Analysis I, Fall '16 (164)

"They that can give up essential liberty to obtain a little temporary safety
deserve neither liberty nor safety"   — Benjamin Franklin. (1706-1790)

Homework List

✈ Jump down to 'This Week.'
✈ Jump to the Class Notes directory.

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Week 1

• Monday, Aug 22 — First day of class
  ¤ Read §1.1, 1.2.
  ◊ Pg 36. #1.1, 1.2, 1.4, 1.5, 1.7, 1.8, 1.9, 1.10, 1.12, 1.13, 1.14.
  
  

Week 2

• Monday, Aug 29
  ◊ Pg 36. #1.15, 1.16, 1.17, 1.19.
  ¤ Read §1.1, 1.3.
  
  

Week ?

• Monday, Sept 5
  ¤ No class - Labor Day
  
  

Week 3

• Monday, Sept 12
  ◊ Assignment to turn in: ¡See bottom of Week 4!
  ◊ Pg 36. #1.21, 1,22, 1.24, 1.25, 1.26, 1.27, 1.31, 1.33, 1.34, 1.35, 1.37, 1.38, 1.40 (can use a TI Nspire or 89/92, XCas, etc.), 1.44, 1.50, 1.52, 1.54, 1.55.
  ¤ Optional reading: “Mathematical Beliefs and Conceptual Understanding of the Limit of a Function,” J. E. Szydlik, JRME.
  
  

Week 4

• Monday, Sept 19 — «Avast ye, mateys! Talk like a pirate for a free Krispy Kreme!»
  ¤ Asynchronous Class Session 1
  
  • §1.4. Integration (Quadrature or Numerical Integration)
    • Read §1.4.
    • Watch and critique: Trapezoidal Riemann Approximations from the Khan Academy.
    • Read and critique: Simpson's Rule from Interactive Mathematics. Especially look at the 'proof'.
    • Investigate TI's numerical integration algorithms.
    • Attempt to numerically evaluate the integral \[ \mathcal{F} = \int_0^1 \frac1x \cdot \cos\left(\frac{\ln(x)}{x}\right)\,dx. \] Compare the values given by TI-84, TI NSpire, Maple, and Wolfram | alpha (Mathematica).
  
  
  • §1.5. Sequences and Series (Sequences)
    • Read §1.5.
    • Watch and critique: Convergent and Divergent Sequences from the Khan Academy.
    • Read and critique: Convergence Of Series from Paul Dawkins at Lamar Univ.
    • Write the negation of "a sequence \(\{a_n\}\) converges" in terms of \(\varepsilon\) and \(N\).
    • Choose a number \(A\in\mathbb{R}\). Change the expression under the square root in \[ r_n = \dfrac{\sqrt{n^2+n}-n}{n} \] so that \(\lim\limits_{n\to\infty} r_n = A\).
  ◊ Pg 36. #1.50, 1.51, 1.52, 1.53, 1.54, 1.55, 1.60ab, 1.61, 1.62, 1.63.
  
  ¤ To turn in Oct 3:
  • Pg. 36: #1.47 b, 1.48, 1.57, 1.59, 1.60 c
  ¤ Optional reading: «History of Integration» by B Kalio
  
  

Week 5

• Monday, Sept 26
  ◊ Pg 36. #1.64 a, 1.66, 1.67, 1.69, 1.70, 1.71, 1.72
  ¤ Read §2.1.
  
  ¤ Extra Credit Problem Set: Sum over \(\mathbb{N}\)
  

Week 6

• Monday, Oct 3
  ◊ Pg 117. #2.1, 2.3, 2.4, 2.5, 2.6, 2.7, 2.9.
  
  

Week 7

• Monday, Oct 10
  ◊ Pg 117. #2.9 (reprise), 2.10, 2.11, 2.13, 2.14, 2.15.
  ¤ Calculate \[ \displaystyle L = \lim_{x\to \pi/4} \frac{\sin(x) -\cos(x)}{\cos(2x)} \]
  1. By factoring;
  2. By using the Sandwich Theorem;
  3. By using l'Hôpital's Rule.
  
  

Week 8

• Monday, Oct 17
  ¤ Asynchronous Class Session 2
  1. §2.2. Continuity
     • Read §2.2.
     • Describe the differences between the "elementary calculus definition" of continuity and our "analysis definition."
     • Explain how to prove Thomae's function is continuous on the irrationals.
     • Explain why \(f\)'s discontinuity in Thm 2.14 (pg 57) must be a jump discontinuity.
     • Define \(S(x)\) by \[ S(x) = \begin{cases} x & x<1 \\ 0 & x = 1 \\ x-2 & x>1 \end{cases}. \] Why does \(S\) not contradict Thm 2.18 (pg 58)?
     • Look up «Bernstein polynomials», and plot the first several over \([0,1]\).
     • Explian the difference between Continuity and Uniform Continuity.
     • Is \(f(x)=x^2\) uniformly continuous on
       1. \([0,1]\)?
       2. \((0,1)\)?
       3. \((0,\infty)\)?
     • Give an example of a function \(h(x)\) that is continuous on \((-1,+1)\), but not uniformly continuous there.
     
     
  2. §2.3. Differentiation
     • Read §2.3.
     • Define \(H(x)=\begin{cases} 1 & x\ge0 \\ 0 & x<0 \end{cases}\).
       
       
       1. Compute \(\lim\limits_{x\to0+} \dfrac{H(x)-H(0)}{x-0}\).
          
          
       2. Compute \(\lim\limits_{x\to0-} \dfrac{H(x)-H(0)}{x-0}\).
          
          
       3. What is \(H^{\prime}(0)\)?
     • Write a description of Darboux's Intermediate Value Theorem for Derivatives appropriate for an elementary calculus class.
     • Why can't \(H(x)\) from above have an elementary antiderivative?
     
     
  3. Pg 117; #2.16, 2.19, 2.23, 2.24, 2.25, 2.27, 2.29, 2.34, 2.35, 2.37 (prove informally).
  
  

Week 9

• Monday, Oct 24
  ◊ Pg 117. #2.38, 2.39 (using Darboux sums), 2.40 (choose two of the properties), 2.43 (a good 'comp problem'!). Look at: 2.42.
  
  

Week 10

• Monday, Oct 31 — Allhallond-Eue!
  ◊ Pg 117. #2.44, 2.45, 2.47, 2.48.
  ¤ Midterm test.
  
  

Week 11

• Monday, Nov 7 — Election Day Eve; don't forget to vote!
  
  ¤ Asynchronous Class Session 3
  1. §2.5 Sequences, Series, and Convergence Tests
     • Read §2.5.
     • When we defined limits before, we needed \(a\) to be a limit/accumulation point of the domain of \(f\). Why do we *not* need this restriction for the limit of a sequence/series?
     • Give a formal definition for "A Property \(P(n)\) holds eventually iff ...".
     • What is a « lion in the desert » proof?
     • Give an example of a sequence of rational numbers that converges, but not to a rational number.
     • Write verbal definitions of \(\limsup\) and \(\liminf\).
     • Determine the number of terms needed so that \(\displaystyle\sum_{k=1}^n \frac1k = \) (your age).
     • Raabe's convergence test is useful when the ratio and root tests fail. Write Raabe's test as a theorem in the same fashion as the ratio and root tests.
  
  
  2. §2.6. Pointwise and Uniform Convergence
     • Read §2.6.
     • Explain why each partial sum in Example 2.23 (pg 105) is continuous. Why is the pointwise limit discontinuous?
     • Describe how Thm 2.7 (pg 109) fits with our theme of "commutative operators".
     • Determine the maxima for \(f_n\) of Example 2.25 (pg 110).
     • Look up Bernstein polynomials. How do they relate to Thm 2.70 (pg 112)?
  
  
  ◊ Pg 117. #2.50, 2.52, 2.53, 2.56, 2.57, 2.58, 2.62, 2.63, 2.64, 2.65.
  ¤ Optional Reading: "Revival of Infinite Series" in D. E. Smith's History of Mathematics, Vol 2.

Week 12

• Monday, Nov 14
  ◊ Pg 117. #2.66, 2.67, 2.68, 2.71, 2.72, 2.73, 2.74.
  ¤ We know that a \(p\)-series \(\displaystyle\sum_{k=1}^\infty \frac{1}{k^p}\) converges for \(p>1\) — where \(p\) is a constant.
  1. Investigate \(\displaystyle\sum_{n=1}^\infty \frac{1}{n^{1+1/n}}\). \(\quad\) Hint: Show \(n^{1+1/n} < 2n\), then use a comparison test.
  2. Investigate \(\displaystyle\sum_{n=1}^\infty \frac{1}{n^{1+2\ln(\ln(n))\,/\ln(n)}}\). \(\quad\) Hint: To simplify the denominator: first \(n^{a+b}=n^a\cdot n^b\), then substitute \(n=e^a\), so that \(a=\ln(n)\). And do a bunch of algebra.
  3. What can you conlude about the following
     Conjecture: If \(a_n>0\) and \(a_n\to0\), then \(\displaystyle\sum_{n=1}^\infty \frac{1}{n^{1+a_n}} \) converges.
  
  

Week 13

• Monday, Nov 21
  
  ¤ Asynchronous Class Session 4
  1. §2.6. Pointwise and Uniform Convergence
  • Power Series
    1. Suppose a power series \(\displaystyle S(x)=\sum_{k=0}^\infty a_k x^k\) converges for a specific value \(x^*\!\). Prove that \(S(x)\) converges uniformly for all \(x\) s.t. \(|x|\le|x^*|\).
    2. Fill in the details in the proof of Theorem 2.72 (pg 114).
    3. Determine the radius of convergence for the following power series:
       1. \(\displaystyle \sum_{k=1}^\infty \frac{x^n}{n+\ln(n)} \)
          
          
       2. \(\displaystyle \frac{\,x^2}{1\cdot2} - \frac{\,x^3}{2\cdot3} + \frac{\,x^4}{3\cdot4} - \cdots \)
          
          
       3. \(\displaystyle \sum_{k=1}^\infty (-1)^{k+1} \frac{x^{2k-1}}{2k-1} \) (Find an elementary function that this series represents.)
  • Verify the expansion of the following elliptic integral: \[ \int_0^{\pi/2} \frac{dt}{\sqrt{1-k^2\sin^2(t)\,} } = \frac{\pi}{2}\left[ 1 + \left(\frac{1}{2}\right)^2 k^2 + \left(\frac{1\cdot3}{2\cdot4}\right)^2 k^4 + \left(\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\right)^2 k^6 + \cdots \right] \] for \(|k|\le 1\).
    Hint: Use the binomial series to expand \[ \frac{1}{\sqrt{1-k^2\sin^2(t)\,}\,} = 1 + \frac{1}{2}\cdot k^2\sin^2(t) + \frac{1\cdot3}{2\cdot4}\cdot k^4\sin^4(t) + \frac{1\cdot3\cdot5}{2\cdot4\cdot6}\cdot k^6\sin^6(t) + \cdots \]
  • YANF: Yet Another Nasty Function:
    Let \(f_1(x)\), \(f_2(x)\), \(f_3(x)\), \(\dots\) be a sequence of sawtooth functions defined on \(\mathbb{R}\) such that
    1. \(f_n\) is made up of line segments of slope \(\pm1\)
       
       
    2. \(f_n(x) = 0\) for \(x = \pm m\cdot 4^{-n}\) for \(m = 0,1,2,\dots\)
       
       
    3. \(f_n(x) = \frac12\cdot4^{-n}\) for \(x = \frac12\cdot4^{-n} + m\cdot4^{-n}\) for \(m = 0,1,2,\dots\)
       
       
    Define \(\displaystyle f(x) = \sum_{k=1}^\infty f_k(x) \).
    1. Show that \(f\) is continuous on \(\mathbb{R}\).
    2. Show that \(f\) fails to have a derivative anywhere on \(\mathbb{R}\).
    
    

    Plot of \(f_1, f_2, f_3\)	Plot of \(S3 = f_1+f_2+f_3 \)

    Maple worksheet to generate the plots.
  
  

Week 14

• Monday, Nov 28
  ◊ Case Study - TI Calculators.
  ◊ Gauss and Gauss-Kronrod Quadratures.
  
  

Week 15 — Final Exam Week    ♞  (This Week)

• Monday, Dec 5
  ¤ Final exam.
  ¤ Final Exam solutions are due today.
  ◊ Project presentations are today. Grading rubric.
  
  

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