Post #28

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

Implicit Derivatives

The only difference between implicit derivatives and regular derivatives is that implicit derivatives include dy or y', the actual derivative of y.

y=x+2 y'=1

In an implicit derivative, you are always asked to solve for y'.

Example:

x^2+2y=0

1. Take derivative of both sides first.

2x+2y'=0

2. Then solve for y'.

y'=(-2x)/2

Some examples include:

4x+13y^2=4 y'=(-4/26y)

cos(x)=y y'=-sin(x)

y^3+y^2-5y-x^2=4 y'=2x/((3y+5)(y-1))