Determining the Properties of the Parent Function Using the First and Second Derivatives

OVERVIEW:

1. Calculus Commercials: How Many Ways Can You Define the Derivative?
2. Mathematical and Computer Science Definitions of "AND" and "OR" Notes (slide 10)
   
3. Determine the Properties of the Parent Function Using the First and Second Derivatives (slide 2 to 9)
   

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1. Calculus Commercials: How Many Ways Can You Define the Derivative?

A continuation of last class' lesson, but first, we watched calculus commercials, which can be accessed here:

http://apcalc2008.blogspot.com/2008/12/gauntlet-is-thrown-down.html

The class agreed that the time restriction of 30 seconds to define the derivative was challenging, but there are our finished products! Note that people have commented on our projects.

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2. Mathematical and Computer Science Definitions of "AND" and "OR"

Let
A = go to the store
B = wear the hat

One day, you aren't sure if you should go to the store or wear the hat. So you think and think and think, but in terms of mathematical and computer science definitions, you notice that their definition of AND and OR are different from the AND and OR of English. The OR used in math is the inclusive OR. The OR used in English is the exclusive OR.

Here is a list of all the possibilities:

1. You go to the store and wear the hat. (A = true; B = true)
   
2. You go to the store but not wear the hat. (A = true; B = false)
   
3. You don't go to the store but wear the hat. (A = false; B = true)
   
4. You don't go to the store and you don't wear the hat. (A = false; B = false)
   

A U B = satisfy the conditions of A OR B
A ^ B = satisfy the conditions of A AND B


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3. Determine the Properties of the Parent Function Using the First and Second Derivatives

Using the derivative rules, we found the roots of the function.

Remember, k is a constant, not a variable!

We let k equal a negative number, like -1 000 000, in f' and discovered that to the left of 1/k, f is increasing, so f' there is positive.

We let k equal a positive number, like 1 000 000, in f' and discovered that to the right of 1/k, f is decreasing, so the f' is negative.

Since f is increasing to the left of 1/k and decreasing to the right of 1/k, we can imagine that at 1/k, there is a local max.

Remember, k is a constant, not a variable!

In the previous slide, we evaluated f' when 1/k>0.

In this slide, we evaluated f' when 1/k<0. style="font-style: italic;">

Since f is decreasing to the left of 1/k and increasing to the right of 1/k, we can imagine that at 1/k, there is a local min.

We have now obtained as much info as we can from the f'. On to f''!

Using the derivative rules, we determined f''.

We determined the roots of f'', so we can determine the sign of f'' on either side of the roots; this info can be used to determine f's concavity to either side of the roots.

We let k equal a negative number, like -1 000 000, in f'' and discovered that to the left of 2/k, f'' is positive, so f is concave up.

We let k equal a positive number, like 1 000 000, in f'' and discovered that to the right of 1/k, f'' is negative, so f is concave down.



Remember, k is a constant, not a variable!

In the previous slide, we evaluated f'' when 2/k>0.

In this slide, we evaluated f'' when 2/k<0. style="font-style: italic;">


Synthesizing all that we know, there is that list of properties f has.

• We know where f is increasing or decreasing at certain intervals.
• We know f's concavity.
• We know where f changes sign.
• We know f's local extema.

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END NOTES:

• Homework: the rest of 5.5 Antiderivatives
• Looking forward to the rest of the presentations tomorrow!
• Ending our scribe for 2008 is Francis.
• Reminder tomorrow's class will range from 10 to 25 minutes, with the Holiday Inn Gym Riot, starting at 10:30 AM, hosted by the DMCI Student Council. We will be called down to the gym between 10:15 AM to 10:30 AM. See you there! =)