
If the age of anyone on the 31 December (note: the last day of the year) is added to the year of their birth the sum will >> always = the current year, exact.
If the age of anyone is calculated before their birthday of the current year then the sum of their age + year of birth will >> always = the current year - 1 year (the current year less 1 year).
Example 1:
Anthony Hopkins (an actor, by way of example - see wikipedea, if interested) was born 31 Dec 1937.
This year, at current date - 6 Dec 2022 - Anthony is 84 years old.
84 + 1937 = 2021.
This year, on 31 Dec 2022 - Anthony will be 85.
85 + 1937 = 2022. (the current year, exact)
Example 2:
Next year, on 1 Jan 2023 Anthony will - still - be 85.
85 + 1937 = 2022.
Next year, on the 31 Dec 2023 Anthony will be 86.
86 + 1937 = 2023. (the current year, exact)
Scientists at the university of Innsbruck have discovered over 30,000 viruses by using the high-performance computer cluster 'Leo' and sophisticated detective work. The viruses hide in the DNA of unicellular organisms. In some cases, up to 10% of microbial DNA consists of built-in viruses.
The Mercator projection (/mərˈkeɪtər/) is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and south as down everywhere while preserving local directions and shapes. The map is thereby conformal. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator but accelerates with increasing latitude to become infinite at the poles. As a result, landmasses such as Greenland, Antarctica, Canada and Russia appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.
Example images
There are many ways to get across the point of distortion in the Mercator map and other equirectangular variants. Here are some you may enjoy. I have seen most of these on Reddit but original sources are unknown. If missing, I am happy to add creator credit if the origin is identified.
source: https://axbom.com/world-map/
