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In the remaining exercises, it is assumed you will be using a spreadsheet.Do Exercise 13 with $n=16$.

(a) 20.052734(b) 20.615234(c) 20.333008

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 4

Approximation of Areas

Integrals

Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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In Exercises $13-16,$ refe…

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In the following exercises…

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In Exercises $17-26,$ find…

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In Exercises $13-26,$ find…

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In Exercises $21-26,$ find…

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for this problem, we are asked to determine F prime of A for a equals 124 and seven. So what we want to do is look at our figure up above and figure out what the tangent lines at each one of those points will be. And then we just want the slope of the tangent lines. So we can very clearly see that the tangent line for a equals one would be just a horizontal line. So f prime, there will be zero. Then we want f prime of two to again, is while it is on that sort of just flat area there, which means it's derivative must be zero. We want f prime of four. So we want to draw a tangent line there. We can see that that is a straight line segment. So we can see that we have a rise of one over a run of two. So we can see then that F prime of four is going to equal one half. Then we want f prime of seven. So if we look at seven that are at X equals seven, that would be at the peak of this curved bit there. So the tangent line at X Equal seven would just be another horizontal line. We can see that it's flat at that point, so you have then that at prime of seven will equal zero.

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