Investigate and approximate the area under the function [math] f(x) = 5-x^2 [/math] from x = 0 to x = 2 Answer the questions below it to test your understanding.
Test your understanding: 1.) The Left Hand Sum will always give an underestimate of the actual area under f(x). How can you tell? What can you say about the Right Hand Sum? 2.) Notice [math]f'(x) = -2x <0[/math] on the interval [math][0,2] [/math], so f(x) is [b]decreasing[/b] . Based on your answer to (1) above, explain how Left and Right Sums can give a good bound for the actual area under a decreasing function. What changes if the function was [b]increasing[/b]? 3.) The Actual Area under f(x) from x=0 to x=2 is [math]22/3[/math] [b]exactly[/b], or 7.3333(rounded). Experiment with the number of rectangles. a.) How many rectangles is needed for the [b]Left Sum[/b] to be within 0.1 of the actual area (7.3333)? What about for the [b]Right Sum[/b]? and the [b]Midpoint Sum[/b]? b.) Which type of Sum would you rather program in a computer to calculate the estimated area under a function? Why?