Week+01+Aug18-22

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First Day of Class- August 18th 2008
Today we reviewed the average rate of change of a function over an interval and the instantaneous rate of change of a function at a particular point. The importance of both graphical and algebraic interpretations of functions and concepts was emphasized, as well as the possibility of describing functions using data and words. The slides used during class are here.

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August 19th 2008
The focus of today's class was to review some of the functions learned during Pre- calculus. Exploration 1.2 ,part of our homework last night, asked us to graph five different functions and copies of these graphs are shown below.

The function is decreasing when x = 1. The rate of change at x = 1 is slow because the y values are decreasing by relatively small amounts for each increase in x.



This trigonometric function is neither decreasing nor increasing when x = 1.



This quadratic function is increasing quickly when x = 1



This function is increasing quickly when x = 1.



This function is slowly decreasing when x = 1.

Following the completion of this exercise, the class brainstormed special features of the six classifications of functions: Power Functions Polynomial Functions Rational Functions Exponential Functions Logarithmic Functions Trigonometric Functions. Stay tuned for more student produced review in the future.

August 20th 2008
The lesson today introduced the definition and notation for the derivative of a function at a point. The derivative of function f at x = c is written f '(c) and it's value is equal to the instantaneous rate of change of the function at the point wher x = c. Geometrically, f '(c) is equal to the slope of the tangent line to the graph of function f at the point where x = c. These concepts are discussed in the lesson notes linked below.

August 21st 2008
The second concept of calculus to be introduced is the concept of the definite integral. The definite integral is the area between the graph of a function and the x axis on a defined interval of x. Later in the course we will learn how to find the exact value of the definite integral but, for now, an estimate of its value is sufficient. The definite integral can represent many different quantities;for example, the definite integral of a velocity-time graph is the distance traveled over the given time period. The investigation and notes from class are in the slideshow below. media type="custom" key="1710271"

August 22nd 2008
Following the introduction to the definite integral, the lesson today discussed a better way to approximate the area under a curve than by the tedious method of counting squares. Using trapezoids is a more accurate and less time consuming way to approximate area and the general formula in terms of one variable is called the Trapezoidal Rule. The graphic from class is shown below.