Reference no: EM133753897
Case: In order to model real-world scenarios and transformations with limits, it's important for us to understand functions, domains, and ranges. Additionally, proficiency in rate of change and slopes helps with analyzing dynamic processes and data trends. Knowledge of extrema enables us to optimize problem-solving and time management. With this understanding, please answer the following questions and show stepwise calculations. Please explain your reasoning wherever necessary.
Task 1. Interpret the following graph in detail:
Domain and Range graph
(i) Identify the domain and range.
(ii) Does this graph represent a function and a one-one function. Why/Why not? Provide a detailed explanation/justification.
Task 2. Before working on this task 2, please read the following readings:
Reading section 1.2 Functions of the following textbook will help you in understanding the concepts better.
Imagine that the export of Avocados from Indonesia is described by the relation:
E(P) = P - 10000, P ?10000 where P represents the production (in thousand) of Avocados.
On the basis of the above scenario, answer the following questions:
(i) Draw the graph of E(P).
(Use graphing tool for drawing the graphs, use a scale where each unit represents one thousand on both the X and Y axes (Ex: consider 1= 1000)).
Using the graph:
(ii) Determine if E(P) is a function of P.
(iii) Find the domain and range of E(P).
(iv) Find how much export is done for 70 and 20 thousand of production.
(v) What are dependent and independent variables in this problem?
Task 3. The following graph (representing f and g) illustrates the relationship between the weights (y in tons) of two animals and their respective lengths (x in feet).
(i) In the event of intersection on the graph, determine the rates of change in length concerning weight for both categories. What conclusions can be drawn from this?
(ii) Select any two points on each of the graphs f and g (designated as C and D on f, and E and F on g, excluding O and A), and calculate the slopes of the lines CD and EF connecting them. What insights can be inferred about their slopes within the context of the problem? Please discuss your findings.
Task 4. Use the following graph to explain the local extrema of the function at the given points. Explain clearly how they differ from maximum and minimum values of function. Determine the intervals of all extrema shown in the graph (mention the intervals with the names Ex: (A, B) and specify whether they are increasing or decreasing).
Task 5. Before working on this task 5, please read the following readings:
Reading section 1.4 Function Notation (pages 61-62) of the following textbook will help you in understanding the concepts better.
In the highly prosperous nation of 'W', the income tax system is structured as follows:
a. Individuals earning up to $2200 are taxed at a flat rate of 10% of their income.
b. For those with incomes exceeding $2200 and up to $8945, the taxation scheme is bifurcated:
The first $2200 of income is taxed at 10%.
The remainder, above $2200 and up to $8945, is taxed at a rate of 18.5%.
c. If an individual's income surpasses $8945, the taxation policy is delineated as follows:
The initial $2200 of income is taxed at 10%.
The subsequent income, above $2200 and up to $8945, is taxed at 18.5%.
Any income exceeding $8945 is taxed at a rate of 30%.
Based on the above scenario, answer the following questions:
(i) Represent the above rule that country W has made as a piecewise function mathematically using the symbol ? or any other relevant symbol.
(ii) Take any income that comes in each slab from the country W and calculate the tax for each segment.