AP Calculus AB : Geometric understanding of graphs of continuous functions

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

Example Question #1 : Continunity As A Property Of Functions


 is differentiable for which of the following values of

Possible Answers:

Correct answer:


 is not differentiable at  and  because the values are discontinuities.  is not differentiable at  because that point is a corner, indicating that the one-side limits at  are different.  is differentiable:the one side limits are the same and the point is continuous. 

Example Question #2 : Functions, Graphs, And Limits

                                                                             Correct graph

Consider the graph of  above. What can we say about  when  ?

Possible Answers:

None of these are correct.

Two or more of these are correct.

 is discontinuous at  because there is a sharp turn at .

 has a removable discontinuity at .

 has a horizontal tangent at .

Correct answer:

 has a horizontal tangent at .


Note that , indicating that there is a horizontal tangent on  at . More specifically, the derivative is the slope of the tangent line. If the slope of the tangent line is 0, then the tangent is horizontal.

The other two are incorrect because sharp turns only apply when we want to take the derivative of something. The derivative of a function at a sharp turn is undefined, meaning the graph of the derivative will be discontinuous at the sharp turn. (To see why, ask yourself if the slope at  is positive 1 or negative 1?) On the other hand, integration is less picky than differentiation: We do not need a smooth function to take an integral.

In this case, to get from  to , we took an integral, so it didn't matter that there was a sharp turn at the specified point. Thus, neither function had any discontinuities. 

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