MAT 1120 — §103 & 104, Fall '15 (154)

Let us reflect that, having banished from our land that religious intolerance under which mankind so long bled and suffered, we have yet gained little if we countenance a political intolerance as despotic, as wicked, and capable of as bitter and bloody persecutions.

— Thomas Jefferson (1743-1826) in his First Inaugural Address.

Homework List

✈ Jump down to '♞ This Week.'

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Week 1

• Monday, Aug 17 — First day of class
  ◊ Register for Wiley Plus using the correct Course ID: §103 - 11 am or §104 - 12 pm
  ◊ Begin the first Wiley Plus assignment; due Friday by 5:00 pm.
  ¤ Chapter 5 Review Exercises. Pg. 309, #1, 3, 5, 13; 17, 23, 29, 49.
  
  
• Tuesday, Aug 18
  ¤ Chapter 6 Review Exercises. Pg. 345, #1, 3, 7, 11, 17, 23, 39; 51, 53a, 59, 73, 99.
  ◊ Take a look at Paul Dawkins’ “Calculus Cheat Sheet.”
  
  
• Wednesday, Aug 19
  ◊ Group Work: Derivative Review problems; No. 1, 5, 9, ... (every other odd).
  
  
• Friday, Aug 21
  ❎ Quiz 1 is next Friday! — the first chance to demonstrate your excellence!
  ¤ Group Work: Derivative Review problems; No. 2, 6, 10, ... (every other even).
  
  

Week 2

• Monday, Aug 24
  §7.1, pg 360. #1, 5, 9 (every other odd), ..., 77; 79, 81, 89, 91, 93, 114.
  
  
• Tuesday, Aug 25
  §7.1, pg 360. #3, 7, 11 (every other other odd), ..., 73.
  ◊ Check the Quick Reference Card Collection for trig identities and derivatives, etc.
  
  
• Wednesday, Aug 26
  §7.2, pg 368. #3, 9, 19, 23, 27, 29, 35, 39, 41; 43, 49, 61, 85. Challenge problem: 58.
  
  
• Friday, Aug 28
  🎯 Quiz 1 is today! — the first chance to demonstrate your excellence!
  §7.3, pg 374. #3, 13, 15, 20, 29, 47; 53, 61.
  
  

Week 3

• Monday, Aug 31
  ❗Beginning today, the noon class (§104) will meet in WA 310.
  §7.4, pg 384. #1→ 17 (odd); 27.
  
  
• Tuesday, Sept 1
  §7.4. Finish the problems from the Partial Fractions worksheet.
  
  
• Wednesday, Sept 2
  §7.4, pg 384. #21, 23; 55, 57, 63, 69, 71, 83, 87, 89, 90.
  
  
• Friday, Sept 4
  §7.4. Finish the problems from the Trig Substitutions worksheet.
  
  🎓 Challenge Problem.   Integrate \(\displaystyle\int \frac{dx}{1+x^2}\) in two ways:
  
  
  1. With the trig substitution \(x=\tan(\theta)\).
  2. With partial fractions using \(\dfrac{1}{1+x^2} = \dfrac{A}{x-\imath} + \dfrac{B}{x+\imath}\).
  
  

Week 4

• Monday, Sept 7 — Labor Day; no classes.
  
  
• Tuesday, Sept 8
  §7.5, pg 392. #1, 3, 5, 9; 11, 19, 21, 25.
  ◊ Problems (cf., the Numerical Integration slides)
      The \(erf\) function is defined by \(\displaystyle erf(x) = \frac{2}{\sqrt{\pi}}\,\int_0^x e^{-t^2}\,dt\). This integral has no elementary antiderivative.
  1. How big must \(n\) be so that the trapezoid rule approximates the integral for \(erf(5)\) with an error \(\le 10^{-5}\)?
  2. How big must \(n\) be so that Simpson's rule approximates the integral for \(erf(5)\) with an error \(\le 10^{-5}\)?
  3. What is an upper bound for the error using the midpoint rule to approximate the integral for \(erf(5)\) with \(n=10\)?
  4. What is an upper bound for the error using Simpson's rule to approximate the integral for \(erf(5)\) with \(n=10\)?
  
  
• Wednesday, Sept 9
  ◊ Problems (cf., the Numerical Integration slides)
  
  1. Use whatever technology to approximate the following integrals using the stated sums.
  1. \(\displaystyle\int_{0}^{1}\frac{4}{1+x^2}\,dx\) using left sum, right sum, midpoint sum, trapezoid rule, Simpson's rule with \(n = 100\). (Exact Answer: \(\color{purple}{\pi}\))
  2. \(\displaystyle\int_{0}^{1} e^{-x^2} dx\) using left sum, right sum, midpoint sum, trapezoid rule, Simpson's rule with \(n = 100\). (Exact Answer: \(\color{purple}{0.7468241328\dots}\))
     
     
  2. In the following exercises, use the error estimate formulas to give upper bounds for the errors if the given integral is approximated by (a) a trapezoidal sum with \(n = 10\) and (b) Simpson's rule with \(n = 10\).
     1. \(\displaystyle\int_{0}^{2} 2x\,dx\)
     2. \(\displaystyle\int_{0}^{3}\ x^3\,dx\)
     3. \(\displaystyle\int_{1}^{5}\ \ln(x)\,dx\)
     4. \(\displaystyle\int_{0}^{2}\ \sin(x)\,dx\)
        
        
  3. In the following exercises, use the error estimate formulas to give the minimum number \(n\) needed to approximate the given integral by the given kind of sum.
     1. \(\displaystyle\int_{0}^{2} x^4\, dx\quad\) Trapezoid sum; error \(≤ 0.0005\)
     2. \(\displaystyle\int_{0}^{3} x^5\, dx\quad\) Trapezoid sum; \(3\) decimal places
     3. \(\displaystyle\int_{0}^{2} x^4\, dx\quad\) Simpson sum; error \(≤ 0.0005\)
     4. \(\displaystyle\int_{0}^{3} x^5\, dx\quad\) Simpson sum; \(3\) decimal places
        
        
  
  
• Friday, Sept 11
  🎯 Test 1 is next Friday, Sept 18!
      Get started on your note sheet!
  💻 Our first group project, "Definite Integrals & Computing Accuracy," is due next Friday. Each group of two people will submit one report (it maybe submitted via email).
  ¤ To determine \(P\) and \(Q\), copy and paste the text below into Maple, replace phone1, phone2 with your phone numbers (no area code, dashes or spaces), and then hit \(\fbox{\(Enter\)}\).
       ph1,ph2 := phone1, phone2:
       randomize(ph1+ph2):
       P=rand(3..9)();
       Q=rand(3..9)();
     Include this code in your project report.
  ◊ A brief set of Maple notes on ApproximateInt relevant to the project. (Look in the Quick Reference Sheet collection for a larger reference.)
  
  

Week 5

• Monday, Sept 14
  §7.6, pg 401. #1, 3, 5, 9, 11, 23, 27; 35, 39, 43, 47; 49, 55, 61.
  
  
• Tuesday, Sept 15
  §7.6, pg 401. #7, 21, 25; 37, 41.
  
  
• Wednesday, Sept 16
  ¤ Study for the test!
  ◊ Chapter 7 Review Exercises, pg 408.
  
  
• Friday, Sept 18
  🎯 ¡Test 1 is today!    (Test 1 Topics list)
       You can bring:
  • pencils (Check the erasers!)
  • a calculator (Check the batteries!)
  • one \(8.5''\!\times11''\) sheet of paper with any notes/formulas/etc.
  ¤ Your group project report, "Definite Integrals & Computing Accuracy," is due today.
  
  

Week 6

• Monday, Sept 21
  §8.1, pg 421. #19, 21, 25, 27, 29, 31, 35; 39, 43.
  
  
• Tuesday, Sept 22
  ◊ §8.1 Type Problems:
  1. A horizontal line \(y=c\) intersects the curve \(y=8x-27x^3\). Find the number \(c\) such that the areas of the regions between the curve and the line \(y=c\) for \(x\ge0\) are equal. (See figure.)
     
     
  2. A paper drinking cup filled with water has the shape of a cone with height \(h\) and semivertical angle \(\theta\). A ball is placed carefully in the cup, thereby displacing some of the water and making it overflow. What is the radius of the ball that causes the greatest volume of water to spill out of the cup?
     
     
  3. A semicircle with radius 1 is given by \(y=\sqrt{1-x^2}\). The areas between the semicircle and the line \(y=c\) for \(-1\le x\le 1\) define three regions. Find the value of \(c\) that minimizes the sum of the three areas.
  
  
• Wednesday, Sept 23 — Autumn finally catches up to Fall semester!
  §8.2, pg 427. #1, 3, 5, 9, 11, 15, 19, 21, 23; 25, 27, 35, 36.
  
  
• Friday, Sept 25
  ¤ Quiz next Friday.
  §8.2, pg 427. #13, 17, 22, 24; 39, 43, 47, 54, 63, 70 (with \(k=\frac{1}{2}\)).
  
  

Week 7

• Monday, Sept 28
  ¤ Astroid Arclength and Archimedes' Trammel:
  1. Find the arclength of the standard astroid given by \(x^{2/3}+y^{2/3}=1\). Graph the astroid.
  2. Find the arclength of the standard astroid given by \([\cos^3(t),\sin^3(t)]\) for \(t=0..2\pi\). Graph the astroid.
  3. How do the two astroids above compare?
  
  
• Tuesday, Sept 29
  §8.5, pg 456. #1, 2, 3, 5, 9; 11, 35, 45. Look at 24.
  ¤ Work in Physics graphic
  ◊ Redo Example 4 (pg 451) on
  1. Venus where gravity at the surface is \(8.8 m/s^2\);
  2. Mars where gravity at the surface is \(3.72 m/s^2\);
  3. Jupiter where gravity at the surface is \(23.1 m/s^2\).
  
  
• Wednesday, Sept 30
  §8.7, pg 470. #1, 5, 7, 9; 19, 23; 25, 27, 33.
  
  
• Friday, Oct 2
  ¤ Quiz today.
  
  

Week 8

• Monday, Oct 5
  §8.8, pg 478. #1, 3; 5, 17.
  
  
• Tuesday, Oct 6
  ¤ A Maple worksheet with SequenceGrapher
  ◊ Instructions for graphing sequences on a TI-84.
  §9.1, pg 495. #1→11 odd; 13, 17, 21, 25, 29, 37, 43, 48, 59.
  
  
• Wednesday, Oct 7
  §9.2, pg 502. #1→7 odd, 9→13 odd, 23, 25, 30; 41; 48, 49.
  
  
• Friday, Oct 9
  §9.3, pg 510. #1→11 odd; 13→33 every other odd.
  
  

Week 9

• Monday, Oct 12
  §9.3, pg 510. #15→33 every other odd, 37, 46, 47, 50.
  
  
• Tuesday, Oct 13
  
  
• Wednesday, Oct 14
  🎯 ¡Test 2 is today!    (Test 2 Topics list)
       You can bring:
  • pencils (Check the erasers!)
  • a calculator (Check the batteries!)
  • TWO \(8.5''\!\times11''\) sheets of paper with any notes/formulas/etc.
  
  
• Friday, Oct 16 — Fall Break; no classes Thursday and Friday
  
  

Week 10

• Monday, Oct 19
  §9.4, pg 518. #1→9 odd, 39, 41, 61, 69, 103, 107.
  
  
• Tuesday, Oct 20
  §9.4, pg 518. #15→23, 29, 31, 43, 47, 52, 67, 77, 85.
  
  
• Wednesday, Oct 21
  §9.5, pg 518. #1→23 odd, 27, 29.
  
  
• Friday, Oct 23
  ¤ Project day! — no formal class today; work on your report.
  ◊ Download a Maple Koch Snowflake worksheet
  
  

Week 11

• Monday, Oct 26
  §10.1, pg 544. #1→15 odd; 17, 23, 29.
  ¤ Why is «false» the answer to #47?
  
  
• Tuesday, Oct 27
  §10.2, pg 550. #1→15 odd; 17, 19, 23; 27, 35, 43, 47.
  
  
• Wednesday, Oct 28
  §10.3, pg 557. #1→19 odd.
  
  
• Friday, Oct 30
  ◊ Carefully construct concise criticism for the calculations contained in the video we considered today! (See a math transcript.)
  
  

Week 12

• Monday, Nov 3 — Did you reset your clocks Sunday morning? Daylight Saving Time is over until March 16, 2016.
  §10.3, pg 557. #21, 27, 29, 32, 33, 41.
  ¤ Challenge Problem: Find a Taylor series for the function \(f(x)=e^{-1/x^2}\) for \(x\ne0\) and \(f(0)=0\).
  
  
• Tuesday, Nov 4
  ¤ Finish the Taylor Series Worksheet.
  
  
• Wednesday, Nov 5
  §10.4, pg 563. #1→7 odd; 9, 11, 19.
  
  
• Friday, Nov 7
  🎯 Quiz today over §10.1→ 3.
  
  

Week 13

• Monday, Nov 9
  §10.4, pg 563. #13, 15, 17.
  ◊ Determine the value of \(n\) needed to approximate \(f(x)=\sin(x)\) to within \(5\) decimal places for \(x=0.8\).
  
  
• Tuesday, Nov 10
  §10.5, pg 575. #1, 3, 5; 11, 18.
  
  
• Wednesday, Nov 11
  §10.5, pg 575. #2, 4, 6, 7; 23.
  ❎ Suppose \(f\) is a \(2\pi\)-periodic function that has the \(3\)-term fourier series graphed below.
  
  
     Which function below is the original?
  1. \(f_1(x) = (x/\pi)^2\)
     
     
  2. \(f_2(x) = \big|\,x\,\big|\)
     
     
  3. \(f_3(x) = \begin{cases}-1 & -\pi\le x \lt 0 \\ +1 & \phantom{-}0 \le x \lt \pi \end{cases}\)
     
     
  4. \(f_4(x) = \begin{cases} 1+x & -\pi\le x \lt 0 \\ 1-x & \phantom{-}0 \le x \lt \pi \end{cases}\)
     
     
  
  
• Friday, Nov 13 — Paraskevidekatriaphobia alert!
  ❎ Test 3 is next Friday, Nov 20!
  §11.1, pg 589. #1, 5, 7; 19.
  §11.2, pg 594. #5, 9.
  ¤ Drawing slope fields:
  • Online program
  • TI-83/4 program, instructions
  ◊ Draw slope fields including several solution curves for
  1. \(y^\prime = 0.3\cdot y\) Exponential growth
  2. \(y^\prime = 0.3\cdot y\cdot(1-y)\) Logistic growth
  3. \(y^\prime = 0.3\cdot y\cdot(1-y)(y-0.4)\) Logistic growth with threshhold
  4. \(y^\prime = 0.3\cdot y\cdot(1-y) - 0.025\), \(y(0)=0.1\) Logistic growth with harvesting
  5. \(y^\prime = 0.3\cdot y\cdot(1-y) - 0.05\), \(y(0)=0.1\) Logistic growth with harvesting
  
  

Week 14

• Monday, Nov 16
  ◊ Matching slope fields to differential equations: worksheet.
  §11.3, pg 602. #1, 2, 3 → 9 odd; 11; 19.
  
  🎓 Challenge Problem.   Euler's method works nicely for more variables. The Lotka-Voltera equations describe a predator-prey system. Let \(F\) be the predator and \(R\) be the prey. The Lotka-Voltera equations are \[ \left\{\begin{array}{r l} \frac{d}{dt}F\mspace{-15mu} &= \delta R F - \gamma F \\ \frac{d}{dt}R\mspace{-15mu} &= \alpha R - \beta R F \end{array}\right. \] where \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are positive constants.
  Euler's method for the system is \[ \left\{\begin{array}{l} \;t_{new} = t_{old} + dt \\ F_{new} = F_{old} + F^\prime(F_{old},R_{old})\cdot dt \\ R_{new} = R_{old} + R^\prime(F_{old},R_{old})\cdot dt \end{array}\right. \] Let \(\alpha=4\), \(\beta=2\), \(\gamma=3\), and \(\delta=3\). Draw a "slope field" for the system using the \(x\)-axis for \(R\) and the \(y\)-axis for \(F\) with slopes \(m\) given by \[ m = \dfrac{dF/dt}{dR/dt}. \] Draw several solution curves. What do you observe?
  
  
• Tuesday, Nov 17
  §11.3
  §11.4
  
  
• Wednesday, Nov 18
  §11.4, pg 607. #1, 3, 15, 25; 31.
  ◊ Review of 9, 10, & 11.
  ¤ The review exercises (pg 529, 578, & 655) make good practice problems for the test.
  
  
• Friday, Nov 20
  🎯 ¡Test 3 is today!    (Test 3 Topics list)
       You can bring:
  • pencils (Check the erasers!)
  • a calculator (Check the batteries!)
  • Three \(8.5''\!\times11''\) sheets of paper with any notes/formulas/etc.
  
  

Week 15    ♞  (This Week)

• Monday, Nov 23
  §11.5, pg 616. #1, 3, 11; 19, 21.
  
  
• Tuesday, Nov 24
  §11.7, pg 634. #1, 5, 9; 23, 28.
  ◊ Problem 1 from Population Models (slide 17).
  

Here endeth the lessons...


Week 16

• Monday, Nov 30
  ◊ Review
  
  
• Tuesday, Dec 1
  ◊ Review
  
  
• Wednesday, Dec 2
  ◊ Review
  
  

Final Exam Week

• Monday, Dec 7
  🎯 11:00 M(T)W F Classes: Final Exam from 9:00 AM-11:30 AM
  
  
• Tuesday, Dec 8
  🎯 12:00 M(T)W F Classes: Final Exam from 9:00 AM-11:30 AM
  
  
• You can bring to the final exam:
  • several pencils (Check the erasers!)
  • a calculator (Check the batteries!)
  • Any notes/formulas/&c., but NO books.
  
  
• Comprehensive Final Exam Topics
  • Test 1 Topics
  • Test 2 Topics
  • Test 3 Topics
  • Final Exam (Extra) Topics

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“On two occasions I have been asked, ‘Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?’ I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.”

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