Assignment 6: COMPLETED EXAMPLE HOW TO USE DERIVATIVES TO SIGNAL IMPORTANT FUNCTIONAL VALUES (Tutorial). Power Point Lesson 13 goes trough applications of all the steps you should need for this assignment, and gives examples and explanations. There is no Lesson 12, so you did not miss it. This exercise draws on your ability to compute 1st and 2nd derivatives using the rules from Lesson 11 (applied in Assignment 5). However, you will now apply derivatives in a very traditional way to help graph a nasty looking function. You can get the assignment's 50 points without the graph, thus missing out on the bonus points. You must correctly find derivatives and apply them to fill in the table, however. Being lazy, I decided to use Excel's power to help. My spreadsheet's "worksheet" computes (for every X value) the Y value according to the given function, the first derivative (dy/dx also called Y"), and the second derivative (dy/dx also called Y"). You are given the equation for Y. You can also calculate all very carefully by hand. 1. I worked out by hand the equation for the first derivative (dy/dx). I also worked out the equation for the second derivative (d2y/dx). I wrote on the worksheet sheet I turned in with the assignment the algebra used to find the two derivatives (the derivatives must be correct to receive credit for them). I also set each derivative equal to zero and solved each for the all X values that make the derivative zero (if your calculations are correct, the X values will be among those given in the top row of the worksheet table). Note: the first and last X values are not to be considered a maximum, minimum, or inflection value. They are just end points of the provided data. 2. In the worksheet table for the first derivative (dy/dx) row write a "0" only in the cell or cells corresponding to the X values making the first derivative zero. This is usually the signal that the X value marks a maximum or a minimum. At this point, you do not know which is which. You do not need to calculate other values of Y' or fill in other cells in the table. The zeros are the signal we want to locate. 3. In the worksheet table for the second derivative (dy/dx) row write a "0" only in the cell or cells corresponding to the X values making the second derivative zero. This usually is the signal that the X value marks an inflection. Only if the function is to the fourth power, will there be more than one inflection point. 4. Here is a summary to help you read derivative signals for the example given below: Step 1: given a third degree function like Y = (2/3)x3 -6.5X2 +15X-47, find its first (Y') and second (Y") derivatives (and write the algebra for them also on your worksheet): Y' = 2x-13X+15 which factors into (2X-3)(x-5)=0 Thus Y' will be zero when X is 1.5 or X is 5. These values are likely the maximum and minimum X values (you do not yet know which is which, however). Y" = 4X-13 set equal to zero (4X - 13) = 0 means X = 3.25. This signals the X value that is an inflection. Also plug X = 1.5 into this second derivative (which gives-7). If Y' = 0 and also Y" is negative at the same X value, then this X is a maximum. But if Y' = 0 and also Y" is positive at the same X value, then the X is a minimum. The numerical value is not the signal. The signal is whether the value is positive or negative. Step 2 and 3, make your table look like below: Calculated values are only needed in the cells indicated. Calculating values Y .... Y8 give you enough plot points to see what your function looks https://se.instructure.com/courses/2986/pages/assignment-6-completed-example?module_item_id=213450 1/2 7/12/23, 12:11 PM like. Assignment 6: COMPLETED EXAMPLE: ECON-5133-W2 - Managerial Economics (2023SU) Analyzing Derivative Signals (values calculated for blank cells do not add useful information) X 1 1.5 2 3 3.25 4 5 6 Y Y Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y' 0 0 -7 0 +7 Step 4: Correctly work out the Y values by plugging the X values into the function and entering them into the table. Step 5: Read the results: When X = 1.5, Y' = 0 and because Y" evaluated at X = 1.5 is negative, this signals that X=1.5 is a Maximum. When X = 5, Y' = 0 and because Y" evaluated at X = 5 is positive, this signals that X = 5 is a Minimum. When X = 3.25, Y" = 0, this signals that X = 3.25 is an Inflection. On the worksheet you downloaded for this assignment, you will need to specifically state which X value is a maximum, which a minimum and which inflection(s). Step 6: For the Bonus points attempt, graph X values in the table on the horizontal and Y on the vertical axis. Mark X = 1.5 correctly on the X axis (label this the maximum point on the plot it should be directly in line with the X = 1.5 value). Correctly mark X = 3.25 on the X axis (label this the inflection and this should be directly in line with the graph's inflection point. Correctly mark X = 5 on the X axis (label this the minimum and this should be directly in line with the minimum on the graph). Label the values on the graph (we will not guess for you). If your graph does not agree with the values you computed and tabled, you have a mistake somewhere in your computations which you must correct, or alternatively, your graph of the function may be wrong. Seek Tutor.com for help. If you used Excel, attach your spreadsheet as a file, and also fill in the table with the needed values. Also submit the other worksheets. As if by magic, you might get up to 20 bonus points. The neater looking and correct the graph is, the more likely you are to get the 20 bonus points. There are two more examples in Power Point Lesson 13, near the end. As usual, the submission link is the last item inside the folder. Multiple files can be attached, and submitted at the same time. Prepares for Course Objectives B and C: Name DERIVATIVE APPLICATION EXERCISE Due (9) (Worth 50 points + 20 point optional bonus) In the following the symbol for the first derivative is dy/dx or Y' and the symbol for the second derivative is dy/dx or Y", all of which are standard symbols for derivatives. Y=x-5X2-8X +20 X -1.333 -.667 0 1 1.667 3 4 5 Y dy/dx dy/dx A function is given above. The table gives several pre-selected values of the independent variable (X). Use all of the (X) values to have enough to answer questions for this assignment. 1. Calculate the Y value corresponding to each X value in the table. Write the Y value and its sign (+ or -) calculated in the cell below its X value. Pick any three of the X values and on a separate worksheet. show how you calculated its Y value. Use these X,Y pairs also if you chose to do the bonus option. 2. Find the critical values among the given X values by: a. Correctly finding the first derivative of the given function. Write dy/dx here. Show how you found the derivative on the worksheet (the derivative is half the credit for this question). b. Using the derivative from (2a) solve it for all the X values that make the derivative equal zero. Show your work on the worksheet and also write "0" in the cell of the dy/dx row for the X value only where dy/dx = 0. There may be none, one, or two such. (the dy/dx = 0 values are the other half credit for this question). 3. Find additional critical values among the X values by: a. Correctly finding the second derivative of the given function. Write dy/dx here Show how you found this second derivative on the worksheet (the second derivative is half the credit for this question). b. Using the second derivative from (3a) solve it for all the X values that make dy/dx equal zero. Show your work on the worksheet and also write "0" in the cell of the dy/dx row for the X value(s) only where dy/dx = 0. There may be one, or two such. (the dy/dx=0 values are the other half credit for this question). Read the signals given by the critical derivative values above. (NOTE: do not consider the first or the last X value as being among the critical values). 4. Are there MAXIMUMS among the X,Y pairs? If so, then the pair's dy/dx = 0 and also its dy/dx will be a negative (-) value for each such X,Y pair. What are the X values and also the dy/dx values? There may be none, one, or two maximums. Write your values below: Value of X Value of its dy/dx 5. Are there MINIMUMS among the X,Y pairs? If so, then the pair's dy/dx = 0 and also its dy/dx will be a positive (+) value for each such X,Y pair. What are the X values and also the dy/dx values? There may be none, one, or two minimums. Write your values below: Value of X Value of its dy/dx 6. Are there INFLECTION points among the X,Y pairs? If so, then the pair's dy/dx may or may not equal zero. However, dy/dx at each will be zero (0) for each such X,Y pair. What are the X values and also the dy/dx values? There may be one, or two inflections. Write your values below: Value of X Value of its dy/dx References: A tutorial at the top of the Blackboard page that contains the twelve versions and Power Point lesson 13 Twenty point optional bonus opportunity. If done, submit along with the above using the same Gold submission link. Using all X,Y pairs tabled above, plot them. Scale the axis's so that the maximum(s), minimum(s) and inflection(s) are clearly obvious. Indicate numerical values on each axis. Label each maximum, minimum, and inflection point as such on the plot along with its X and Y values. EXAMPLE: Suppose at X = 3.00 and Y = 11.00, you decide this point is an inflection. Then write (3,11) INF at this point on the graph. You may plot this graph with Excel (but you must still label the various critical points). Your plot must show each maximum, minimum, and inflection point found in the six previous steps (if labeling is omitted, we will not guess which is what - you must tell us). The scoring rubric and scoring for this bonus opportunity will be shown as a separate exercise in My Grades.