Theoretical Issues
https://www.maa.org/taxonomy/term/40475/all
enAnimating Nested Taylor Polynomials to Approximate a Function
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/animating-nested-taylor-polynomials-to-approximate-a-function
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The authors provide a condition for a function to have nested \(n\)-th degree Taylor polynomials with varying centers, which can approximate the function visually.</em></p>
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Thu, 11 Jul 2013 18:55:03 +0000
lang128063 at https://www.maa.orgOn A Mean Value Theorem
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/on-a-mean-value-theorem
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><em>The author proves a somewhat different Mean Value Theorem and uses it to obtain various facts about the curve.</em></div></div></div>
Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95035 at https://www.maa.orgOn Antiderivatives of the Zero Function
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/on-antiderivatives-of-the-zero-function
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The author discusses an intuitive direct proof of the fact that functions with zero derivative must be constant, which turns into a rigorous proof by simply invoking the completeness of the real numbers.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95707 at https://www.maa.orgDivergence of Series by Rearrangement
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/divergence-of-series-by-rearrangement
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Assume that a series with non-negative terms converges to \(S\). If from this assumption, a rearrangement of the series can be shown to converge to a different value \(S'\), then the original series must have diverged. The author uses this result to show that a number of series diverge.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95045 at https://www.maa.orgArc Length and Pythagorean Triples
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/arc-length-and-pythagorean-triples
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>A family of curves whose lengths are closely related to Pythagorean Triples and therefore rational</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95740 at https://www.maa.orgThe \(n\)th Derivative Test and Taylor Polynomials Crossing Graphs
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-nth-derivative-test-and-taylor-polynomials-crossing-graphs
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The author describes \(n\)<sup>th</sup> derivative test, a generalization of 2nd derivative test, then applies it to give a quick proof of a condition for determining when the graph of a function and its Taylor polynomials intersect.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95048 at https://www.maa.orgThe Convergence Behavior of \(f_{\alpha}(x)=(1+\frac{1}{x})^{x+\alpha}\)
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-convergence-behavior-of-falphax1frac1xxalpha
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><em>Descriptions of the behavior of a family of functions</em></div></div></div>
Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95749 at https://www.maa.orgConvergence-Divergence of \( p\) -Series
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/convergence-divergence-of-p-series
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article is a discussion of convergence and divergence of \(p\)-series without using the Integral Test.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95051 at https://www.maa.orgA General Form of the Arithmetic-Geometric Mean Inequality via the Mean Value Theorem
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-general-form-of-the-arithmetic-geometric-mean-inequality-via-the-mean-value-theorem
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><em>A weighted arithmetic/geometric mean inequality is proved.</em></div></div></div>
Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95779 at https://www.maa.orgDisks, Shells and Integrals of Inverse Functions
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/disks-shells-and-integrals-of-inverse-functions
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><i>Most calculus students take it for granted that the shell and disk methods for computing the volume of a solid of revolution must always give the same result.</i></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com94733 at https://www.maa.orgInvestigating Possible Boundaries Between Convergence and Divergence
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/investigating-possible-boundaries-between-convergence-and-divergence
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Once students master the Integral Test, it is useful to show that there cannot be a series on the boundary between convergence and divergence.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95054 at https://www.maa.orgAn Optimization Oddity
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/an-optimization-oddity
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><i>"This problem also fits quite nicely in the area of convex geometry, from which we draw the following proof..."</i></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com94736 at https://www.maa.orgOn the Indeterminate Form \(0^0\)
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/on-the-indeterminate-form-00
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><em>This article gives an explanation of why most textbook examples of the form \(0^0\) have a limit of 1.</em></div></div></div>
Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95060 at https://www.maa.orgA Generalization of the Mean Value Theorem for Integrals
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-generalization-of-the-mean-value-theorem-for-integrals-0
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>A generalization of the Mean Value Theorem for Integrals in terms of Riemann sums</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com94753 at https://www.maa.orgOff on a Tangent
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/off-on-a-tangent
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><em>The authors investigate families of curves whose tangent lines at a fixed \(y\)-coordinate go through the origin.</em></div></div></div>
Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95064 at https://www.maa.org