Objective SWBAT find distance traveled, use rectangular approximation method (RAM), volume of a sphere, and cardiac output.

Transcription

1 5.1 Estimating with Finite Sums Objective SWBAT find distance traveled, use rectangular approximation method (RAM), volume of a sphere, and cardiac output. Distance Traveled We know that pondering motion problems leads to considering slopes of curve, but also, these problems involve areas under curves. For example, a train moving at a steady rate of 75 mph from 7-9 am. What is the total distance traveled? Well, we know to apply the distance formula of D = RT, and we find the answer to be 150 miles. But suppose, you are Isaac Newton trying to make a connection between this formula and the graph of the velocity function. You might notice that the distance traveled by the train is of the rectangle whose base is the time interval [7,9] and whose height at each point is the value of the constant velocity function v = 75. This is no accident, since the distance traveled and the area in this case are both found by multiplying the. This same connection between distance traveled and rectangle area could be made no matter the speed or time interval. But, what is the train had a velocity v that varied as a function of time? The graph would no longer be a horizontal line, so the region under the graph would no longer be. Questions like these were why Newton and Leibniz focused so much on Calculus; especially finding areas under curves. They imagined the time interval being into many tiny, each one so small that the velocity over it would essentially be. Geometrically, this would be equivalent to slicing the irregular region into strips, which would be indistinguishable from a narrow. The two argued that, just as the total area could be found by the areas of the strips, the total distance traveled could be found by summing the small distances over tiny time intervals.

2 Example 1 Finding Distance Traveled when Velocity Varies A particle starts at x = - and moves along the x-axis with velocity v(t) = t^2 for time t > 0. Where is the particle at t = 3?

3 To make is easier to talk about approximations with rectangles, we now introduce some new terminology. Rectangular Approximation Method (RAM) In example 1 we used the to approximate the area under the curve. The name suggests the choice we made when determining the of the approximating rectangles: We evaluated the function at the mid-point of subinterval. If instead we have evaluated each function at the endpoint, we would have obtained the approximation, and if we had used the endpoints we would have obtained the approximation. No matter which RAM approximation we compute, we are adding the products of the form, or in this case,. As we can see, in the above graphs, LRAM is smaller than the true area and RRAM is larger. MRAM appears to be the closest of the three approximations. However, observe what happens as the number n of subintervals increases:

4 There is a program in your calculator called RAM is what was used to calculate the table. All three sums approach the same number (in this case ). Example 2 Estimating Area Under the Graph of a Nonnegative Function The graph of f(x) = x^2 sin x on the interval [0,3]. Estimate the area under the curve from x = 0 to x = 3. Volume of a Sphere Although the visual representation of RAM approximation focuses on area, remember that out original motivation for looking at sums of this type was to find the distance traveled by an object moving with a nonconstant velocity. Example 3 Estimating the Volume of a Sphere Estimate the volume of a solid sphere of radius 4.

5 Cardiac Output The number of liters of blood your heart pumps in a fixed time interval is called your. For a person at rest, the rate might be 5 or 6 liters per minute. During strenuous exercise the rate might be as high as 30 liters per minute. It might also be altered significantly by disease. A technique to measure the cardiac output is to inject a dye into a main vein near the heart. Then the concentration can be measured every few seconds as the blood flows past. The trick is to divide the number of by the under the dye concentration curve. Example 4 Computing Cardiac Output from Dye Concentration Estimate the cardiac output of the patient whose data appear in the table above and the given graph. Give the estimate in liters per minute. Homework