2'1 The Derivative and the Slope of a Graph

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2.1 The Derivative and the Slope of a Graph
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To determine the rate at which a graph rises or
falls at a single point, you can find the slope
of the tangent line.
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Slope of a graph

• The problem of finding the slope of a graph at a
  point becomes one of finding the slope of the
  tangent line at the point.

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A more precise method of approximating the
tangent lines makes use of a secant line through
the point of tangency and a second point on the
graph.
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Slope of a Secant Line
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• You obtain better and better approximations of
  the slope of the tangent line by choosing the
  second point closer and closer to the point of
  tangency.
• Using the limit process, you can find the exact
  slope of the tangent line at (x,f(x)).

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Definition of the Slope of a Graph
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The equation that is derived from the limit
process that represents the slope of the graph of
f at the point (x,f(x)) is called the derivative
of f at x. It is denoted by f (x), which is
read as f prime of x.
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Definition of the Derivative
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In addition to f (x), other notations to denote
the derivative of y f(x) are
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Not every function is differentiable.

• A function will not be differentiable at a point
  in the following situations.
• Vertical Tangent Lines,
• Discontinuities, and
• Sharp Turns in the Graph.

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Continuity is not a strong enough condition to
guarantee differentiability.

• If a function is differentiable at a point, then
  it must be continuous.

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Differentiability Implies Continuity

• If a function is differentiable at x c, then it
  is continuous at x c.

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• http//clem.mscd.edu/talmanl/MOOVs/MovingSecantLi
  ne/MovingSecantLine.MOV
• http//clem.mscd.edu/talmanl/MOOVs/MovingSlopeTri
  angle/MovingSlopeTriangle.MOV

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http//www.howardcc.edu/math/MA145/2.1/2.1.htm