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Copy file name to clipboardExpand all lines: CalculusII.tex
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\setboolean{calc2included}{true}
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\part{Calculus II}
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\ifthenelse{\boolean{bsc}}{
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\apexchapter{Applications of Integration}{chapter:app_of_int}
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We begin this chapter with a reminder of a few key concepts from \autoref{chapter:integration}. Let $f$ be a continuous function on $[a,b]$ which is partitioned into $n$ equally spaced subintervals as
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\[a=x_0 < x_1 < \cdots < x_{n-1}<x_n=b.\]
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Let $\Delta x=(b-a)/n$ denote the length of the subintervals, and let $c_i$ be any $x$-value in the $i^\text{ th}$ subinterval. \autoref{def:rie_sum} states that the sum
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\[\sum_{i=1}^n f(c_i)\Delta x\]
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is a \textit{Riemann Sum.} Riemann Sums are often used to approximate some quantity (area, volume, work, pressure, etc.). The \textit{approximation} becomes \textit{exact} by taking the limit
\autoref{thm:riemannSum} connects limits of Riemann Sums to definite integrals:
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\[\lim_{n\to\infty} \sum_{i=1}^n f(c_i)\Delta x = \int_a^b f(x)\ dx.\]
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Finally, the Fundamental Theorem of Calculus states how definite integrals can be evaluated using antiderivatives.
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This chapter employs the following technique to a variety of applications. Suppose the value $Q$ of a quantity is to be calculated. We first approximate the value of $Q$ using a Riemann Sum, then find the exact value via a definite integral. We spell out this technique in the following Key Idea.
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\keyidea{idea:app_of_defint}{Application of Definite Integrals Strategy}
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{Let a quantity be given whose value $Q$ is to be computed.\index{integration!general application technique}
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\begin{enumerate}
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\item Divide the quantity into $n$ smaller ``subquantities'' of value $Q_i$.
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\item Identify a variable $x$ and function $f(x)$ such that each subquantity can be approximated with the product $f(c_i)\Delta x$, where $\Delta x$ represents a small change in $x$. Thus $Q_i \approx f(c_i)\Delta x$.
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%% A sample approximation $f(c_i)\Delta x$ of $Q_i$ is called a \textit{differential element}.
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\item Recognize that $\ds Q\approx\sum_{i=1}^n Q_i = \sum_{i=1}^n f(c_i)\Delta x$, which is a Riemann Sum.
This Key Idea will make more sense after we have had a chance to use it several times. We begin with Area Between Curves.%, which we addressed briefly in \autoref{sec:FTC}.
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\input{text/07_Area_Between_Curves}
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\input{text/07_Disk_Washer_Method}
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\input{text/07_Shell_Method}
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\input{text/07_Work}
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\input{text/07_Fluid_Force}
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}{}
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\ifthenelse{\boolean{isEarlyTrans}}{}{%
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\apexchapter{Inverse Functions and L'H\^opital's Rule}{chapter:diff_conc}
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% todo write a Inverse Functions prerequisite section reviewing logs
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My to do:
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5.2: use something other than $\Delta x$ to represent a partition
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obsolete pre_package
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check units in center of mass calculations
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\exerciseset -> \begin{exerciseset} 423 times
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\example -> \begin{example} 596 times
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egrep -c '\dft\W' text/*tex | grep -v ':0$'
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vspace to addvspace, [small|med|big]skip to [small|med|big]break ?
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Typos:
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1.5#19: x>0 should be x\ne0
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3.1 Theorem 3.1.1: $I$ should be a *finite* closed interval.
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3.4#3,4 p157: "Is is" should be "Is it" (twice).
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Example 3.5.2 Solution item 2: $(x^2+x-6)^2$ should be $(x^2-x-6)^2$.
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4.1#2: Newtons’ should be Newton’s.
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5.5 p251, line 7: the top corners of each trapezoid *lie* on the graph.
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6.3 p298: "vise-versa" should be "vice versa".
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6.8 p344 line 6: $\lim_{t\to0^-}\left(-\dfrac1t+1\right)$ should be $\lim_{t\to0^-}\left(-\dfrac1t-1\right)$.
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6.8 p349 Figure 6.8.12: The graph labels should be switched.
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7.5 p387 para 2: Three occurrences of “Newton” should be “newton”.
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7.5#1: Newton-meter should be newton-meter.
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7.5#27: “an truncated” should be “a truncated”.
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8.2 p422 line 7: $\left(\dfrac12\right)^2$ should be $\left(\dfrac12\right)^n$.
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8.2 p423 Definition 8.2.3: If a<0 and p is non-integral, then (an+b)^p may not be defined. You could assume that it's a p-series only when it's always defined, but it may be easier to just assume that a>0 and an+b≠0.
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8.4 Example 8.4.3#2: The summation should begin at 2.
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8.5 p452 Theorem 8.5.2: "partial sums" should be "partial sum".
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8.5 p452 Example 8.5.2: "sum" should be "sums"
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8.5 p453 line -6: The final inequality with $L$ is in the wrong order.
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8.7 p474 line -3: Maclaurin series should be Maclaurin polynomial.
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8.7 Theorem 8.7.1: The proof needs I to be an open interval. The maximum is then over all z between x and c, and requires that f^{(n+1)} be continuous. Changing the interval for the maximum simplifies Examples 8.7.3#1,2 & 8.7.5#3 and Exercises 8.7#23-25. 8.7#26 has other problems.
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8.7#21-24: The error bounds in the solution do not need \pm.
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8.7#26: The nth derivative is not bounded in n. Correct solution: The $n^\text{th}$ derivative of $f(x)=\sqrt x$ has a maximum on $[3,4]$ of $(2n-3)!!(-1)^{n+1}3^{1/2}6^{-n}$. Thus $\abs{R_n(3)}\leq \frac{3^{1/2}2^{-2}}{3^n n(n+1)}$. When $n=5$, this is less than $0.0001$.
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8.8 p488: Analyticity lets you go from a function to its power series. To go the other direction, you need the converse theorem. Something like: "If a function is equal to some power series on an interval, then that power series is the Taylor series of the function." Eg, Example 8.8.5 gives a power series for e^x cos x, but we don't have a way to know that it is the Taylor series.
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8.8 p494 line 8: "terms shown above *gives* the approximation"
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9.4 #5 The solution should specify the comparison ``for all $n\ge 3$''.
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8.4 # 23, 24, 31, 33: ditto
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8.8 Example 8.8.4, line -2: When -1<k<0, the interval of convergence is (-1,1]. (Indeed, (1+x)^k isn't defined when x=-1 and k<0.)
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9.4 p540 para 2, line 4: "It's rectangular" should be "Its rectangular".
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9.4 p542 para 4, line 3: 2\textsuperscript{nd} quadrant should be 1\textsuperscript{st} quadrant.
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9.5 Example 9.5.7: The parameter starts as t, but then switches to theta.
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9.5 # 7b: In the solution, you may want to have $(2/\pi)$ and $(\pi/2)$ in parentheses.
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12.8 Example 12.8.7 p757 line 5: these are simultaneously 0 only when $w=0$, with consequent changes in plurality in the next sentence.
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12.8 Example 12.8.7 p757 para4: This gives a volume of approximately 20,347.
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14 p840 line 9: "the one is well" should be "then one is well".
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14.3 p865 line 5: ``A region is simply connected\ldots'' should be ``A connected region is simply connected\ldots''.
Copy file name to clipboardExpand all lines: errata/Errata.tex
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\section*{Digital Math Resources}
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We are pleased to announce that the 3d files are now available online, linked from the book's distribution page at \url{http://arts-sciences.und.edu/math/calc-1-texts.cfm} (the page currently links to \url{https://sites.und.edu/timothy.prescott/apex/prc/}, but that may change in the near future). This makes it possible to use the mobile app ``3D PDF Reader'' by Tech Soft 3D to view the files on an Android or iOS device. Appropriate links to the apps are provided on the page. A single 3d file is several hundred kilobytes, so this could zap your data allotment if you're not careful.
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We are pleased to announce that the 3d files are now available online, linked from the book's distribution page at \url{https://arts-sciences.und.edu/academics/math/calc-1-texts.html} (the page currently links to \url{https://sites.und.edu/timothy.prescott/apex/prc/}, but that may change in the near future). This makes it possible to use the mobile app ``3D PDF Reader'' by Tech Soft 3D to view the files on an Android or iOS device. Appropriate links to the apps are provided on the page. A single 3d file is several hundred kilobytes, so this could zap your data allotment if you're not careful.
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