Post #33

Average Speed

First of all, remember that a slope is the y value, or dy, of a derivative.

Example:

A ball is flung from a little child. It's path is projected as y=4.9t2m in "t" seconds. What is the average speed of the ball from 0 to 3 seconds?

1. Set up equations and intervals: (f(b)-f(a))/(b-a) 4.9t^2 [0,3]


2. Plug in a and b values for t: f(b)=4.9(3)2=44.1 f(a)=4.9(0)2=0

3. Plug into main equation and solve: (44.1-0)/(3-0)=14.7m/s

Average speed is used for many different things, from finding the speed at which a cannonball was launched out of a cannon from how fast a cheetah runs in a straight line trying to catch it's prey. The concept behind average speed is a fairly simple concept that many people understand right away. You're basically finding the slope of the equation using calculus and algebra. If I ask someone what the average speed of a ball from [3,4] if it's path was graphed as y=x.

y=(4) y=(3) (4-3)/(4-3)=1

Graph Interpretation


Every AP exam will include at least one graph on it's short answer section that requires the test-taker to interpret certain requirements from it in order to receive credit for that question.

Interpreting a graph is very easy but in order to do it properly one must understand certain properties.

1. If the original graph is increasing, the slope is positive.
2. If the original graph is decreasing, the slope is negative.
3. An interval with a positive slope on the first derivative means that there is a downward concavity on that interval in the second derivative.
4. An interval with a negative slope on the first derivative means that there is an upward concavity on that interval in the second derivative.
5. Upward concavity (bowl-shaped) is positive.
6. Downward concavity (umbrella-shaped) is negative.
7. There is a horizontal tangent where the slope=0 on the original graph.
8. Wherever there is a horizontal slope there is either a maximum or a minimum value.
9. A x intercept on the first derivative is either a maximum of a minimum on the original graph.