Calculus & Exam Section 6

Methods One can calculate derivatives and integrals via two different methods: Analytically – Lots of scary math. Graphically – Some drawing and some not-so-scary math.

Graphical Integration In order to do graphical integration: Split up the area under the curve into small rectangles. Calculate the area of each of the rectangles. Sum all the areas of the rectangles.

Graphical Integration Example Integrate the function from x = 0 to x = 4: area = .375+1.25+1.625+2+2+1.625+1.25+.375 = 10.5 2.000 2.000 1.625 1.625 1.25 1.25 .375 .375 The actual area under the curve is 8 when calculated analytically. The more rectangles you use the better your precision.

On to the exam problem…

The Problem You were asked to use graphical integration to graph a stock with the following inflows and outflows assuming the initial value of the stock to be 100: 5 10 15 20 40 80 Flows (units/time) Inflow Outflow

First Things First First, calculate the net flow by graphically subtracting the outflow from the inflow: 80 Inflow Outflow Flows (units/time) 40 Net Flow 60 – 50 = 10 60 – 80 = -20 60 – 70 = -10 30 – 40 = -10 50 – 40 = 10 70 – 40 = 30 40 – 30 = 10 60 – 30 = 30 20 – 20 = 0 20 – 40 = -20 -20 5 10 15 20

Integration Now, chop up the Net Flow function with rectangles and calculate the area for each: 80 Flows (units/time) 40 Net Flow 32 1.25 x 25 = 31.25 31 1.25 x 25 = 31.25 1.25 x 0 = 0 1.25 x 0 = 0 1.25 x 0 = 0 16 1.25 x 12.5 = 15.63 16 1.25 x 12.5 = 15.63 1.25 x 0 = 0 12 1.25 x 10 = 12.5 12 1.25 x 10 = 12.5 12 1.25 x 10 = 12.5 12 1.25 x 10 = 12.5 -25 1.25 x -20 = -25 -25 1.25 x -20 = -25 -16 1.25 x 12.5 = 15.63 -16 1.25 x -12.5 = -15.63 -20 5 10 15 20

Graphing The next step is to calculate the y-value for the given point by adding the area of the integral (area of the rectangle) to the y-value of the previous point. For example, for x = 1: y = 100 + 0 = 100 and so on…

Graphing Step By Step 80 Net Flow Flows (units/time) 40 32 31 0 0 16 0 0 16 16 12 13 12 13 -25 -25 -16 -16 -20 5 10 15 20 106 + 12 = 118 118 + 12 = 130 130 + 31 = 161 161 + 16 = 177 100 + 0 = 100 100 + 0 = 100 94 + 12 = 106 177 + 0 = 177 177 -16 = 161 100 -25 = 75 34 + 16 = 50 50 + 32 = 82 82 + 12 = 94 75 -25 = 50 50 -16 = 34 34 + 0 = 34 100 5 10 15 20 100 200 Stock (units)

Analytical Solution The following is the graph of the analytical solution… for the math geek in all of us: y = 100 y = 10x - 25 y = -20x + 150 y = 5x2 – 70 x + 275 y = 5x2 + 180 x -1450