Thursday, March 18, 2010
IB Math Sl (Class 65)
Lesson Title
Lesson 19: Optimization (4)
Overview
In today’s class students work many optimization problems.
Textbook Sections
Vocabulary
function
independent variable
dependent variable
with respect to
rate of change
limit
derivative
difference quotient
derivative from first principals
Key Attitudes
Willingness to work as a group to help meet individual and group goals.
Enduring Understandings
Change is fundamental to understanding functions.
Mathematical relationships can be represented in 4 main ways: Graphical, Numerical, Algebraic, Verbal (written and oral).
Essential Question
What has to be true about the value of a derivative in order to have a maximum or minimum value?
If the derivative of a function is zero, does this always represent a maximum or minimum?
What information does the first derivative give me about the original function?
What information does the second derivative give me about the original function?
Key Knowledge
The derivative of a function can be used to find the optimal solution to a problem.
If the derivative of a function changes sign around a critical point, the function is said to have a local (relative) extremum at that point.
If the derivative changes from positive (increasing function) to negative (decreasing function), the function has a local (relative) maximum at the critical point.
If the second derivative is positive at a point, the graph is concave up. If the second derivative is positive at a critical point, then the critical point is a local minimum.
If the second derivative is negative at a point, the graph is concave down. If the second derivative is negative at a critical point, then the critical point is a local maximum.
An inflection point marks the transition from concave up and concave down. The second derivative will be zero at an inflection point.
The second derivative may be used to determine local extrema of a function under certain conditions. If a function has a critical point for which f′(x) = 0 and the second derivative is positive at this point, then f has a local minimum here. If, however, the function has a critical point for which f′(x) = 0 and the second derivative is negative at this point, then f has local maximum here. This technique is called Second Derivative Test for Local Extrema.
Key Skills
I can make and label a diagram to represent a situation.
I can identify the key variables in a situation.
I can mentally model what is going on in a situation.
I can create equations relating the key variables in a situation.
I can create an equation between the two main variables in a situation.
I can find and use a derivative to determine an optimal solution for a situation.
I can use multiple representations to determine if a function has a relative maximum or a relative minimum on a given interval.
I can use multiple representations to determine if a function is concave up or concave down on a given interval.
I can use multiple representations to determine if a function has an inflection point on a given interval.
Turn-In (#-1)
PS 19 and or Past Paper
Handouts
No Handouts Posted
Assignment
PS 19
Disclaimer- The assignment as stated in class is the official assignment. Every effort is made to keep this posting accurate, but you should refer to what was stated in class as the final word.
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