Calculus Unit 6 Guide: Derivatives & Their Applications

Studocu is not sponsored or endorsed by any college or university Learning Guide Unit 6 -MATH 1211 Calculus (University of the People) Studocu is not sponsored or endorsed by any college or university Learning Guide Unit 6 -MATH 1211 Calculus (University of the People) Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

MATH 1211-01 Calculus - AY2023-T3  Dashboard  My courses  MATH 1211-01 - AY2023-T3  2 March - 8 March  Learning Guide Unit 6 Learning Guide Unit 6 Learning Guide Unit 6 Overview Unit 6: Applications of Derivatives, Extreme Values of Functions, and the Mean Value Theorem Topics:  Related Rates  Linear Approximations Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

 Maxima and Minima  Mean Value Theorem  Shape of a Graph Learning Objectives: By the end of this Unit you will be able to: 1. Calculate the rate of change of another variable using the Chain Rule 2. Write the linearization of a given function. 3. Identify critical points of a function and local and absolute extrema. 4. Describe the significance of Rolle’s Theorem and Mean Value Theorem 5. Determine the shape of the graph Tasks:  Peer assess Unit 5 Written Assignment  Read the Learning Guide and Reading Assignments  Participate in the Discussion Assignment (post, comment, and rate in the Discussion Forum)  Complete and submit the Written Assignment  Make entries to the Learning Journal  Take the Self-Quiz  Take the Graded Quiz Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

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Introduction So far, you have learned what a derivative is and how to find the derivative of a function. In this Unit, you will see at least five different practical applications of derivatives, one being related rates. For example, consider two busses 5 miles apart. They are moving on parallel highways at a constant speed of 55 mph, one heading east and the other heading west. One of the questions we are faced with is how fast the distance between two buses is changing when they are 13 miles apart. This may be important if any safety measure is required to be taken. These problems are resolved by using methods of related rates. Derivatives are applied when we needed to approximate a function, locally, by a linear function, say at a point x=a, In other words, by tangent lines. If is the function then we approximate this function by its tangent line near . Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

Derivatives also help us understand the shape of a graph: where the graph has a maximum value (hilltop) and a minimum value (valley); intervals where the graph is increasing or decreasing; in which interval the graph is concave up and concave down; and where the critical points inflection points are located. You will use Rolle’s Theorem and Mean Value Theorem and apply derivatives to solve practical applications such as these. See figure 4.31, 4.35, and figure 4.37 for a clear picture. Reading Assignment 1. Herman, E. & Strang, G. (2020). Calculus volume 1 . OpenStacks. Rice University.  Read Chapter 4, sections 4.1 through 4.5, pages 346-354, 359-368, 371-380, 384-392, and pages 395-407 o Review the completed examples to understand the concepts and then complete some problems on your own. Review and practice will contribute to your success in this course.  You should attempt the problems in the book as indicated, and you are allowed to use the book and other resources to understand how to answer the problems. You are welcome to ask about these in the discussion forum. o 4.1 #’s 1, 5, 7, 17, 29 o 4.2 #’s 50, 54, 57, 77, 87 o 4.3 #’s 101, 106, 109, 111, 115, 131 o 4.4 #’s 157, 161, 163, 172, 176, 190 Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

o 4.5 #’s 205, 207, 210, 217, 225, 233, 241 2. IMA Videos. (2012, August 1). Applications of Derivatives [Video]. YouTube.  This Derivative Application video speaks where can we use the concept of derivative calculus we learned in previous videos https://youtu.be/CGQ6saBqfUg 3. The Organic Chemistry Tutor. (2018, March 1). Related Rates [Video]. YouTube.  This video tutorial explains how to solve the distance problem within the related rates https://youtu.be/_sbq6yZ9Kt8 4. The Organic Chemistry Tutor. (2016, February 20). Relative Extrema, Local Maximum and Minimum, First Derivative Test, Critical Points- Calculus [Video]. YouTube. https://youtu.be/0jiVRRKmGoE  This video provides examples and problems about how to relative extrema of a function using the first derivative test. 5. The Organic Chemistry Tutor. (2016, September 28). Rolle's Theorem Explained and Mean Value Theorem For Derivatives - Examples – Calculus [Video]. YouTube. https://youtu.be/0jiVRRKmGoE Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

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 This calculus video tutorial explains the concept behind Rolle's Theorem and the Mean Value Theorem For Derivatives. Discussion Assignment In the discussion forum, you are expected to participate often and engage in deep levels of discourse. Please post your initial response by Sunday evening and continue to participate throughout the unit. You are required to post an initial response to the question/issue presented in the Forum and then respond to at least 3 of your classmates’ initial posts. You should also respond to anyone who has responded to you. Discussion prompt: In our practical life related rates are used in many areas including the medical field to measure tumor growth, rec Create/write an original related rate problem with a theme in mind. Your problem should utilize a mathematical theorem, cone, trig ratios, etc.) like the related rates problems that we solved in our Reading and Written Assignm Your Discussion should be a minimum of 200 words in length and not more than 300 words. Please include a word count. Following the APA standard, use references and in-text citations for the textbook and any other sources. Written Assignment Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

1. An airplane is flying towards a radar station at a constant height of 6 km above the ground. If the distance s between the airplane and the radar station is decreasing at a rate of 400 km per hour when s=10 km., what is the horizontal speed of the plane? Make sure your answer includes units. 2. A boat is being pulled into a dock by a rope attached to it and passing through a pulley on the deck, positioned 6 meters higher than the boat. If the rope is being pulled in at a rate of 3 meters/sec, how fast is the boat approaching the dock when it is 8 meters from the dock? Make sure your answer includes units. 3. A girl is flying a kite on a string. The kite is 120 ft. above the ground and the wind is blowing the kite horizontally away from her at 6 ft/sec. At what rate must she let out the string when 130 ft of string has been released? Make sure your answer includes units. 4. Find the Linearization of at . Provide your answer as ? 5. Use Linear Approximation to estimate . Provide your answer in 2 decimal places. Do not use a calculator. Show work for credit. 6. Calculate the locations of maximums and minimums of the following functions: Show work in details. Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

  7. Find the exact x-value where the function ( ) attains a maximum value. An estimated answer or a calculator answer will not earn any credit. 8. Using the Mean Value Theorem and Rolle’s Theorem, show that has exactly one real root. 9. If and for , how small can possibly be? 10. The graph of a function is given below. Determine the intervals on which the function is concave up and concave down. Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

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Learning Journal The Learning Journal is a tool for self-reflection on the learning process. In addition to completing directed tasks, you should use the Learning Journal to document your activities, record problems you may have encountered, and to draft answers for Discussion Forums and Assignments. The Learning Journal should be updated regularly (weekly), as your instructor will assess the learning journals as part of your Final Grade. Learning Journal: For any problem that you couldn’t solve at first, suggest a strategy you might try to tackle the problem and show  Describe any strategic gaps you were unable to bridge  list three helpful insights that may help another person trying to tackle the problem The Learning Journal entry should be a minimum of 500 words and not more than 750 words. Use APA citations and references if you use ideas from the readings or other sources. Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

Self-Quiz The Self-Quiz gives you an opportunity to self-assess your knowledge of what you have learned so far. The results of the Self-Quiz do not count towards your final grade, but the quiz is an important part of the University’s learning process and it is expected that you will take it to ensure understanding of the materials presented. Reviewing and analyzing your results will help you perform better on future Graded Quizzes and the Final Exam. Please access the Self-Quiz on the main course homepage; it will be listed inside the Unit . Graded Quiz T he Graded Quiz will test your knowledge of all the materials learned thus far. The results of the quiz will count towards your final grade. Please access the Graded Quiz on the main course homepage; it will be listed inside the Unit. After you click on it, the quiz’s introduction will inform you of any time or attempt limits in place. Good luck! Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

Checklist  Peer assess Unit 5 Written Assignment  Read the Learning Guide and Reading Assignments  Participate in the Discussion Assignment (post, comment, and rate in the Discussion Forum)  Complete and submit the Written Assignment  Make entries to the Learning Journal  Take the Self-Quiz  Take the Graded Quiz Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

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