Math 1525 Excel Lab 9 Fall 2 This lab is designed to help ou discover how to use Excel to identif relative extrema for a given function. Example #1. Stud the data table and graph below for the function f(x) = 2x 3-3x 2-12x +6 x -3-39 -2.5-14 -2 2-1.5 1.5-1 13 -.5 11 6.5 -.5 1-7 1.5-12 2-14 2.5-11.5 3-3 3.5 13 4 38 f(x)=2x 3-3x 2-12x + 6 3 2 1-3 -1-1 1 3-2 -3 - x Verif the following facts about the graph: 1. f(x) is increasing on the interval [-3, -1] and on the interval [2, 4]. 2. f(x) is decreasing on the interval [-1, 2]. 3. f(x) reaches a relative maximum value when x = -1. What is the value at this point? 4. f(x) reaches a relative minimum value when x = 2. What is the value at this point? 5. On the interval graphed, the absolute maximum value of the function is = 38. Locate this point on the graph. 6. On the given interval, the absolute minimum value of the function is = -39. Locate this point on the graph. 7. The graph has two concavities. It is concave up on [1/2, 4] and concave down on [-3, 1/2] 8. The graph has a point of inflection at x = 1/2. Note that all of the information in steps (1) - (8) was found just b looking at the graph and/or the data table. No calculations were necessar. 1
Now let's look at the same graph again, but with the graphs of the first and second derivatives included. Note that these derivatives are found b hand. x ' ' ' -3-39 6-42 -2.5-14.5-36 -2 2 24-3 -1.5 1.5 1.5-24 -1 13-18 -.5 11-7.5-12 6-12 -6.5 -.5-13.5 1-7 -12 6 1.5-12 -7.5 12 2-14 18 2.5-11.5 1.5 24 3-3 24 3 3.5 13.5 36 4 38 6 42 f(x) = 2x 3-3x 2-12x + 6 f(x), f'(x), and f''(x) 6 5 3 2 1-3 -2-1 -1 1 2 3 4-2 -3 - x ' ' ' f'(x) = 6x 2-6x - 12 f''(x) = 12x - 6 Verif the following facts about this new graph: 1. The graph of f'(x) > (ie, it lies above the x-axis) whenever the graph of f(x) is increasing. 2. The graph of f'(x) < (ie, it lies below the x-axis) whenever the graph of f(x) is decreasing. 3. The graph of f'(x) = (ie, lies ON the x-axis) at the points where f(x) has a relative extrema. 4. The graph of f''(x) > (ie, lies above the x-axis) whenever the graph of f(x) is concave up. 5. The graph of f''(x) < (ie, lies below the x-axis) whenever the graph of f(x) is concave down. 6. The graph of f''(x) = (ie, crosses the x-axis) when f(x) has a point of inflection. 2
Example #2. If the function ou are working with is not continuous (ie, has one or more points of discontinuit), we can still use a graph of investigate the properties of the function. Stud the data table and graph given below for the function f(x) = (2x)/(x-3) on [, 6]. Before beginning the graph, ou must notice that the function is undefined at x = 3 so that it will have a vertical asmptote there. The asmptote will not be graphed, but ou ma add it later b hand if ou wish. The x-values that ou select for our data table must be chosen ver carefull. Stud the pattern of points chosen for this example graph. x.5 -.4 1-1 1.5-2 2-4 2.5-1 2.9-58 *Note the space left 3.1 62 between the x-values 3.5 14 2.9 and 3.1. If ou do 4 8 not leave this space, 4.5 6 Excel will connect 5 5 these two points. 5.5 4.4 6 4 =(2x)/(x-3) 6 2 2 4 6-2 - -6 Verif the following facts about the graph of this function: 1. f(x) is decreasing on both parts of the graph. What would this tell ou about the first derivative of f(x)? (Ans: the first derivative is alwas negative) 2. There are no relative extrema for this function. 3. f(x) is concave down when x < 3 and is concave up when x > 3. What would this tell ou about the second derivative of f(x)? 4. Important: There is NO point of inflection for this graph because the change in concavit occurs at an asmptote - not at a point on the graph. 5. Notice that as x 3 -, the -values of the function -> negative infinit. Notice also that as x 3 +, the -values of the function -> positive infinit. The last two facts tell us that x = 3 is a vertical asmptote. 6. Notice that as x positive infinit, the -values of the function ->. Notice also that as x negative infinit, the -values of the function also ->. The last two facts tell us that = is a horizontal asmptote. Sketch in this asmptote b hand. 3
Problems to be turned in for Lab 9: Show all work and discuss the answers in complete sentence as if ou were explaining the results to another student. You ma tpe in Word and paste data and graph form Excel if ou wish to tr it. You just highlight data and graph in Excel and cop. Next ou click on our Word file and Paste Special- Microsoft Excel Object ( delete the check on float over box. ) and then click on ok. 1) Given the function F(x) = x 3-1x 2-22x +, Graph F(x) using Excel over the interval [-6, 6] and then find the first and second derivative b hand (do not graph) and verif (showing all work) where F(x) a) Increases and decreases b) Has local maximum and minimum c) Is concave up and/or concave down d) Has points of inflection e) Has Global max and min over the interval 2) Given f(x) = x 5 - x 3, find f '(x) and f ''(x) and graph all three functions on the same axis over the interval [-5, 5 ]. Write an short paragraph to a friend explaining how to tell which graph is which if the can not see the original equations used. Look at Example #1 to guide ou in our writing. 3) Given the function F(x) = 4x / (x - 6), Graph F(x) using Excel over the interval [ -2. 2]. (Using Example #2 as a guide ) Do the first and second derivative b hand and verif (showing all work) where F(x): a) Increases and decreases b) Has local maximum and minimum c) Is concave up and/or concave down d) Has points of inflection e) Has vertical asmptotes f) Has horizontal asmptotes 4) B hand Sketch the best graph ou can using the following information: g) F(-5) = 2, F() is undefined, F(3) = 2, F(8) = 7, F(1) = 2, F( 12) = 4 h) F'(x) < on (, 3 ], [ 8, 1 ] i) F'(x) > on [ -5, ), [ 3, 8], [ 1, 12 ] j) F'(x) = for x = 3 k) F'(x) is undefined at x = 8 l) Indicate the interval where F''(x) > m) Indicate the interval where F''(x) < n) Indicate where F''(x) = 4
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