3.7 Higher Differentiation

There were three main concepts in this section:

There were three main concepts in this section:
1. learning how to take second, third, fourth (etc) derivatives
2. New Notation for Higher Differentiation (dy/dx)
3. Relating the 2nd derivatives to motion and acceleration

Second, Third and Fourth Derivatives

Example:

Higher Derivatives:












Using dy/dx notation
Example:














Another Example:
y = sin (x)
y ′ = cos (x)
y ′′ = -sin (x)
y ′′′ = -cos (x)
y(4) = sin (x)
y(121) = cos (x)

The Second Derivative explains how the slope is changing. It tells us whether the graph is getting steeper or flattening out. In motion, the second derivative is used to measure acceleration.
s′ = v - the derivative of the distance equation is the velocity
v′ = a - the derivative of the velocity equation is the acceleration
s′′ = a - the second derivative of the distance equation is also the acceleration

Motion Example:
Initial Velocity--after one second-- Final Velocity--------- acceleration
10 mph------------------------------------ 20 mph--------------- +10 mph/sec
10 mph------------------------------------ 0 mph----------------- (-10) mph/sec
-10 mph---------------------------------- (-20) mph------------- (-10) mph/sec

Speed vs. Velocity
Speed = the magnitude of the velocity vector. It lacks direction.

When the change in velocity and the acceleration have the same sign (positive or negative), then the speed is increasing

Example:
Find the velocity and acceleration equation:



Find when the particle is speeding up:
-First find when the particle is at rest.
v = 3(t – 1)(t – 3)
-the velocity is equal to zero at 1 second and then at 3 seconds
-Then, test the intervals of velocity
- between 0 and 1: positive
- between 1 and 3: negative
- greater than 3: positive
-Finally, find when the acceleration is going to be 0
- a = 0 at 2 seconds
- Compute when acceleration is going to be positive or negative
- below 2 seconds: negative
- greater than 2 seconds: positive
-When the signs match, the particle is speeding up
So: 1 < t < 2 and t > 3

Here is a link for more information:
http://www.intmath.com/Diff/9_HiDe.php


SO there's the end of the blog for today folks.
-Lauren

Here's something to make you all very happy that it's Thursday:


DOESN'T IT MAKE YOU FEEL ALL WARM AND FUZZY INSIDE??


JAMES, have fun with the next one.