Derivatives can be represented'''

1
Derivatives can be represented...

• graphically as the slope of the tangent line to a
  curve at a point or as the slope of the line a
  curve seems to approach under magnification

2
Derivatives can be represented...

• verbally as the instantaneous rate of change

3
Derivatives can be represented...

• physically as speed or velocity

4
Derivatives can be represented...

• symbolically as the limit of the difference
  quotient

5
Why is the formal limit definition of derivative
important?

• necessary to students complete understanding of
  derivative
• for success in future mathematics study
• a mathematical idea or procedure or fact is
  understood when it is part of an internal network
• number and strength of the connections between
  the parts of this network determine the degree of
  understanding of a concept

6
Why is the formal limit definition of derivative
important?
Understanding Derivative
7
Why is the formal limit definition of derivative
important?
Understanding Derivative
8
How can technology/conjecturing be used to
strengthen these connections?

• Given a particular function, draw the derivative
  function
• Make a conjecture about the general derivative
  function
• Use Excel in a dynamic way to verify/adjust their
  conjecture
• Use Computer Algebra System for further support

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Graph the derivative function
Given
Conjecture
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Verify/Adjust Conjecture

• Using Excel
• Student created Excel spreadsheet will reinforce
  the concept of the limit definition of derivative

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Verify/Adjust Conjecture

• Using a Computer Algebra System
• Maple
• Mathcad
• Mathematica
• Derive
• TI-89/92
• TI-Interactive

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Verify/Adjust Conjecture
13
Assessment

• Discuss each of the following questions with your
  group and then provide a written response to
  each.
• 1. In the limit definition of derivative, what
  role does the difference quotient play in the
  computation of the derivative function?
• 2. In the limit definition of derivative, what
  role does the limit as approaches 0 play in the
  computation of the derivative function?

14
Assessment
3. Generally speaking, the derivative of a a)
constant function is _____________________. b)
linear function is _______________________. c)
quadratic function is ____________________. d)
cubic function is _______________________. e)
quartic function is ______________________. f)
exponential function is ___________________. g)
sine function is _________________________. h)
cosine function is ______________________. i)
natural logarithm function is ______________. j)
absolute value function is ________________.