Mayfield City School District



Math 4 HonorsName ___________________________Lesson 6-5: Using Derivatives to Analyze GraphsDate __________________________Learning Goals:I can use derivatives and their graphs to identify properties of functions.I can determine properties of derivatives from the graph of a function.I.Throughout this course as well as your previous math courses, you have learned about several key characteristics used to describe the graph of a function. Brainstorm with your group members what these characteristics are & list them below.Describe how you would identify the characteristics by looking at the graph of the function.Describe how you would identify the characteristics from the function rule.Based on what you know about the derivative thus far, what characteristics can be identified using the derivative? How would you identify them?OVER Page 2II.The following passages come from a calculus textbook and address the characteristics you listed on the front of this page as well as some new ones. ON YOUR OWN, read the paragraphs carefully and make any notes you feel necessary. Answer the questions as you go along.-1905052705What is meant by “more negative”?What is meant by “less negative”?85725-9525Page 3What is the difference between a relative max or min and an absolute max or min?Can a cubic polynomial have an absolute maximum? Explain.OVER -2762260Page 433337574295 Write a “newspaper description” of graph II.426720077470 9334501905 4762562865-666754064031051502857528575038100If 2286005564505a graph is concave up, where are the tangent lines?If 2286005564505a graph is concave down, where are the tangent lines?What is significant about the point of inflection?3486150-19050Example: Use the terms defined in the previous readingPage 5to describe the function in Figure 12. III.The next section addresses what are called the First and Second Derivative Rules. Again, ON YOUR OWN, read the paragraphs carefully and make any notes you feel necessary. Answer the questions as you go along.1905060960161925156210Describe in your own words, what this rule lets us find out about the graph of a function.Does this rule apply when the derivative is equal to zero? Explain.OVER 190500Page 6Describe in your own words, what this rule lets us find out about the graph of a function.What is the significance when ?438150865505209550179705Complete the following table. Use #1 as your guide. 2228850118745 2.676275151130 3.3971925162560 4.19050-9525Page 7f(x)x-381000476251171575101600OVER HOMEWORK: 1400175148088Day 1 Assignment - Complete all problems on this page.Page 8142875565154200525113030952501333501905095255429257620 20. Using vocabulary from this lesson, describe the graph below with as much detail as possible.437197595250HOMEWORKPage 9Day 2 Assignment: Complete both sides of this page.1.Show all work for the following. NO CALCULATOR UNTIL PART e!a.Find . Determine the intervals on which f is increasing & decreasing. Hint: NLA!b.Use your NLA to determine the coordinates of the relative maximum and relative minimum.c.Find . Determine the intervals on which f is concave up & concave down. Hint: NLA!d.Use your NLA to determine the coordinates of the point(s) of inflection.e.Check your answers to parts a – d using your calculator. (You do not have to sketch the graph.) Sketch & label the graph of the function that has the properties described.354330067945-3873549530 3743325102869389572513716041910022860-38100635205740027305OVER Page 10142875831851943100264160Exercises 35 – 43 refer to Figure 15, which contains the graph of , the derivative of the function .142875156210200025127636 43. Suppose What is the equation of the tangent line to the graph of at the point (4, 1)? ................
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