THE DERIVATIVE AND THE TANGENT LINE PROBLEM

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THE DERIVATIVE AND THE TANGENT LINE PROBLEM

• Section 2.1

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When you are done with your homework, you should
be able to

• Find the slope of the tangent line to a curve at
  a point
• Use the limit definition to find the derivative
  of a function
• Understand the relationship between
  differentiability and continuity

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The Tangent Line Problem

• How do we find an equation of the tangent line to
  a graph at point P?
• We can approximate this slope using a secant line
  through the point of tangency and a second point
  on the curve.

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Find the equation of the secant line to the
function at and

• Y -5x 19
• Y 5x - 11
• There is not enough information to solve this
  problem.

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A secant line represents the

• Instantaneous rate of change of a function.
• The average rate of change of a function.
• Line tangent to a function.

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Definition of the Derivative of a Function

• The derivative of f at x is given by
• provided the limit exists. For all x for which
  this limit exists, f is a function of x.

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Definition of Tangent Line with Slope m

• If f is defined on an open interval containing c,
  and if the limit
• exists, then the line passing through f with
  slope m is the tangent line to the graph of at
  the point
• The slope of the tangent line to the graph of f
  at the point c is also called the slope of the
  graph of f at

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Find the slope of the graph of at

• 4
• 9
• 1
• Does not exist

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Alternative limit form of the derivative

• The existence of the limit in this alternative
  form requires that the following one-sided limits
  
• and
  
• exist and are equal.
• These one-sided limits are called the derivatives
  from the left and from the right, respectively.
  It follows that f is differentiable on the closed
  interval if it is differentiable on and
  if the derivatives from the right at a and the
  derivative from the left at b
  
• both exist.

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Evaluate the derivative of

• -1
• 0
• 1
• Does not exist

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THEOREM Differentiability Implies Continuity

• If f is differentiable at
• then f is continuous at