AP Calculus AB : Definite integral as a limit of Riemann sums

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

Example Question #1 : Definite Integral As A Limit Of Riemann Sums

True or False: If  is a negative-valued function for all ,  

Possible Answers:

False

True

Correct answer:

True

Explanation:

This is true. Since  is negative-valued, its graph is below the -axis, and the Riemann sums used to evaluate the area between  and the -axis have a negative value for height.

Example Question #72 : Integrals

 is a continuous function on the interval  and is differentiable on the open interval .  If , then which of the following statements MUST be true:

Possible Answers:

 over the interval .

 over the interval .

 at some point , where .

 at some point , where .

 over the interval .

Correct answer:

 at some point , where .

Explanation:

According to Rolle's Theorem, if a function is continuous over a closed interval  and differentiable on the open interval , and if , then there has to be some value  such that , where .  To put this another way, if a function is continuous and differentiable over a certain interval, and if the value of the function is the same at both endpoints of that interval, then at some point in between those endpoints, the function is going to have a slope of zero (i.e. its first derivative will be zero).  This does not apply to the second derivative, nor does it require that the slope of the first derivative be zero over the entire interval.

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