The Derivative

1
The Derivative
2
Objectives

• Students will be able to
• Use the Newtons Quotient and limits process to
  calculate the derivative of a function.
• Determine the slope of the tangent line to a
  given curve when shown a graph.
• Find the equation of the secant line through two
  points on a function.
• Find the equation of the tangent line to a given
  function through a specified point.
• Solve problems involving the derivative.

3
Newtons Quotient and limits process for finding
the derivative

• Add h (where ) to a and compute f(a
  h).
• Compute the change in the function value
  .
• Calculate and simplify Newtons quotient
  
• for .
• Calculate the limit

4
Example 1

• Estimate the slope of the tangent line to the
  curve in the graph to the right.

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Example 2

• Estimate the slope of the tangent line to the
  curve in the graph to the right.

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Example 3

• Estimate the slope of the tangent line to the
  curve in the graph to the right.

7
Example 4

• Find each of the following for the function
  below

8
Example 5

• Find each of the following for the function
  below

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Example 6

• Find each of the following for the function
  below

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Example 7

• For the function below, find the equation of the
  secant line through the points x 3 and x 5.

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Example 8

• For the function below, find the equation of the
  tangent line through the point x 3.

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Example 9

• For the function below, find the equation of the
  secant line through the points x 25 and
• x 36.

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Example 10

• For the function below, find the equation of the
  tangent line through the point x 25 .

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Example 11

• Find the x-values where the function to the right
  is not differentiable.

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Example 12

• For the function shown in the sketch to the
  right, give the intervals (or points) on the
  x-axis where the rate of change of the function
  f(x) with respect to x is positive, negative, and
  zero.

16
Example 13

• Identify which graph below represent velocity and
  which represents distance.

Graph 1
Graph 2
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Example 14

• The cost in dollars of producing x tacos is ,
• for .

1. Find the exact cost of producing the 101st taco.
2. Use the marginal cost to approximate the cost of
   producing the 101st taco.