[Not following these instructions will be my first indication of your attention to detail.]
In class: log on to your machine with your standard VU email username and password and open Internet Explorer.
Login to the Villanova home page (click on the upper right "login" icon and use your standard VU email username and password; edit the lower right link list to add citrixweb: http://citrixweb.villanova.edu and Mat1500 home: http://www.homepage.villanova.edu/robert.jantzen/courses/mat1500/ ) .
Then check out our My Classrooms classroom site linked to your list of courses at the center of the web page, and visit the link to our course homepage from it
[ http://www.homepage.villanova.edu/robert.jantzen/courses/mat1500/ ],
and read the on-line links describing aspects of the course
(no need yet to look at the MAPLE command worksheet or examples or tutorial).
Then click on the yellow VUTree Applications icon on the desktop (to access our math software MAPLE9.5 through VUTree, Academic Applications, Math&Statistics, Maple9.5).
bob will quickly show you the computer environment.
Homework: Explore the 2 CDs that come with your textbook. Explore the on-line resources.
Think about paper handout on algebra. Read page 0 [To the Student] of your textbook.
If you are motivated, read section 0: "A Preview of Calculus" in your textbook.
Fill out your paper schedule form (get a copy in class or print it out earlier) to return in class Thursday.
think about paper handouts on understanding functions and functional relationships;
In class: 1.1: 52; normal first class postage [usps]; HW:
1.1: 1, 3, 5, 11, 17, 20, 21, 27, 35, 39, 41, 51, 55, 61.
optional: 17 (here is example 2 as a template: xls, g-calculator, maple);
Maple can be accessed anywhere via the web using citrixweb, see the Maple FAQ link;
see if you can set up the citrix client on your computer (if not already there) and enter to open Maple this weekend. If you are curious, do sections 1-4 and 11 of the New Users Tour accessible from the Maple Help menu.
With your nearest neighbor partner, read text and type in (share typing somehow) all inputs and reflect on what you see for Chapter 1: Sections 0-4 and Chapter 2: Sections 0-4 (skip 3D plotting);
after class finish on your own and read 1.5 and 2.5 also, then try problems 1.6: 8, 10, 11 and 2.6: 13, 14, 15;
in any case after class try the citrixweb on your own (home/laptop) computer to open Maple and then download this file: MAPLE command list for1500 to a C:\ drive folder and open it and do "EditMenu/Execute/Worksheet" to it and just scroll through the example file quickly to be aware that it is there for you to consult when you need to in the future;
course syllabus handout.
Useful MAPLE plotting commands:
> plot(f(x),x=-10..10);
> plot(f(x),x=-10..10,y=-10..10);
> plot([f(x),g(x)],x=-10..10, color=[red,blue]);
> fsolve(f(x)=g(x),x=0..1);
[Always use plot before fsolve to know where to look for solutions. Always check your expression in pretty math mode first (toggle with the x icon on the lower tool bar) before plotting it to know that it is what you intended and not something else.]
Happy Labor Day weekend; Tuesday when you return to normal:
read 1.4, then execute and scroll through the selected examples in the Maple worksheet 1.4; then do 1, 3, 7, 15, 23, 31 [not to be collected; do this to your own satisfaction, ask questions in class if you have any difficulties, you may use a graphing calculator but it is good practice to try with MAPLE];
for 31 the sequence command is useful as in this example:
> f := x -> x^2-c*x;
> seq(f(x),c=[-2,-1,0,1,2]);
> plot([%], x = -2..2);
MAPLE's default sequence of colors is red, green, yellow, blue, purple, cyan and then repeats, but you can also use the color option (see day 6) to choose your own color sequence, and the option thickness = 2 can help see the colors better.
partnering data handout, find a partner for Maple HW; chapter 1 worksheet due M-F next week;
read instructions carefully please;
1.6 thru p.71 (no trig): 3, 5, 9, 11, 13; 15; 19 [Hint: you cannot invert this function, so think...what value of x can you insert into g to get an output of 4?], 21, 24 [Hint: set equal to y, multiply thru by denominator, then solve for x], 27; 35, 40, 43* [ base a log: log[a](x), natural log: ln(x) in pallette; put answer to problem instruction question in MAPLE worksheet as a comment], 47, 51;
handout on "the only base you really need: e".
1.6 (trig): 59, 63, 67 (exact not numeric!), 69, 71, 73*
[same instructions as before, use option "scaling=constrained" in plot command to make units same on both axes, use a square viewing window:
> plot([...],x=-Pi/2,..Pi/2,scaling=constrained);
Why are the horizontal and vertical ranges automatically the same here...?];
which range of angles does your graphing calculator choose for arccsc and arcsec? can it even plot these? does it even have these functions defined?;
trig/arctrig handouts (ignore derivative comments for now); note textbook appendix D review of trigonometry.
handout on inverse function solving process;
in class problem solving (after class homework) 1.R (Review): 5, 10(move 3 key points at grid intersections, interpolate roughly), 18, 22, 25, 26, 28a(maple: plot [P(t),900],click on crossing point),b,c; 1.6: 45, 52b (hint: first get x in one place using identities).
Weeks 3 and 4: come by and find me in my office, tell me how things are going.
This is a required visit. and maple1.mws due next week M-F.
> seq(g(x), x=[1.5,1.1,1.01,1.001]); to evaluate an expression g(x) for a sequence of multiple inputs (always use decimal point values like 1.0 to force floating point evaluation but if the expression g(x) involves π you must use evalf to force it to approximate that as well
> evalf(%);
2.2: 1, 3, 5, 9, 10 ("explain" means in nonmathematical terms in the context of the word problem); 11* [remember: exp(1/x) in MAPLE, answer question with text comment, check with limit command]; 13, 15 [Maple seq command makes this easy]; 23, 25, 31; 35 [Maple seq command makes this easy];
> limit(f(x),x=0);
> limit(f(x),x=0,right);
2.2: 21 try the maple command "factor" on the numerator and denominator separately and see what common factors they have; once you cancel them, does the remaining quotient have problems at x = 1? what is the exact limit therefore? confirm with the "limit" command;
2.3: 44 and 45 use the maple "limit" command (|x| = abs(x) in Maple); then reason through each limit to understand why you get these results, and confirm with a plot;
2.3: 38 (what are the max and min values the product can take at a given x value?);
2.3: 58 what is the domain of this function? what does the "limit" command give for x = -2? what limit or limits describe the behavior of the function at the edge of its domain?
Test 1 Wednesday Sep 29, Thurs Sep 30, Friday Oct 1? which date preferable?;
2.5: 1, 3, 5, 7, 9; 11, 13; 17, 19, 21, 33, 39, 40, 41, 47, 51* [follow this MAPLE example],
optional fun problem 63 (make a typical graph of s versus t for each trip, consider the difference function graph);
2.6 (omit subsection on precise definitions): 1, 2, 3, 5, 7; 9, 11, 13, 17, 21, 31, 35c (multiply by conjugate technique), 37, 39, 43, 47, 49, 55a;
35* [do an appropriate plot for part a) and plot your guess as a constant function together with the function to compare in an appropriately wide enough window to see the squeezing together, then use the seq command (day 11) to get your floating point values for part b), then use the limit command to evaluate the limit exactly for part c)],
55b* [graph appropriately (plot the terminal velocity and the 99% terminal velocity value lines together with the function), then use fsolve to find the desired time, make sure this time is in your graphing window, adjust if necessary];
> Limit(f(x),x=-infinity);
> value(%);
Fall Equinox (Solstice) today! [Stewart Figure 9 on page 61] (more details).
> plot([(x^2-1)/(x^2+1),1],x=-50..50,-4..4);
in class 54b, plot with its limiting value as a constant function in Maple, how long does it take for the concentration to get within 1 percent of its limiting value? (how many days is this?);
now try the Maple HW problem:
55b, repeat plot, how long and how many characteristic times (1/9.8 s) does it take for the value to get within 1 percent of the limiting value? half a percent?
2.7: 1, 3, 5a,b, 9, 11, 15, 17, 23, 24;
5c* [plot two expressions together, then zoom in, make comment about whether it looks right or not];
2.8: 1, 3, 5, 7, 11* [a) use seq command to numerically "evaluate'' the derivative limit, b) zoom in on graph using boxed axes (click on the plot and then on the boxed axes icon on toolbar) to read off rise and run within box to estimate slope, evaluate quotient in worksheet not with a calculator], 13, 15, 19, 21, 23, 29, 31.
T: MLRC 5:30: test 1 problem session
in class problems 2.4: 1-40 (exclude 27-30) cover sections 1-7 (good review to understand them all).
MAPLE function derivatives (CalcLabs sections 4.2,4.3 have more details):
> f : = x-> (x^2+x)/(x^3-1);
> f(x);
> D(f)(x);
> plot([f(x),D(f)(x)],x=a..b,color=[red,blue]);
an evaluated MAPLE function is just a MAPLE expression, so you can also use the diff command:
> diff(f(x),x);
in either case, as in hand calculations, after applying rules of differentiation one must usually simplify the result:
> simplify(%);
> D(f)(2); easy to evaluate derivative of a Maple function
> subs(x=2, diff(f(x),x) ); awkward to evaluate derivative of a Maple expression
[if anyone wants to go over their test with me problem by problem, stop by].
For Friday: 2.9: 27, 33a, 35 [Hint: at endpoints use the one sided secant line slope, at interior points the average of adjacent secant line slopes (no technology needed, can be done by hand! since numbers so simple)], 37 [reasons = what goes wrong with the limit], 45 [Hint: f(x)= sgn(x) x^2.];
CalcLabs 4.7: 1a*, 2*; maple2.mws due the week after break; don't wait till the last minute, follow instructions. 3 people still have not come to my required 5 minute office visit.
Fall Break. Enjoy.
3.1: 3-31 odd (before attempting differentiation: multiply out, divide thru, convert to power notation, evaluate powers), 37*, 39, 45, 49;
> with (Student[Calculus1]);
> DiffTutor(x*(1-x)^2); [put your expression in parentheses here]
3.2: 38d [do two ways: dividing thru first without quotient rule but combining fractions in final result, then with quotient rule, compare].
3.4: 1, 3, 5, 9, 19, 21, 25a, 29, 31, 37 [Hint: express tan as sin/cos, then introduce factor 1 = 3*(2t/6t) and associate factors in parentheses with sin ratios], 43 [divide top and bottom by theta, express tan as sin/cos]; recall trig handout and appendix D trig review.
complete maple2.mws reading the on-line instructions for each problem and adding brief comments that put each problem in context and respond to any questions raised in the book statement.
32* [define s to be a Maple function S of t, then plot [S(t),D(S)(t)] for t = 0..2*Pi, answer questions c,d,e in maple comments, remember equilibrium position means s = 0, speed is absolute value of velocity],
33 (make a diagram), 34a,b [W is a constant too]
34c*[define a MAPLE function F using 3/5 as the coefficient instead of 0.6 to keep exact arithmetic, then after clicking on the graph to determine the value of theta, use fsolve to find it to high precision, and then use solve to find it exactly, and give its numerical approximation with evalf, be sure to comment a response to the question asked],
3.5: 1, 5, 7, 11 [rewrite rule first, negative power], 15, 17, 23, 25.
3.5: 29, 37, 38, 47a,b [rewrite rule first, negative power, check with a plot of y and dy/dx], 51 [sin(x)^2 not sin(x^2)!], 55a, 57a,c, 59, 63, 68 [remember p.41, March 21: t = 80, ans: 2min 54sec; May 2: t = 141, ans: 1min 24sec, convert to minutes and seconds for comparison with watches!],
69*[Make a Maple function S to represent s in terms of t, then plot [S(t),D(S)(t)], and then make some comments about features of the graph of s reflected in the graph of s'], 74* [first define f as a MAPLE function; for part a, use the simplify command, after which one can solve part b exactly (in terms of fourth roots), even by hand; answer the questions of the book with MAPLE comments].
3.6: 1, 7, 15, 23, 25(can check by plotting with tangent line), 27* [display plot with tan line], 37, 57, 63;
Bring arctrig review sheet handout with you next class;
> with(plots):
> implicitplot(x^2-x*y+y^2=3,x=-10..10,y=-10..10);
> implicitdiff(x^2-x*y+y^2=3,y,x);
[[optional challenge for those with extra interest only (this is the target problem again but with implicit differentiation) worth extra credit of one quiz: 69. Hint: Write eq for tan line y = m x +b at point (x0,y0) on ellipse, so that x0^2 + 4y0^2 = 5 (eq1); use this condition to simplify the y intercept. Impose that (-5,0) lie on tan line to get second condition (eq2) for (x0,y0) and multiply through by 4 y0 to get rid of fractions and then solve for x0. Then solve eq1 for y0. Then backsub values into tan line eq and find y value at x=3.]]
25, 39, 45, 49a,b*,c* [use the single line below for the plot, be sure to answer c) in a comment referring to the plot], 59;
> plot([f(x),D(f)(x),(D@@2)(f)(x)],x=a..b,color=[red,blue,green]);
> with(Student[Calculus1]);
> CurveAnalysisTutor(f(x));
MLRC problem session 5:30pm.
3.10: [for word problems always use the d?/d? derivative notation, never prime]
1, 3; 7, 11, 19 (see example 3), 33 [for b) this problem describes information about an obvious triangle (draw it); which two sides of this triangle lead to the simplest functional relationship between the angle and a variable side length that we must use as a starting point? Hint: if one side of the triangle is constant, it should be used in the ratio used to define the angle, and the variable side should be in the numerator so it is not also composed with the reciprocal function in the relationship.].
[in class: 30, 34, 35].
3.11 (differential approximation): 21, 29, 35 [now use differentials], 41, 45, 46, 49.
4.1: 1, 3, 5, 7, 11; 15, 21, 23, 29; 31, 33, 35 (quadratic formula!),
41 [like example 7: either use product rule or rewrite first by multiplying out and then differentiate; then combine into a single fraction, then factor quadratic numerator (use maple factor command?); compare with plot:
> plot(surd(x,5)^4*(4-x)^2,x=-1..6); or on graphing calculator with this same window],
45, 49, 59, 63 (use log diff, expanding ln of RHS), 69* [see example 10 in worksheet].
70 [we already know that F '(arctan(mu)) = 0 from previous HW 3.5.34; evaluate F at endpoints & crit point, why is crit value the smallest of the three values?],
73 [multiply out before differentiating, then factor afterwards];
In class paper handout: This is a do-able realistic calculus problem:
Applied Project 4.1 Part 1 (page 288): express D(alpha) given in the equation in terms of alpha alone using Snell's law to solve for beta in terms of alpha using the arcsin function and the value of k = 4/3 quoted in the text; plot for all angles between 0 and π (Pi), click on minimum to get approximate radian angle and convert to degrees; make a MAPLE function Dev(alpha) since D is already used by the derivative, then use fsolve and D to find the critical number and value to high accuracy in radians, then convert both to degrees to get the two angles given in the text [or define the Maple function directly in terms of degrees by replacing D(alpha) by 180/Pi*D(Pi/180*alpha)];
try finding the critical point by hand for homework if you are interested;
optional for rocket scientists: 72* [data fitting plus max/min template given].
derivatives of Maple functions:
> f:=x->x^4-x^2;
> D(f)(x);
> (D@@2)(f)(x);
55*[do what the textbook asks you to do, guestimate intervals of concavity and state your guesses before checking the graph of the second derivative, then compare with a comment],
MLRC problem session 5:30pm; Test 3 thru 4.2 available afterwards in paper at session or at my office door or via web; due back Friday, Dec 3; don't leave it till the last minute. You have a week in addition to your T-day vacation to work on it; you may NOT COLLABORATE, you may use your book or notes and you will have to use MAPLE or a G-Calculator; if you get stuck you may only discuss it with me in my office (don't wait till it is too late);
8 class days left after today!;
no homework: work on test 3!;
in class, explicit ratios only: 4.4: 5, 9, 11, 13, 21, 23, 27, 31, 35.
4.4: prods: 41, differences 45; powers: 53, 57; in context: 69, 70;
optional for those who like a challenge: 71.
read 4.6: do 15* [produce several graphs that show all the details that one graph cannot supply, use MAPLE to find the (x,y) for the critical points and points of inflection, finally make an exaggerated hand sketch "road map" labeling all points with 3 significant digit accuracy that summarizes your results to be handed in on Monday as a quiz grade; print out your maple worksheet and staple it to your roadmap]
[but I won't be in to pick them up in the evening, so you can still work till Saturday (Sunday?), but you have to move on to the graph problem to be handed in monday, so try to finish Saturday]
you can do this all by hand and get exact results (like arccos(1/4) for the location of the inflection points):
4.5: 37 [Note: odd function so pick period -Pi<=x<=Pi; use sin(2x)=2sin(x)cos(x) to find zeros of f and f '', use cos(2x)=2cos(x)^2-1 to find zeros of f ', but keep trig(2x) formulas for differentiation].
1b, 7, 17 [example 3], 28, 29, 31;
maple substitute quiz due today;
maple4.mws complete and due by the end of exam period
50 [units: c(v) = gal/hr, G(v) = c(v)/v = gal/hr/(mi/hr) = gal/mi, so you need to express G'(v) in terms of c'(v) to see what the condition is that G is minimized; hint: for a point on the curve c(v), what does the ratio c(v)/v represent?].
> with(Student[Calculus1]):
> f := x-> x^3 -3x +1;
> plot(f(x), x=-3..3);
> NewtonsMethodTutor(f(x),2);
[If you are interested in the solution of the car loan monthly interest rate problem, see problem 41 and the worksheet carloan.mws].
4.R: Browse the concept check, true-false quiz and exercises in this chapter review and attempt problems for which you think you need practice; and when preparing for final exam, do the same for the other chapters, at least chapter 3;
in particular, apart from the mechanical kinds of problems note:
4.R: 9, 11; 45 [note that f '(x) is quadratic in x50, what is the sign of this derivative? first find the roots of the quadratic expression], 47, 48; 57 [how would you define dimensionless variables x and y?, 59 [draw the p-N (price-number) plane with the information from the first two sentences: point and slope; use to write an equation for the linear relationship between price and number from the data; how does revenue R depend on p and N? (product!) What is the minimum value of p (domain) allowed by the seat capacity? Does this turn out to matter? Can one lower the price below this amount?].
teaching evaluations;
course summary handout [rules of differentiation];
MAPLE upgrades on all assignments still possible until end of exam period, when Maple 4 is due;
brain food teaser: 4.R.76 [not final exam material; hint: make a possible diagram of the difference distance in the s versus t plane, or the difference velocity in the v versus t plane, how is an inflection point of the difference distance (equal accelerations) characterized in either of these diagrams?].