Tangent Line Approximations

1
Tangent Line Approximations

• All functions can be approximated by a linear
  function using a tangent line.

2
Tangent Line Approximations

• Function values can be approximated by the
  Tangent Line function values.
• The closer the point on the tangent line is to
  the point on the function, the better the
  approximation.

3
Tangent Line Approximations

• Find the equation of the tangent line.
• Calculate f(3).
• Calculate T(3).
• Compare the results.
• Which calculation was easier?
• Calculate f(2.1) and T(2.1).
• Compare the results.

4
Differentials

• Differentials ? Derivative

5
Differentials

• A derivative the slope of a function.
• A derivative the slope of the tangent line to a
  curve at a specific point.
• A derivative
• The differentials, dy and dx are the parts
  forming the derivative.

6
Differentials

• ?y and ?x indicate the vertical and horizontal
  change of a secant line through two points on a
  curve.
• dy and dx indicate the vertical and horizontal
  change of the tangent line through ONE point on a
  curve.

7
Differentials

• dx a small positive number
• (the smaller the better)
• dy f (x) dx

8
Differentials

• Find dy at x 2, given dx ?x.1
• Find ?y at x 2, given dx ?x .1
• Which calculation is easier?