Calculus Archives - Skilled Papers

Hello, please no plagiarism no ai, you don’t have to show your face, just record your voice and the paper that you are going to be solving the problems on, im gonna you use your video as an example and i am gonna recreate it showing my face and stuff, thank you!
Project 2 Due Sunday, May 12 by 11:59pm
Project Objective:
• Show the graphical representation of the derivative of a function given the graph of the function.
• Identify facts about the shape of the graph of a function, given the graphs of the first and second derivatives of that function.
• Use the graphical implication of derivative as an instantaneous rate of change to solve an applied problem.
Assessment Criteria:
Projects assess two major areas: knowledge of business calculus concepts as they apply to real life problems AND soft skills such as effective communication and time management skills. Your knowledge and understanding are assessed based on your explanation and presentation provided in the recorded video clip.
Let’s get started!
For this project you will create a video presentation. In the video presentation you will explain the concepts and mathematical processes involved in solving a given problem. First read and complete Steps #1 and #2 for each task (in these steps you will prepare and gather all the information and calculations that are needed for your video presentation), then complete Step #3 of these tasks which is to create the video presentation that covers Task #1 and Task #2.
Task #1
Purpose: Utilize chain rule for marginal analysis and to use the graphical implication of derivative as an instantaneous rate of change to solve an applied problem.
Step #1
Among other factors, CPI and PPI are two commonly used measures of inflation. As your first step read pages 6-8 of the following news release (https://www.bls.gov/news.release/pdf/ppi.pdf) from the Bureau of Labor Statistics to understand what CPI and PPI represent and how they compare. You will be asked to explain using your own words (without reading from script) what these two measures represent.
Step #2
Review the questions below. If you are uncertain about the answers, return to your textbook and review differentiation rules such as chain rule (Sec 3.4), the first and the second derivatives and their graphical representation (Sec 4.1 and 4.2). You should have a solid understanding on how the derivative of a function looks like graphically. You should have a solid understanding on what the first and second derivatives say about the shape of the graph of the underlying function. You should have a solid understanding on how to find the derivative of a function combination and a function composition. Let’s look at the questions.
Question #1: A cost function is defined by
C(x)=5+
f(x)
−
−
−
−
√
where
f(x)
is a linear function. Find the marginal cost function.
Hints for Question #1:
◦ Do not replace f(x)
, knowing that the radicand is a function of x
is sufficient to find the marginal cost.
◦ To find the marginal cost, you need to know what marginal cost represents. Refer to the textbook, section 2.7 page 162 to review marginal analysis.
◦ Once you have a clear understanding about marginal cost, review the techniques of differentiation as they relate to function combinations (notice that C(x)
is a function addition of two other functions) and function compositions (notice that f(x)
−
−
−
−
√
is a function of a function, which is known as a function composition).
◦ In answering this question, first explain, using your own words, what marginal cost implies. Identify the functions involved in the combination that forms C(x)
. Explain the differentiation method to differentiate sum of two functions. Talk about a method of differentiation (hint: Chain Rule) that is used to differentiate function compositions of form F(x)=f(g(x))
, use this method to differentiate f(x)
−
−
−
−
√
. Show detailed steps in finding the marginal cost function.
Question #2: Suppose that CPI is defined by a function
f(x)
. Explain what
f
′
(x)
represents? Below is an example CPI curve defined by a function
f(x)
. On the graph s
how the graphical representation of
f
′
(x)
.
I attached the graph below
Question #3: If the annual rate of change of the CPI is reported to be decreasing, what does this say about the shape of the graph of the CPI? Draw an example CPI curve such that the annual rate of change of the CPI is decreasing.
Hint: Read this question carefully. What does the rate of change (instantaneous rate of change) of a function represent? the first derivative of the function? the second derivative of the function? or the function itself? We are talking about an increase or decrease in the ‘instantaneous rate of change’ function.
Question #4: If the annual rate of change of the PPI is reported to be increasing, what does this say about the shape of the graph of the PPI? Draw an example PPI curve such that the annual rate of change of the PPI is increasing.
Hint: Read this question carefully. What does the rate of change (instantaneous rate of change) of a function represent? the first derivative of the function? the second derivative of the function? or the function itself? We are talking about an increase or decrease in the ‘instantaneous rate of change’ function.
Step #3
Now you are ready to start creating your video presentation.
◦ In your video presentation first introduce yourself, I need to see who is presenting.
◦ In your video presentation discuss very briefly what CPI and PPI represent as measures of inflation. Do not read from script!
◦ In your video presentation explain and answer all of the questions below. For Question 1 don’t forget to explain, using your own words, what marginal cost implies then show step-by-step how to find the marginal cost function. Questions 2, 3, and 4 require graphing and explaining your graph as you answer each question. Below are the questions for your reference:
Question #1: A cost function is defined by
C(x)=5+
f(x)
−
−
−
−
√
where
f(x)
is a linear function. Find the marginal cost function.
Question #2: Suppose that CPI is defined by a function
f(x)
. Explain what
f
′
(x)
represents? Draw an example CPI curve and assume the CPI curve is defined by a function
f(x)
. On your graph show the graphical representation of
f
′
(x)
.
Question #3: If the annual rate of change of the CPI is reported to be decreasing, what does this say about the shape of the graph of the CPI? Draw an example CPI curve such that the annual rate of change of the CPI is decreasing.
Question #4: If the annual rate of change of the PPI is reported to be increasing, what does this say about the shape of the graph of the PPI? Draw an example PPI curve such that the annual rate of change of the PPI is increasing.
Task #2
Purpose: Use the graphical implication of derivative as an instantaneous rate of change to solve an applied problem.
Step #1
First watch the following video. As you watch the video jot down any concepts that you have learned in this course that relate to microeconomics.

Step #2
Review the questions below. If you are uncertain about the answers, return to your textbook and review the graphical representation of first and second derivatives of a function (Sec 4.1 and 4.2). You should have a solid understanding on how the derivative of a function looks like graphically. You should have a solid understanding on what the first and second derivatives say about the shape of the graph of the underlying function.
Question #1 Look at the Cost function graph given below. What do the slopes of the given tangent lines represent?
The graph is attached below
Question #2 Refer to the graph in question 1, as production increases do the slopes of the tangent lines increase or decrease?
Question #3 Refer to the graph in question 1, as production increases, does production process become more efficient or less efficient for this company?
Step #3
◦ In your video presentation identify an example concept that you have learned in this course that may relate to microeconomics.
◦ In your video presentation address the questions given in step 2.
Video Presentation Requirements:
• Video presentation should not exceed 30 minutes.
• Introduce yourself in the beginning of the video presentation (I need to know who is presenting, I need to see your face (at least during the introduction))

Access the help on integrals (Desmos) Links to an external site. and review the first example in the section about indefinite integrals. This example demonstrates the method of graphing antiderivative of the sample function f(x) = x^2. The outcome is the graph of F(x) = x^3/3.
In Desmos, graph the first derivative function from your application problem in Discussion 6. Then using the integral notation as shown in the above example, graph its antiderivative so that the maximum value of the antiderivative function is twice as great as the maximum value of your function (c) from Discussion 1.
Using the graph of the antiderivative function, estimate the -intercept and the end value of the function (as x –> + infinity).
Interpret the y-intercept and the end behavior of the antiderivative function in the context of your application problem.

Category: Calculus

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