Chapter 3 Derivatives

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Chapter 3 Derivatives
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3.1 Derivative of a Function
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Differentiate using the
alternate definition.
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Relationships between the
Graphs of f and f

• Because we can think of the derivative at a point
  in graphical terms as slope, we can get a good
  idea of what the graph of the function f looks
  like by estimating the slopes at various points
  along the graph of f.
• We estimate the slope of the graph of f in
  y-units per x-unit at frequent intervals. We
  then plot the estimates in a coordinate plane
  with the horizontal axis in x-units and the
  vertical axis in slope units.
• Look at page 103

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p.105 (1-19)odd, (26-28)
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3.2 Differentiability
How f(a) Might Fail to Exist
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• A good way to think of differentiable functions
  is that they are locally linear that is, a
  function that is differentiable at a closely
  resembles its own tangent line very close to a.
• In the jargon of graphing calculators,
  differentiable curves will straighten out when
  we zoom in on them at a point of
  differentiability.
• What is linearization?
• Lets discuss calculator derivatives.

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Differentiability Implies Continuity
The converse is not a true statement.
Intermediate Value Theorem for Derivatives
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p. 114 (1-37) odd
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3.3 Rules for Differentiation
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Find an equation for the line tangent to the
curve at the point (1,2). Find the
first four derivatives of
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p.124 (1-47) odd
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3.4 Velocity and Rates of Change
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Motion Along a Line
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Look at page 129 on how to read a velocity graph.
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When a small change in x produces a large change
in the value of a function f(x), we say that the
function is relatively sensitive to changes in x.
The derivative f(x) is a measure of this
sensitivity.
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p. 135 (1 -39) odd
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3.5 Derivatives of Trigonometric Functions
Lets prove this using the definition of the
derivative.
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Find the derivative of
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The motion of a weight bobbing up and down on the
end of a string is an example of simple harmonic
motion.
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Lets derive the formula for tangent.
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Find yn if
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p. 146 (1-41) odd
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3.6 Chain Rule
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Power Chain Rule
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Ex (a) Find the slope of the line tangent to
the curve at the point where
(b) Show that the slope of every line tangent
to the curve is positive.
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p. 153 (1-39, 53-69)odd skip (57), also do (56,
58)
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3.7 Implicit Differentiation
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Implicit Differentiation Process
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p.162 (1-57) odd and 54
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3.8 Derivatives of Inverse Trigonometric Functions
Derivatives of Inverse Functions
Lets go the exploration of page 166
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How did we get this as a result?
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Identity Functions
Rules for differentiation
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Ex Ex A particle moves along the
x-axis so that its position at any time
Is . What is the
velocity of the particle when t 16? Ex
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p. 170 (1-33) odd
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3.9 Derivative of Exponential and Logarithmic
Functions
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Ex At what point on the graph of the function y
3t 3 does the tangent line have a slope of 21?
How can we show this?
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• Sometimes the properties of logarithms can be
  used to simplify the differentiation process,
  even if logarithms themselves must be introduced
  as a step in the process.
• The process of introducing logarithms before
  differentiating is called logarithmic
  differentiation.

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p.178 (1-55)odd