2.2, 2.3

2.2 and 2.3

1) Main Points

We can plot a line tangent to the point to find the derivative. When we plot a graph of derivatives we notice that is is different from the graph of the function. If f'> 0 then f is increasing, if f' < 0 then f is decreasing, if f' = 0 then f is constant over that interval. Derivates can also b expressed in tables. From these table we can "guess" a formula for the derivative. The derivative can also be written as dy/dx, which means the difference in y divided by the difference in x. If this was for a graph of distance over time then the derivative would represent a graph of the velocity because distance/time = velocity. The units of the derivative are the units of dy/dx. The derivative of the derivative, dv/dt would be the acceleration, which would graph the changes in the rate of change. The derivative can be used to calculate other things regarding rates of change.


2) Challenges

With derivatives, units can get confused and are also very important. It is easy to functions and their derivatives mixed up. Sometimes it is hard to determine what the derivative represents in a word problem (I need to continue to remind my self that it quantifies a rate of change and can be used to predict what happens next in the function).

3) Reflections

Derivatives are the start of real calculus and are obviously very applicable to and also very useful in real life. You can use the derivative to measure the speed of someone running at constant velocity or you can predict the length of his next couple strides if he is running at an inconstant speed. Derivatives have financial applications. You can use the derivative to predict an increase in sales if you increase a variable such as advertising. The derivative goes beyond just modeling what nature does into predicting and showing how it changes.