MCV4U: Calculus: Culminating Assignment Draft Name: ____________________________ In the 1920’s, a scientist (J.P. Maxfield) at Bell Laboratory did research to determine the most optimal design for...

2 answer below »
I have a calculus assignment which I need done. All the instructions are attached in the files selected. This is a part research, part mathematics problem. Thanks!


MCV4U: Calculus: Culminating Assignment Draft Name: ____________________________ In the 1920’s, a scientist (J.P. Maxfield) at Bell Laboratory did research to determine the most optimal design for recording music on a vinyl disk. In 1948, RCA Victor released a new style of record that competed well against those developed by CBS and Columbia. RCA’s new record eventually became the dominate medium for releasing new singles and may still be found today in jukeboxes all across the world. For your final assignment, you will incorporate what you have learned in the Calculus component of the course into a creative presentation. Objective: Create a project that demonstrates a thorough understanding of one particular aspect of Calculus. Ideally, this project will EXTEND the knowledge you have acquired during the course. Format: The format of this project is flexible; choose a format that best allows you to demonstrate your understanding of your chosen topic. Options include (but are not limited to): • Written report with graphs and diagrams • Visual and/or audio presentation (Powerpoint slides, video, etc.) • Creation of physical models with supporting written work Topics covered in class: • Rates of Change • Limits • Continuity • Derivatives • Higher Order Derivatives • Optimization MCV4U Assignment 2021 https://www.google.ca/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=2ahUKEwj5zKbnoefaAhWHz4MKHXAAAjwQjRx6BAgBEAU&url=https://www.pinterest.com/pin/39336196717770581/&psig=AOvVaw2QEirUTms1VlMWfm9MxIuA&ust=1525358574886688 https://www.google.ca/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=2ahUKEwix-66poufaAhXo34MKHfElA-EQjRx6BAgBEAU&url=https://londonjazzcollector.wordpress.com/record-labels-guide/rca/the-rca-labels/&psig=AOvVaw2QEirUTms1VlMWfm9MxIuA&ust=1525358574886688 Project Ideas: Where is the Math? Research a topic that connects Calculus to one or more of the following: technology, art, music, nature, economics or sports. Clearly identify the connection to what you have learned in class, providing ACTUAL examples. Letter to the Editor Write and answer a fake letter to the Editor of Calculus Times from a businessman, a politician or a scientist that poses a simple problem for which you need the use of Calculus to solve. The problem should be detailed and involved. The answer MUST utilize the skills you have learned in this course. Optimize Me! Optimize a 3-dimensional container. Pick a container in the shape of something NOT covered in class. Your container should only have two measurements since you want to have two variables. You can choose to maximize the container’s volume or minimize its surface area. If possible, use a REAL container and include a picture of it with your project. Real Data Function Modelling Using REAL DATA, analyze some sort of a phenomenon. Tell me something interesting about this phenomenon in terms of functions, their derivatives, etc. • Spread of a real disease such as Covid-19, Malaria, AIDS etc. • Record for a specific event in the sporting world • Rate of oil consumption or production • Life expectancies in different countries My Look-Fors: ✓ Topic: the topic chosen is about the application of Calculus ✓ Background: background on the topic is included ✓ Visual Aid(s): shows time, effort, and creativity ✓ Mathematics: the mathematics in the topic is connected to course work; clearly explained, including real examples of the mathematics studied in class ✓ Complexity & expertise: demonstration that something old was consolidated or something new was learned ✓ Organization: organized, clear, and neat ✓ References: Any references that are used in the completion of your project must be cited using APA format. Include references somewhere near the end of your project/report. For help with APA format, consult the website http://www.citationmachine.net/ and click on the “APA” button at the top of the page. This website works for any type of source material: book, magazine, newspaper, journal, web site, etc. http://www.citationmachine.net/ MCV4U – Culminating Assignment Rubric - Draft Criteria Level 4 (Excellent Work) Level 3 (Good work) Level 2 (Satisfactory work) Level 1 (Could be better) Overall Quality Report is of excellent quality, highly creative, interesting, and visibly appealing. Report is of good quality, somewhat creative, interesting and visibly appealing. Report is of satisfactory quality, not very creative, interesting or visibly appealing. Report is of poor quality and lacking in creativity, interest and visible appeal. Application of Mathematics The mathematics in the report is clearly evident and reference to course work is demonstrated and explained. The report is correct and shows significant insight. The mathematics in the report is somewhat evident and reference to course work is mostly demonstrated and explained. The majority of the report is correct. The mathematics in the report is evident but very little reference to course work is demonstrated or explained. The mathematics in the report is trivial and not explained. No reference or connection to course work is evident. Readability Report has few or no errors in spelling, punctuation, and/or grammar. Report is easy to read. All sources are cited. Report has some errors in spelling, punctuation, and/or grammar. Report is readable. Most sources are cited. Report has many errors in spelling, punctuation, and/or grammar. Report is challenging to read. Few sources are cited. Report is unreadable. Sources are not cited. Complexity & Depth of Understanding Report shows excellent depth of understanding of relevant concepts and principles. Report shows proficient depth of understanding of relevant mathematical concepts and principles. Report shows satisfactory depth of understanding of relevant mathematical concepts and principles. Report shows limited depth of understanding of relevant mathematical concepts and principles. MCV4U Assignment Rubric 2021 MCV4U-01 Culminating Task Biological Applications of Calculus in Real Life Calculating Drug Sensitivity: • One of the practices in medical field include measuring drug sensitivity • It is important for doctors and nurses to understand the characteristics of certain drugs that are being prescribed to patients • It is also crucial to know how well the drug will be absorbed into the bloodstream and how long it will remain active in the body • All of these can be measured by comparing the strength of a drug to a patient’s sensitivity to the drug • *The drug strength for a certain drug is not applied for all drugs • The strength of the drug is usually given by R(M) • M measures the dosage in milligram (mg) For example: ?(?) = ??√?? + ?. ?? We can use the product rule and the chain rule to find the derivative. • The derivative of the function represents the sensitivity of the patient’s body to the drug Assume the drug strength, with a dosage of 325mg, is being calculated  The derivative calculated to be ?′(?) = 3?+40 √2?+40  Substitute 325 mg into M  ?′(325) = 3(325)+40 √2(325)+40 • The sensitivity of a patient's body to the drug with a dosage of 325mg is about 38.64. The graph ?(?) = 2?√10 + 0.5? represents the strength of a certain drug. The derivative of the drug equation, ?′(?) = 3?+40 √2?+40 , represents the sensitivity of patient’s body to the drug.  By comparing the two graphs, it can be seen that the drug strength has changed  By applying the derivative, drug sensitivity for a patient can be accounted for  By knowing the drug sensitivity of a patient to certain drugs with respect to the dosage, doctors and pharmacists are able to test the effects that it will have on the body  Therefore, the drug will be able to work more effectively. Example: Calculating Drug Sensitivity in Real Life using Optimization • The sensitivity corresponds to approximately how much change to expect in R as the result of a unit change in the dosage. • Generally, doctors prefer to prescribe dosages of maximum sensitivity. • The strength of a patient’s reaction to a dose of M milligrams of a certain drug is R(x) = ?1? 2(?2 − ?) where c1 and c2 are positive constants. For what value of M is the strength a maximum, and what is that maximum reaction value? For what value of M is the sensitivity a maximum? What is the maximum sensitivity? Environmental Application of Calculus in Real Life In the field of chemistry, calculus can be used to predict functions such as reaction rates and radioactive decay. We can use the following equation to calculate the average rate of reaction of a reactant product, A: ????? = ?????????????? ?? ? ?? ???? ?2 − ????????????? ?? ? ?? ???? ?1 ?2 − ?1 Or ????? = ∆[?] ∆? • The chemical symbol (A) placed inside the square brackets indicate the entity’s concentration in mol/L. For the decomposition reaction of nitrogen dioxide, NO2(g), a brown gas that can be formed from gasses in tailpipe emissions. It is a major component of smog. The ground-level NO2(g) and one of the products of its decomposition, NO(g), undergo further reactions that increases ground-level ozone, which harms humans and other living things. We can calculate the average rate at which the NO2(g) is consumed over the first 50 s of the decomposition reaction. ????NO2(g) = ∆[NO2(g)] ∆? = [NO2(g)]?=50? − [NO2(g)]?=0? 50 ? − 0 ? = 0.0079 ??? ? − 0.0100 ??? ? 50 ? ????NO2(g) = −4.2 × 10 −5 ???/? ∙ ? We can also calculate the average rate for the disappearance of NO2(g) for each 50 s interval. Over the same period the rate of oxygen production is given by ????O2(g) = + ∆[O2(g)] ∆? = 2.2 × 10−5 ???/? ∙ ? Environmental engineers can use the results from the average rate of reaction and engineering principles to solve environmental problems, such as the decomposition of NO2(g). Example: Calculating Carbon Monoxide Level using the Chain Rule An environmental study of a certain community suggests that the average daily level
Answered 5 days AfterJun 22, 2021

Answer To: MCV4U: Calculus: Culminating Assignment Draft Name: ____________________________ In the 1920’s, a...

Asif answered on Jun 28 2021
154 Votes
APPLICATION OF CALCULUS
Mathematics deals with several operations one of which is optimizing ‘continuous change’. Geometry deals with shapes, algebra deals with generalization of arithmetic operation, whereas calculus deals with ‘continuous change’. Calculus can be segregated into several categories :
Namely:
· RATE OF CHANGE:
Differentiation and Integration can be used to estimate the rate of change in various aspects namely rate
of change of population which might help us t determine the population of an entire city, rate of change of loss or profit in a certain company ,etc.
For example - The population of town is modelled by P(t) = 6t2+110t+3000P(t) = 6t2 + 110t + 3000 where ‘P’ is population and ‘t’ is number of years since 1990.
So if we have to calculate the population of the city in 2021 we have to take t=31 (2021-1990)
So by replacing t with 31 we get the answer as 12,176, which is the estimated population for the town in 2021.
Now we can also calculate the rate of population change in 2005,
For that we need to take the derivative of the equation:
dp/dt = 12t + 110
So in 2005 , t = 15
So, dp/dt = 12*15 + 110 = 290
To compute the average change of population at  t = 15 , we simply choose another point t and so the Average Rate of Change will be :
change of population/time passed = P(t) − P(15)/(t−15) = 6t^2 + 110t + 3000 − 6000/(t − 15) = 6t^2 + 110t−3000/(t−15)
Now we have the average rate of change at t = 15.
To have the instantaneous rate of change at t = 15, we will choose some values of ‘t’ that will get closer and closer to 15 on both sides → t = 14.99, t = 15.01 etc. [1]
· LIMITS:
By using limits we can estimate the instantaneous velocity ie, the velocity of an object at a given moment at a given time.
For example –
If we throw a ball from the roof , we can calculate the velocity of the ball exactly when it reaches the ground which cannot be calculated by using the simple formula
Velocity = Distance / time
So here we can apply calculus as ;
V final = lim time→0 distance/time
In other words, as we measure the time it takes as we get closer and closer to the ground, the time approaches 0 (and so does the distance).
In calculus, this would be asking for the derivative of the distance, s, travelled with respect to the time, t, it takes. Using derivative notation, the final velocity can be expressed as:
V final = ds/dt
The right hand side represents the instantaneous velocity at a given moment in time. And the derivative is defined using limits. [2]
· CONTINUITY:

Precise definition of can be provided by calculus using limits.
A function is only said to be continuous in the interval [a, b] if it is continuous at every point inside the given interval. This definition is also made by assuming that both the functions, f(a) and lim x → af(x) , lim x → a , exist. If the above conditions are not met, then the given function will not be continuous at x = a .

In the above picture it is evident that the function is continuous for the given interval.
· DERIVATIVES:
If there exists a variable ‘y’ which varies with another quantity ‘x’ which can be given as y = f(x) , ie dy/dx = f ‘(x). This represents the rate of change of ‘y’ with respect to ‘x’.
Again, dy/dx x=x0 = f ‘(x0) which represents the rate of change of y with respect to x when x = x 0.
To apply it in mathematics we can look at a simple example which demonstrates the rate of change of circumference of the circle when the radius is increasing at the rate of 0.7cm/s.
Here the rate of change in radius is given as 0.7cm/s
At time ‘t’ , the radius is given as ‘r’
Circumference of the circle = 2пr
Differentiating both sides with respect to ‘t’ , we get ,
dC/dt = d/dt (2пr) = 2п*dr/dt .
now, dr/dt = 0.7
So by substituting the value in above equation we get the rate of change of circumference with respect to time as 2*п*0.7 = 4.396cm/s.
· HIGHER ORDER DERIVATIVES:
Higher order derivatives can also be used in several aspects for example for calculating the instantaneous acceleration or instantaneous force. If velocity = ds/dt then acceleration = d2s/dt2.
For example if s(t) = 10t2 , then v(t) = s ‘(t) =20t and a(t) = s’’(t) or v’(t) = 20.
[3]
· OPTIMIZATION:
Optimization usually estimated the largest value or the smallest value...
SOLUTION.PDF

Answer To This Question Is Available To Download

Related Questions & Answers

More Questions »

Submit New Assignment

Copy and Paste Your Assignment Here