Math 1526 Excel Lab 2 Summer 2012 Riemann Sums, Trapezoidal Rule and Simpson's Rule: In this lab you will learn how to recover information from rate of change data. For instance, if you have data for marginal revenue then you will be able to approximate the revenue values. An example of rate of change is velocity. If the distance function in feet is given by s = 160t - 16t^2 then instantaneous velocity for any time t is found by taking the first derivative. Velocity is v = s' = 160-32t. Velocity is the change in distance divided by the change in time (v = d/t). If you know the velocity then you can find distance by solving the equation and getting d = vt or distance equals velocity times time. Below is a graph of the velocity t v (t) 0 160 0.5 144 1 128 1.5 112 2 96 2.5 80 3 64 3.5 48 4 32 4.5 16 5 0 5.5-16 6-32 6.5-48 7-64 200 150 100 50-50 -100 velocity v = 160-32t 0 0 1 2 3 4 5 6 7 t Insert vertical lines to represent rectangles under your graph (click on a value on the x axis to insert vertical gridlines) and look at the width of each rectangle. Each width is a unit of time which we will call t and t = 1 second. The length (height) of each rectangle will correspond to the v-value for a particular t-value. For instance if t = 1 then v = 160-32(1) = 128 is the length of the first rectangle. If t = 2 then v = 96 is the length of the second rectangle. Remember that we decided that distance could be found by multiplying velocity by time, but geometrically it is the same thing as multiplying the length and width of a rectangle. This would give you the area of the rectangle. Geometrically finding the distance is the same as finding the area of the rectangle whose dimensions are the same values as the time and velocity. If you look back at our graph you notice that the sides of any one rectangle are not the same. Which side do you use? It turns out that we can use both. See the example below: Example 1: Find the change in distance from t =1 sec. to t = 4 sec. using the velocity graph and the function v = 160-32t from the previous page. To do this we will find the area of the rectangles from t = 1 to t = 4. Note, we will be using the three rectangles formed on [1,4], so n = 3 and the width will be (4-1)/3 for this example. To find the area of the rectangles, we used Excel to generate the following table. Column 1 - Enter the number of the subinterval beginning with 1 and ending with n. Column 2 - Enter the left-hand endpoint for each subinterval. Column 3 - Enter the right-hand endpoint for each subinterval. Column 4 - Width of subinterval = right hand endpoint - left hand endpoint Column 5 - Enter the function evaluated at the left hand endpoint. 1
Column 6 - Enter the function evaluated at the right hand endpoint. Column 7 - Calculate the left-hand area of each rectangle by multiplying left hand height by width of subinterval. Column 8 - Calculate the right-hand area of each rectangle by multiplying right hand height by width of subinterval. Now sum all of the left-hand areas and all of the right hand areas. Notice that since the function is decreasing, all of the left hand areas are bigger than the actual area on the graph and all of the right areas are smaller. We therefore conclude that the actual value is probably somewhere between the two values, so we average the left approximation and the right approximation to find the best possible approximation using this method. (b-a)/n = (4-1)/3 = 1 Now use Edit-fill to generate the t-values for the subintervals: Subinterval Left hand Right hand Width of Left-hand Right-Hand Left-hand Right-hand label t-values t-values Subinterval Height Height Areas Areas 1 1 2 1 128 96 128 96 2 2 3 1 96 64 96 64 3 3 4 1 64 32 64 32 The approximations are: 288 192 Average of left and right areas: = 240 This method for recovering information from the "rate of change" is called Riemann Sums. The smaller the width of the rectangles the more accurate the answer. As the width t -> 0, then area of the rectangles fits close under the graph and the answer is closer to the actual value. It can also be used to recover all types of information from rate of change data. For instance if you have marginal revenue data you can easily approximate the change in total revenue as production is changed. Trapezoidal Rule: There are several other ways to approximate actual values from rate of change data. Above we used rectangles fitted to our data, but it is possible to also fit trapezoids to the data and then add the areas of the trapezoids. This often gives a better fit. In your book is a good discussion of the Trapezoidal Rule but in this lab we will just introduce the formula and how to use it. The formula is: Area =( x)(1/2)[ f(x 0 ) + 2f(x 1 ) + 2f(x 2 ) + 2(x 3 ) +...+2f(x n-1 ) + f(x n )] where n is the number of trapezoids used, x is the width and f(x 0 ) is the length of the trapezoid at the beginning value x 0, and f(x 1 ) is the length of the trapezoid at x 1, etc. Example #3: Using the trapezoidal rule and n = 10 and the function v = 160-32t (the function from the first graph in this handout) to approximate the distance traveled in the first five seconds (the interval is [0,5]). (1) First find the width of the subintervals: (b-a)/n = (5-0)/10 = 1/2 (2) Next generate the following table: Column 1 - the t values that form the endpoints of the subintervals Column 2 - the function v = 160-32t evaluated at the endpoint values from column 1 2
Column 3 - the constant multiplier from the trapezoidal formula (1 for f(t 0 ) and f(t n ), 2 for all other f(t) s) Column 4 - the product of the function value from column 2 and the constant from column 3. (3) Calculate the sum of the Product column using the sum command: =sum(highlight cells to sum) (4) Now at the bottom, enter the formula for the trapezoidal formula: =(1/2) * width of subinterval * sum of products t v (t) =160-32t Constant Product 0 160 1 160 0.5 144 2 288 1 128 2 256 1.5 112 2 224 2 96 2 192 2.5 80 2 160 3 64 2 128 3.5 48 2 96 4 32 2 64 4.5 16 2 32 5 0 1 0 Sum of Products = 1600 Trapezoidal Approximation = 400 We, therefore, conclude that approximately 400 feet were traveled in the first five seconds. Simpson s Rule: Simpson s Rule is yet another method to approximate values from rate of change data. In this method you fit parabolas to your data. This gives you the following formula: Area =( x)(1/3)[ f(x 0 ) + 4f(x 1 ) + 2f(x 2 ) + 4(x 3 ) + 2f(x 4 ) +...+4f(x n-1 ) + f(x n )] where n is always an even number and is the number of parabolas used, x is the width and f(x 0 ) is the length at the initial value x 0, and f(x 1 ) is the length at x 1, etc. Example #4: Suppose that Marginal Cost for a certain function is given by the following table. q (quantity) 0 20 40 60 80 100 120 MC 260 255 240 240 245 250 255 Use Simpson s Rule to approximate the change in cost if 120 units are sold. (1) First find the width of the subintervals: n = 6, so (b-a)/n = (120-0)/6 = 20 (2) Now, following the procedure used in Example #3, develop table for this problem. (3) Calculate the sum of the product column using sum command: =sum(highlight cells to sum) (4) The formula at the bottom uses the formula for the Simpson s Rule approximation: =(1/3) * width of subinterval * sum of products q MC Constant Product 0 260 1 260 20 255 4 1020 40 240 2 480 60 240 4 960 80 245 2 490 100 250 4 1000 120 255 1 255 Sum of Products = 4465 Simpson' s Approximation = 29766.67 (Rounded to 2 decimal places for dollars and cents) 3
Based on the given table, the change in cost to produce between 0 and 120 pieces of the product is $29,766.67. This is actually the change in variable cost since the fixed costs are the same. (Problems To Turn In): Be sure to work problems in correct order, staple papers, and type everything. Problem #1: Suppose you are given the following data for marginal revenue and you wish to recover information about revenue. q 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 MR 450 441 433 425 417 409 400 392 384 376 368 359 351 343 335 318 a) First type in the data and then graph the data. Then put gridlines on your graph: Double click on your graph to edit. Then click on Insert-gridlines. Click on Major x-axis gridlines for vertical gridlines and Major y-axis gridlines for horizontal gridlines. b) For this problem, you are going to use the chart that you just generated to find the total revenue if 15 units of the product are sold. For this part of the problem, let q = 5. Go back to the Marginal Revenue graph. Double-click on one of the gridlines. This will call up a dialog box that will allow you to change the scale on the q-axis. Change the major unit scale to 5, since q = 5. This will cause your graph to display a graph with rectangles that are 5 units wide. Now use the method of Riemann Sums (described previously in Example #1 to approximate total revenue if 15 products are sold. c) Duplicate your Marginal Revenue graph (copy and paste works) and repeat part a) using the same data and graph, but change the scale on the q-axis so that q = 3 for each rectangle. How do the answers from a) and b) compare? d) Repeat the above process in step b) one more time. This time change the scale on the q-axis so that q =1. Compare this result with the first two. Which is the better answer? Problem #2: Using same velocity function that was used in Example #2, v = 160-32t, with n = 8: a) Find t for the velocity function from t = 0 to t = 4, then plot the graph of v(t) using t as the scale on the t-axis. Insert gridlines that are t units apart. b) Develop a table similar to the one used in Example #2 to find the change in distance from t = 0 to t = 4 seconds using Riemann sums and n = 8 subintervals. Problem #3: In the table below is data on the amount of aluminum recycled since 1978 in an undisclosed country. Use Excel to generate a table similar to those in Examples 3 and 4 to find the total pounds of aluminum (in billions of pounds) recycled from 1984 to 1994 using the Trapezoidal Rule. 4
Problem #4: In the table below is data on the amount of aluminum recycled since 1978 in an undisclosed country. Use Excel to generate a table similar to those in Examples 3 and 4 to find the total pounds of aluminum (in billions of pounds) recycled from 1988 to 2000 using Simpson's Rule. Table for problem 3 and 4 year 1978 1980 1982 1984 1986 1988 1990 1992 plastic recycled 5 6.2 10 14.6 20 25.8 32 40.2 year 1994 1996 1998 2000 plastic 50 58.2 60 64.3 recycled 5