Tuesday,+February+1+Accumulation+Functions+Day+1

(Wenwen) If we are given the graph of a //derivative// g'(x), we can determine: the change in g(x) since the change in g(x)=the area under g'(x). This area can be calculated using either the left-hand approximation or the right-hand approximation. Of course since one of these is always an over approximation and the other an under approximation, it would always be the safest to employ the midpoint approximation. Nice explanation!!

(Richie) Given the graph of g'(x), the //visual// interpretation of the total amount by which g(x) grows over a set interval is: Whether g'(x) is positive or negative dictates whether the g(x) is increasing or decreasing. Whether g'(x) is increasing or decreasing also dictates whether the g''(x) is positive or negative which in turn shows if g(x) is concave up or down. The area under the g'(x) graph is equal to the total change of the g(x) graph. Good!

(Sami) Since we can't always have a formula for area of a region, we can approximate it using: **the integral What do you mean by this? I don't believe we have defined it. What did you do in your worksheet //today// to get a better approximation for the area? **

(Eliza) A Riemann sum is: a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It may also be used to define the integration operation. The method was named after German mathematician Bernhard Riemann. What does this method involve? What is the geometric interpretation of a Riemann sum? What did you do today with a Riemann sum?

(Leah) When we studied differentials, we insisted that Δx be as small as possible to get a good estimate for Δy. When we are looking for area under a curve, what is the visual reason for Δx to be as small as possible? You want the Δx to be visually as small as possible when finding the area under the curve because that will give you the closest approximation of the curve. If the rectangles are as small as possible, they will not be an overestimate or an underestimate. The rectangles need to be as thin as possible (the x values are really close together) and tall enough to hit the curve. The smaller the Δx is the more exact and the more values you will have in your Riemann sum. Much better! Very good!!

(Julie P.) f ' (x) is defined as : the derivative of a function Whereas, the area under f ' (x) is defined as: the change in f'(x) when the derivative is positive and the function is increasing and vise versa.

Homework: p. 364 # 2, 3, 4, 5, 11, 12; p 376 # 1, 2, 3

Reminder that Valentine's Day is coming up!! Send that special someone a little romantic math shirt: