Applications of Derivatives

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4.4 Antiderivatives

The process of antidifferentiation takes a derivative, and asks, "What would original function would you differentiate to make this derivative?"  You will not be able to find the original function unless a coordinate is provided, as each anti-derivative includes a constant C.  However, we can still find important features of the function, such as where extrema occur, or where points of inflection may occur.  Today we will start looking at graphing functions based on information obtained from the first and second derivatives.

Resources:

  • Notes

Assignment:

  • p192 #25-37, 43, 45, 47, 51

Things you should know after today:

  • given f'(x), determine the antiderivative
  • given f'(x) and a coordinate on f(x), determine the antiderivative inculding the constant term
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4.5 Relating the graphs of f, f` and f``

The process of antidifferentiation takes a derivative, and asks, "What would original function would you differentiate to make this derivative?"  You will not be able to find the original function unless a coordinate is provided, as each anti-derivative includes a constant C.  However, we can still find important features of the function, such as where extrema occur, or where points of inflection may occur.  Today we will start looking at graphing functions based on information obtained from the first and second derivatives.

Resources:

  • Notes

Assignment:

  • p203 #1-29

Things you should know after today:

  • determine where a function is increasing/decreasing
  • determine concavity of a function
  • determine the location of extrema
  • determine inflection points
Attachments:
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Download this file (4.5.notes.pdf)4.5.notes.pdf 4658 kB

4.8 Optimization Problems

One of the applications of derivatives is to use critical points to identify extrema.  This is often related to finding maximum or minimum values in applications.  While you should determine whether the extrema found is a maximum or minimum, sometimes it is not strictly necessary because in the context of the problem, the type of extrema identified by your critical point is obvious.

Resources:

  • Notes

Assignment:

  • p214 #1-20
  • p216 #25-27, 31-35, 37
  • p218 #41, 42, 47-49, 53

 

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4.11 Differentials

We can estimate the amount of change in a function using a derivative.  We can think of the derivative,

as measuring the change in y and the change in x for very small differences in y and x.  This can be used for determining how much a function changes, provided that the numbers are very small.

Resources

  • Notes

Assignment

  • p229 #19-35, 39-45
Attachments:
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