Lab 14: Accumulation and Integration

Transcription

1 Lab 14: Accumulation and Integration Sometimes we know more about how a quantity changes than what it is at any point. The speedometer on our car tells how fast we are traveling but do we know where we are? We know the bank's interest rate on our money, but do we know the balance? At times these answers make all the difference. Of course it helps if we know our starting point! Until now we've been looking at a function and determining the slope of the tangent lines by finding the derivative. Now we're going to turn the process around and start with the derivative. What can we discover? For instance, suppose we know the average values of the derivative function over equal width subintervals of a given interval. We obtain these subintervals by chopping up an interval into equal pieces. The pieces don't have to be equal but that's often a convenient case. In our first example, we know the average velocities for each hour (each subinterval) of a ten hour car trip. What does this information tell us? Let's look at the example in detail. 1. Accumulation of distance when average velocities are known. Open the Accumulation tool in the Integration Kit and select Car Trip in the pop-down list. 1.1 Suppose that the lower graph represents average velocity in miles/ hr on a drive south on Highway 101. How many miles will be traveled in this direction in a ten hour driving day from the starting point at San Francisco south toward Los Angeles? 1.2 Note that the tool wakes up with all positive velocities. What would a negative velocity represent? ("Oops, we forgot the kids at the rest stop! I suppose we'd better go back and pick them up!")

2 1.3 Does negative velocity decrease net distance for the 10 hour day? (By net distance traveled we mean the distance the family is from their starting point not the mileage on the odometer. Sometimes the net distance is called the displacement.) 1.4 How would you interpret a negative net distance? 1.5 Using the default values with which the tool wakes up, is it possible to adjust the average velocity on the last interval on the lower graph so that the accumulated distance is zero? What would this velocity have to be? Is this a reasonable speed for an automobile? (Tool note: Click the cursor in a subinterval in the lower window to adjust the average rate of change for that subinterval. ) 1.6 How can we find the increase in mileage on the tires or on the odometer? In this case we are interested in the total distance traveled as opposed to the displacement or the net distance traveled. Figure out a method for calculating this for the case when there are positive and negative velocities. 1.7 Suppose the family stopped at an amusement park for 2 out of the ten hours. What would the effect be on the total distance for the ten hours? Can we determine the effect on the net distance for the ten hours? Explain.

3 2. Further Explorations with Accumulation as Distance 2.1 Let's work a little more with the Accumulation: Car Trip tool. For each of the following arrangements of horizontal lines in the lower plane, describe in words and/or sketch the resulting graph in the upper plane. a. Lower Plane: All the horizontal lines have exactly the same height above the horizontal axis so that a straight line is formed. Upper Plane: b. Lower Plane: The first half of the horizontal lines are distance b units above the horizontal axis and the second half of the horizontal lines are distance b units below the axis. Upper Plane: 2.2 Carefully describe how you can make the upper curve look like the "peaks" of a pair of isosceles triangles (that are side-by-side and share a point on the horizontal axis) by manipulating the horizontal lines in the lower graph. Use a sketch if needed. (Hint: The "peaks" don't have to be the same size. Because there are 10 intervals, you can use 6 for the first one and 4 for the second.) 2.3 In general the graphs in the upper plane are made up of straight line segments. However for convenience, we will call them curves. Fill in the blanks: a. To construct a decreasing curve in the upper plane, we must use only those horizontal lines in the lower graph whose distance from the horizontal axis is (positive, negative). b. To make a curve that is roughly concave up in the upper plane, we should use horizontal lines in the lower graphs whose height is (increasing, decreasing) as t increases

4 3. Accumulation of value in a share of stock Open the Accumulation: Netscape Stock tool. This tool tracks the accumulation of value in one share of Netscape stock since its initial public offering price of $28 in August of At the time that the stock was offered to the public, interest in the internet was intense. The stock soared to $75/share by the end of the first day of sale. However by the end of the first month, it closed at $49.5/share. The average increase (or decrease) in a share of Netscape stock /month is shown in the lower graph. The value of one share of Netscape stock is shown in the upper graph. As you can see, Netscape stock has earned its reputation as a volatile stock! One thing to note is that in February 1996, Netscape had a 2-for-1 stock split. The stock price shown on the tool after that date is adjusted to be twice the stock price shown on the NASDAQ stock exchange so that we can compare stock prices over time. 3.1 Determine your profit per share if you bought the share at the initial price and you sold the share at 6 months. (In our hypothetical case you don't have to worry about commissions.) 3.2 Calculate the profit per share if you bought the share at 6 months and sold it at the end of the 1 year. 3.3 Suppose that instead of displaying the value of your share, you wished to display the net profit(or loss) on your share. Try the profit button on the tool. Notice that the overall shape of the upper graph is unchanged but that the graph has been moved down. In fact for this tool, the profit is calculated relative to the price of the initial offering. Profit = current share price - initial share price. Has the lower graph been changed? Does it still work for the profit? The profit graph represents the (net, total) accumulation of value in a share of Netscape stock. (Look at the discussion of question 1.3.) Initial values: Except for this example about Netscape stock, each tool assumes that the initial value for the upper curve was zero. If it were not the case, we would have to subtract out the initial value to obtain the net accumulation in the same way we calculated profits from the stock. Look for this idea in future labs as we start to employ the definite integral!

5 4. Additional examples of accumulation There are many examples where the average rate of change is known over subintervals of a given interval (of the dependent variable) and an accumulation can be calculated. Some of these examples are listed here: the energy per second stored by a solar battery in the course of a day, the work done by a variable force, the velocity attained under variable acceleration, and the money accumulated in a slot machine. Select one of these examples and answer the following questions: 4.1. Which example did you select? 4.2. In each case negative values in the rate of change are possible. Explain. 4.3 Invent an additional question that could be asked for the example you selected, one that would show the connection between the real world and the concept of accumulation. 4.4 Give another example of accumulation in the real world. Discuss whether negative rates of change are possible. Also mention conditions that might influence the average rate of change over various subintervals.

6 5. The definite integral as an accumulation Open the Numerical Integration tool. We are going refine our estimates for the accumulation by taking finer and finer subdivisions. Select the second function in the function list: f(x) =xe x. This function is an example of a surge-dissipation function. It can be used to model the build-up of a drug in the bloodstream or the sales of a trendy item. 5.1 Choose from the selections below the graph: n=4 (number of intervals) and left (for the left endpoint). The tool then constructs a rectangle with height equal to the value of the function at the left endpoint and width equal to 2 units (which is a fourth of the total length of 8 units shown) for each of the four subintervals. Then the tool adds up the rectangular areas to obtain an estimate of the accumulated area under the curve. Refine this estimate by selecting higher values of n. Do you see that the sum of the areas are getting closer and closer to the actual area under the curve? 5.2 Do the same procedure using the midpoint of each interval to find the height of each rectangle. Note that you need only select midpoint and then take higher and higher values of n to get finer and finer subdivisions. Imagine subdividing again and again so that rectangles become narrower and narrower. Can you see that the limiting value is the actual area under the curve? The limiting value (if it exists) is called the definite integral from [a,b].