Suppose that a jet takes off, becomes airborne at a velocity of 180 mph and climbs to its cruising altitude. The following table gives the velocity every hour for the first 5 hours, a time during which its velocity increases at a constant rate. Time (hours) Velocity (mph) 0 1 3 4 5 180 40 300 360 40 480 1. The question we would like to answer is how far does the jet fly during some period of time? Since the velocity is changing the distance cannot simply be calculated by using d vt ; however, we can still find an answer. This method will involve constructing rectangles. A graph of the planes velocity is given below. The rule for the velocity function f of the plane is f ( t) 180 60t a. Divide the interval 1,5 hours into 8 equally sized subintervals. How wide is each subinterval? Mark off these subintervals on the graph along the time axis.
b. Consider the first subinterval. Draw the right rectangle on the first subinterval. That is a rectangle whose right side is equal in height to the functions value at that point. What is the width of this rectangle? What is the value of the function at the right edge of the first subinterval? Show how you would use this value and the velocity function to determine the height of this rectangle. c. What is the area of this rectangle constructed on the first interval? Show the calculations used to find your answer. How far did the plane travel in this first ½ hour time interval? d. Similarly find the areas of the remaining right rectangles by completing the table below Interval Number Right Edge Width of Interval Height f() t Area f() t Area Value 1 1.5 0.5 f (1.5) 70 f (1.5) 0.5 135 3 4 5 6 7 8 e. Write an expression for the total area by summing the expressions in the or the Right Hand Rectangular Approximation Method RRAM = Area f() t column. We call this RRAM f. Calculate the total area for RRAM and interpret your answer in context.
g. Is your answer in part g an over or an under estimate for the actual distance the plane traveled? h. What can we do with the number of subintervals to get a more accurate estimate for the actual distance traveled? In the previous activity, we saw that the distance traveled could be approximated by summing the areas of rectangular regions. We can see that as we let the number of rectangles approach infinity that the time intervals approach zero and the sum of the areas of the rectangles would approach the true distance traveled. You might notice that with a linear function finding this area could be done without looking at summing rectangular regions by using known geometric formulas for areas of triangles, rectangles, and trapezoids. However, when the function is not linear, we will rely on finding this area using rectangles with smaller and smaller intervals.. Suppose we want to approximate the area of the shaded region T in the first quadrant, bounded above by the graph of f ( x) x 0,4. See the figure below., and below by the x axis on the interval T
a. Our strategy is to approximate the area of the region T by using rectangles. On the graph below, partition the interval 0,4 into four equally sized subintervals. Use the right endpoints of the intervals as inputs for height and construct the right rectangles on each of the subintervals. Find the total area of these rectangles (show the calculations used). Recall that this is called RRAM. Will this total area be less than or greater than the actual area? Does this depend on whether the function is increasing or decreasing? b. On the graph below, again partition the interval 0,4 into four equally sized subintervals. This time use the left endpoints of the intervals as inputs for the height and construct rectangles on each of the subintervals. Find the total area of these rectangles (show the calculations used). This is called LRAM. Will this total area be less than or greater than the actual area? Again, does this depend on whether the function is increasing or decreasing?
c. Obviously, the actual area of the region T lies somewhere between the values in parts a and b. On the graph below, again partition the interval 0,4 into four equally sized subintervals. This time use the midpoints of the intervals as inputs for the height and construct rectangles on each of the subintervals. Find the total area of these rectangles (show the calculations used). This is called MRAM. The three methods you used in problem are referred to as the Left Rectangular Approximation Method (LRAM), the Right Rectangular Approximation Method (RRAM), and the Midpoint Rectangular Approximation Method (MRAM). See the figure below for an illustration of each of these methods. Notice that the height of the rectangle is the output value of the function at the location used left, right, or midpoint of the subinterval. Notice that if a function is increasing over a given interval, then LRAM is a lower estimate of the area under the graph, while RRAM is an upper estimate. 3. What can you say about the estimates using LRAM and RRAM if the function is decreasing? Illustrate with a sketch if necessary.
4. The notation LRAM ( x ) means divide the given interval for the graph of 4 y x into 4 equally spaced subintervals, use the left endpoint of each of these intervals as an input value for the function to determine the height of the rectangle and then find the total area of these rectangles. a. Describe the meaning of RRAM ( x ) on 30 0,4. b. Open the sketchpad file RAM.gsp and you will see the screen below. Drag point N to change the number of rectangles. Find the value for RRAM ( ) 30 x. Is this an over estimate or an under estimate of the area under the curve? RRAM ( x ) = 30 d. Use Sketchpad to find LRAM x. Is this an over estimate or an under estimate? ( ) 30 LRAM ( x ) = 30 e. Based on your results from parts c and d above, take a guess at the exact value for the area under the curve f ( x) x 0, 4. on
The following TI-83/84 program computes the sum for n subintervals, evaluating the function stored in Y1 at a given point x in each of the subintervals. You first must enter the function into Y1 and then run the program. Enter this program into your graphing calculator or get a copy of it from your teacher. Enter 0 to find LRAM, 0.5 for MRAM, and 1 for RRAM. c. Verify the values you found in a-c using the RAM program.
5. Suppose we want to approximate the area of the region bounded above by the graph of on,5. 1 f ( x) x 3 a. Graph the function and shade the bounded region whose area you will be approximating. b. Complete the table below. Use the RAM program to find these values. n LRAM RRAM 4 8 16 c. Based on the values in the table what is your approximation for the area? d. Find the exact area of this trapezoidal region using the formula for the area of a trapezoid. A h b b 1 1
6. a. Estimate the area of the region bounded above by the graph of g, below by the x-axis on the interval 4,4 by using either RRAM 8 or LRAM 8. Show your work. b. Estimate the area of the same region described in part a using MRAM 4. Show your work.