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Monday 3/1
First we worked on three problems of the day all concerned with the Fundamental Theorem of Calculus. These can be seen in the graphics below. Following these problems we talked about working with the fundamental theorem and word problem, specifically when teh function presented is given as a rate function. We had experienced problems of this type when working particle motion problems when the velocity function was presented and we were asked about distance traveled or the position at any given time. (Caren and Larry cycling to school was an AP problem which came to mind!) The problems presented to us today had the rate function defined either graphically or algebraically and we had to apply the Fundamental Theorem to that rate functions and interpret the result.

Problem 1. A bicyclist is pedaling along a straight road with velocity,v, given in the graph below. Suppose the cyclist starts 5 miles form a lake, and that positive velocities take her farther from the lake and negative velocities towards the lake. When is the cyclist farthest from the lake and how far away is she then?

Problem 2 Water is leaking out of a tank at a rate of R(t) gallons /hour, where t is measured in hours. a) Write a definite integral that expresses the total amount of water that leaks out in the first two hours. b) The graph of R(t) is shown below. On a sketch, shade in the region whose area represents the total amount of water that leaks out in the first two hours.

c) Give upper and lower estimates of the total amount of water that leaks out in the first two hours.

Problem 3 A news broadcast in early 1993 said the average American's annual income is changing at a rate of r(t) = 40(1.002)^t dollars per month, where t is the number of months from January 1, 1993. How much did the average American's income change during 1993?

Problem 4 The number of cars in a parking lot is given by f(t), where t is the hours from 8 am. The graph of f '(t) is shown below. If there were 30 cars in the parking lot at 10 am, how many cars are int eh parking lot at 3pm?

Problem 5 On a particular day, suppose that t hours after midnight, the outside temperature is changing at a rate r(t) = 5 cos(0.5t^2+ 2) degrees Fahrenheit per hour. If the temperature is 72 degrees F at 10 am, what was the temperature at 6 am?

Problem 6 The temperature of a tank of water is changing at a rate of r(t)= e^(srt(t)) - 6 degrees F/hr, where the time t is in hours. Suppose the temperature of the water at t = 0 is 56 degrees F. a) What is the temperature of the water when t = 4? b) On the interval [0,4], where is the temperature a minimum?

Tuesday 3/2
Having answered any problems associated with yesterday's work, we moved on to some AP Problems on the Fundamental Theorem. Two of the problems and their solutions can be seen below under Wednesday's class.

Wednesday 3/3
Review for the test, focusing on the AP problems assigned yesterday.