1.3 and 2.1.

1.3 and 2.1.

1)Main Points
Chapter 1.3 deals with rates of change and how to calculate them, especially if they are not constant. The average rate of change can be modeled as the Change in the dependent variable / the change in the independent variable. For a function to be increasing, the values of F(x) must increase as x increases. Chapter 2.1 discusses using the derivative to calculate the instantaneous rate of change as opposed to the rate of change over a long period time. The instantaneous velocity of an object can be determined by taking the limit of the average velocity of the object at smaller and smaller time intervals. The instantaneous rate of change of a function f at a is called the "derivative" of f at a, this is written as f'(a). The derivative is the same thing as a) the slope of the graph at that point and b) the slope of a line tangent to that exact point. The derivative can be calculated mathematically by subtracting a very small change in the dependent variable by a the same change in the independent variable. For example: you know that the rate of change is the change in y divided by the change in x. And you know that y=f(x). So with two points, a and (a+.0001) you can in the derivative of f(x) at a by doing {f(a+.0001)- f(a)/ (a+.0001) - a}.

2)Challenges

On a distance over time graph it is easy to forget that the slope between two points is the average velocity. The concept of limits is something that I've only had a little bit of exposure to but it makes sense when calculating something at in instant in time. It will be important for me to keep in mind that it is not an exact number. Also it is sometimes difficult to recognize the tangent line of a curve in an exact spot.

3)Reflections


The use of derivative seems to have lots of practical applications regarding speeds of objects and many inconstant rates of change. The book used the example of a grape fruit thrown up into the air and calculated its average speed. The derivative could have been applied in this situation calculate the instantaneous speed of the grapefruit instead of just the average speed. There are many real world applications for this chapter, probably more so than linear functions because usually things in the world don't change at constant rates.