Calculus 1 Project

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Question Description

Hi There Two attached's One is the project guidelines and the other is the functions please select Only One function and work on it

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Scanned with CamScanner Calculus I Project #1: What Do Derivatives Tell Us? 100 points To be successful on your project you must: ➢ Read and follow instructions carefully. ➢ Review the grading rubric carefully to ensure you are familiar with the expectations for each section. ➢ Write clearly, using appropriate terminology and accurate mathematical notation. College-level writing is expected, as is the use of correct grammar. ➢ Submit a neat, professional report typed (double-spaced) using your choice of word processing software (including a mathematical notation package) and including printouts and diagrams from your choice of graphing application, computer algebra software, or statistical software (as applicable). The use of appropriate technology is expected throughout the project. o In particular, embedded “complete” graphs or charts and/or computer printouts will be expected as part of the report. Hand-drawn graphs are not acceptable. o Hand calculations (if applicable to the project requirements) should be scanned and included as an appendix at the end of the report. . In this project you are going to focus on using derivatives and limits, with some other mathematical ideas you already knew from algebra and precalculus. Your professor is going to give you two functions to work with. You will choose one function and follow the set of instructions below. You should plan to document your work by using a capture program to record the results from the use of technology, and then include the captured images in your final submission. The first task will be to investigate properties of your chosen function. You will need to include the equation of your function (and other mathematical functions, equations, and notation) in your final report using an equation editor. [1] Identify and include in your report the domain of your function. Use technology to graph your chosen function.[2] Use technology to determine the x and y intercepts of your function (including captured images in your report). Any intercepts should be labeled on your graph and identified in your report. For each function identify the x-values where discontinuities occur. These should be identified as removable or non-removable. The intervals of continuity should be given in your report. Following that, identify any asymptotes that exist. Vertical asymptotes should be written as equations using the editor. Horizontal asymptotes should be written using limit notation and the editor. If asymptotes do not exist, that should be noted. After this, identify the end behavior of the function using the equation editor for formatting, if you have not done this with asymptotes already. Next you will need to compute the first and second derivatives of your chosen function using technology (again, document the results in your report with captured images). You should use an equation editor to write, with correct formatting, your first and second derivatives. Create a complete graph[2] of the first derivative. Use the technology to determine the x intercepts of the first derivative (where the first derivative is zero). They should be labeled on the graph and identified below the symbolic representation of the respective first derivative function along with the titles of the axes, units, etc. that are part of a complete graph. Then, create a complete graph[2] of the second derivative. Use the technology to determine the x intercepts of the second derivative. They should be labeled on the graph and identified below the symbolic representation of the respective second derivative function along with the titles of the axes, units, etc. that are part of a complete graph. Calculus I Project #1: What Do Derivatives Tell Us? 100 points Next you will create a sign diagram for the first and second derivatives for your function. Be sure to label the critical points on the first derivative sign diagram. Indicate where the function has a positive derivative and where it has a negative derivative. On the second derivative sign diagram, label the points of inflection for your function. Below that, show where the functions would be concave up or concave down. Now discuss the relationships between the graphs of your function and its two derivatives. Discuss how the intercepts on the graphs of derivatives are related to the graphs of their respective functions. Similarly, discuss how the positive/negative values of the derivatives are related to the behavior of the original function. [1] MathType, which interfaces with MicroSoft Word and other word processing packages is available on many of the computers on campus. Also, MicroSoft Word and Google Docs have a built in equation editor that you can use to format your equations / functions. In MicroSoft Word just choose Insert from the top menu bar and then Equation. [2] A complete graph means that the axes are labeled with their respective variables; the axes should also have titles with units; the X and Y intercepts should be labeled; the equation of the graph should be labeled; and graph should have a title. Remember a graph tells a story and these labels help to tell that story. TECHNOLOGY (minimum): • • • • Word Processor Equation Editor Tech: derivatives Copy images into final document (screen print; Snipping Tool; image capture software) SUMMARY (read directions carefully): • • • • • • • • • • Choose ONE function (5 points) Equation of function o Domain (10 points) Graph your function (5 points) Intercepts o Label on graph o Include in report (10 points) Discontinuities o Removable or non-removable o Intervals of continuity (5 points) Asymptotes: vertical, horizontal, oblique, other (5 points) End Behavior (if not an asymptote) o Equation editor (15 points) First and second derivatives o In report using equation editor o Use technology to compute o Screen capture to document use of technology (10 points) Graph of first derivative o Compute and label intercepts (10 points) Graph of second derivative o Compute and label intercepts Calculus I • • Project #1: What Do Derivatives Tell Us? 100 points (10 points) Sign Diagram o First Derivative: Critical Points, Positive/negative o Second derivative: Inflection points, Concave up/down (15 points) Discussion o How are the three graphs related? o How do intercepts of derivatives relate to the graph of your function? o How do signs of derivatives relate to the graph of your function? HINTS: • • • • Use an equation editor Use Snipping Tool or equivalent to capture required information from computer algebra system for inclusion in your report Identify and use a single computer algebra system, such as MATLAB, desmos.com, or wolfamalpha.com Use same horizontal scale for all graphs of your function. Vertical scales should be appropriate for the function, the first derivative, and the second derivative. ...
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Tutor Answer

dresva
School: UC Berkeley

Here is the solution https://files.fm/u/5wuuxf8mLet me know if anything else is needed.

Contents
1 Function

2

2 Graph of the Function

2

3 Intercepts of the Function

3

4 Graph of the Intercepts of the Function

3

5 Continuity

4

6 Asymptotes

4

7 End behavior of the function

4

8 First and Second Derivatives

5

9 Intercepts of the First Derivative

6

10 Graph of the First Derivative

7

11 Intercepts of the Second Derivative

7

12 Graph of the Second Derivative

8

13 Sign Diagram for First Derivative

9

14 Sign Diagram for Second Derivative

9

15 Discussion

10

A Appendix

11

1

1

Function

The chosen function is f (t)
Equation of f (t) is:
f (t) = 50

t2 + 2t + 4
t2 + 4t + 8

Let D(t) be the denominator of f (t)
D(t) = t2 + 4t + 8 = (t + 2)2 + 4
∴ D(t) 6= 0

∀t ∈ R

Domain = (−∞, ∞)

2

Graph of the Function

2

!

3

Intercepts of the Function

The following MATLAB code computes the x and y intercepts of f (t)

1

f = @(t) 50*(t.ˆ2+2*t+4)./(t.ˆ2+4*t+8);

2
3

a = solve(f);

4

a =a(imag(a)==0);

5

b = f(0);

6
7

v =['The x intercept is',num2str(a)];

8

v0 = ['The y intercept is ' ,num2str(b)];

9

disp(v);

10

disp(v0);

Command Window

∴ There is no x intercept, the y intercept is 25
Point of interception is (0, 25)

4

Graph of the Intercepts of the Function

3

5

Continuity

D(t) = t2 + 4t + 8 = (t + 2)2 + 4
∴ D(t) 6= 0

∀t ∈ R

∴ f (t) is continuous on the interval (−∞, ∞)

6

→ No discontinuities exist

Asymptotes

D(t) 6= 0

∀t ∈ R

→ No vertical asymptotes

To find the horizontal asymptote ha :


ha = limt→∞ f (t) = 50 × limt→∞

ha = 50 ×

t2 + 2t + 4
t2 + 4t + 8

!



2
4
 t 1 + t + t2 


= 50 × limt→∞  

8 
4
 2
t 1+ + 2
t t

1+0+0
= 50
1+0+0

A horizontal asymptote exists at y = 50

7

End behavior of the function

It was calculated during finding the horizontal asymptote of f (t)
limt→∞ f ...

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Anonymous
Goes above and beyond expectations !

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