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PRECALCULUS (SECOND SEMESTER)
2/20–pg. 313–Whole page
2/19–Sinusoids y = A sin (Bx) +C; A (Amplitude), “C” (vertical displacement/translation) ; “B” affects the period, which we barely discussed in class this morning.
You may use a graphing calculator to investigate how the period works…what relationship is there between the period and ‘B”. Read pg. 353 in your book; do 1-16, photocopied page.
2/7–Test tomorrow on basics of trigonometry–see study guide from old book (photocopied). Review the graphs of the six trig functions (see below, 1/31) as well as the inverses of sine, cosine and tangent; also trig values of special angles and their multiples; remember quadrants. Bring a scientific calculator; there will be bonus questions on the test.
2/6–Photocopied pg. 17^2 (old book)–further practice on inverse trig functions: Class Exercises–7,8; written exercises, 5-8 (do not use a graphing calculator)
2/5–pg. 385: 1-12 (omit cos¯¹ (x)). Remember: trig functions must be restricted in order to get their inverses since trig fns are periodic. We discussed inverse sine and inverse tangent tomorrow and will revisit them tomorrow. Please reread the section, look for related topics on line, graphs inverse functions in calculator..ask yourself, “how do we get the inverse? how is the inverse related to the original function? Domain? Range? In what quadrants are the inverse functions defined?”. The answer to an inverse trig function is an angle Inverse trig fn (value)= angle; trig (angle)=value (domain and range swapped)
2/2–do pg. 236 (photocopied)..13-18; Also read 4.7 (Inverse Trig Functions)..Remember that if you need additional help in understanding some of the topics, additional practice or if you want to read ahead, “google” the topic in question or go to Khan Academy–it doesn’t hurt to read ahead–Semper paratus!
02/01–HWK–Do pg. 285: Class exercises (top of page): 3-7 (some required a calculator); Written Ex: 3-12. We will go over these probs B4 our minor assignment tomorrow.
1/31–Participation Grade tomorrow–I suggest you review the unit circle and the graphs we discussed this morning, paying attention to special features (domain, range, even/odd, symmetry, vertical asymptotes, etc.) HWK: pg. 400–33-38; we will also go over the hwk assigned on 1/30
1/30–pg. 279; 1-4 (“Written Exercises”–objective: reference angles)
1/29–pg. 349: 67-70; pg. 366: 29-40 (Read about reference triangles).
1/26–In your textbook–pg. 348: 43-48; pg. 347 (previously assigned during my absence–1-42, all–if you haven’t done these probs, I strongly suggest that you do them over the weekend). Remember the chart you completed re trig functions of special angles, review them. How can I quiz you on the chart? Well, I can call on someone to pick a random angle and call on someone else to pick a trig function at random….so please review the values…all you need really is the first quadrant, then look for the reference angle in Q2, Q3, or Q4 and attach the appropriate +/-).
1/25–In the old book, photocopied pg. 274; 33-42.I will reference the unit circle or the graphs of sine and cosine; I recommend that you don’t use a calculator since you won’t be allowed to use a calculator when you see this kind of prob on an assessment. Minor assignment tomorrow: converting degrees to radians; radians to degrees; area of a sector, arc length, apparent size..all the problems on the assignment tomorrow have been done in class and /or assigned for homework (Also t/f, multiple choice)…graphing/scientific calculators OK.
1/23–pg.266–(photocopied)18-20; 24..Addtionally, on the back of the handout with the unit circle and the angle labeled in radians there is a chart with the six trig functions of special angles, please complete the chart and look for patterns…we’ll discuss tomorrow.
01/10–Length of an arc (s= Θr, if Θ is in radians; (Θ/360)* (2πr), if Θ is in degrees). Area of a sector (a sector of a circle is the region of the circle bounded by two radii and an arc of the circle, think “slice of pizza”) If Θ is in radians: (Θ/2)* r² or (sr/2)..Which formula to use? Well, it depends on information given in the problem. Remember: anytime you are using radians, think Π. Homework: pg. 265, 3-10 (photocopied, old book). Maybe quiz tomorrow–converting from degrees t radians and vice versa; co terminal angles. Check your book, 328 #71.
01/9–photocopied pg. 262: 17-29, odd; also in your text: pg. 325–25-33 (odd)
01/08–I need to assign you a #, so please bring your book (Campos, Mostafa); also remind me about fees (Madison, Olivia), For homework:
pg. 325: 9-16, you want the degrees to cancel, so multiply by (π/180º); for 17-24, multiply by 180º/π, cancelling the π. For #23, “2” means ” 2 radians” (remember that π radians ≅ 3.14 radians–think “unit circle”). Follow ex. 2. If you have difficulties, we will address those in class tomorrow, as the topic was barely introduced today.
01/05-Please bring textbook issued today so that I may assign a #; you must also sign your book in ink. Now, for your hwk:
Read 4.1; do 1-8 , pg. 325 Do probs by hand and confirm with calculator (see Ex. 1). Have a good weekend!
2/20–For my friend Greer and other interested parties: Test tomorrow 18 MC + 2 FR for a total of 36 points (I decided to use two questions Neel and Suraj suggested; sorry, Khaliflower, peutre la prochaine fois?). No calculators. MC questions are conceptual, use transferable test-taking techniques learned in Mr. Lampo’s test prep class (how to eliminate answers, how to maybe not work a problem all the way if your work is beginning to match one of the choices and you are heading in the right direction, pattern recognition, etc). Review the whole chapter–properties of integrals, average value, Riemann sums, trapezoidal rule (Ricardo), slope fields, u-subs (Andrew Ruiz), integration of even/odd functions where the lower and upper limits are opposites, finite integral with u-sub (change limits of integration if doing a substitution). Oh, did I say FTC? Yeah, big time.. Bruno, even though we are focusing on Ch 4, there may be a couple of items from previous chapters, that’s the nature of FR. PLEASE DILIGENT AND BE ON TIME AS YOU WILL PROBABLY NEED THE WHOLE CLASS PERIOD.
2/7–Rainy evening…stay home and work on cal probs. Test manana–Review slope fields, Riemann sums, meaning of the integral, FTC, Average Value of a Function, formulas on pg. 244, properties of definite integral (pg. 270). Format: MC. T/F, FR (yes, Zunair, it may include any topics already covered, EVT, tangent line, related rates…that’s what a good FR is supposed to do, ask Kha). If you have been paying attn in class and doing your hwk , you should be OK (not sure about Hassan Malik or Zunair, the hwk “bros”..expect to be wrekt!). More of a thinking, conceptual test, a la AP…read carefully and write beautifully. If absent tomorrow, please make up on Fri. If afflicted by the flu, don’t worry; take care, get well soon and we’ll talk upon your return.
2/6–pg. 297: 42-54 (even); 56, 66-76 (even) + 75, 80, 90
2/5–Read 4.5–follow examples and guidelines given on pg. 293..it will take us a couple of days to master this topic–do not despair–patience, recognition, chain rule will get you through. HW: 2-34, even (Thank you, Taylor, for reminding me to post).
2/1–Bruno, tomorrow we will go over hwks assigned on 1/31 and 1/30 (finally!)..if you have any problems, please ask your Chinese friends in China. We may also do some probs from the Stewart book as well as two FR questions (practice, non graded) supplied by my FR assistant, Kha Le.
1/31–Now that we have learned (or have we?) the average value of a function, as well as fine tuning FTC (what can I say? I gotta be optimistic), address the probs assigned on 1/30.
1/30 (pls tell Megan Marsalone what date it is)–pg. 285 (Second period–read or “google” about Average value of a function (Khan Academy is a good source as is Paul’s Notes); third period, Neel proved) Do or attempt pg. 285–46, 48, 50, 61, 63
1/29–pg. 286: 69, 73, 75, 81, 82, 83, 87, 88, 91, 92, 96
1/26–Let’s address any lingering questions from 1/25 (Ryne, Ricardo et al) ; additionally: pg. 286: 2,4; 16, 24, 26, 36, 40, 42, 44, 46, 48. Read 279; 281.
1/25–READ 4.3–Pay attn to definitions, theorems and boldfaced terms Hwk: pg. 273 #6 (use limit def–> use formulas on pg. 254); 15-42, multiples of 3; 45, 46 (most of these problems may be done by using the “geometry” approach).
01/ 23–Assignment, hand out (match integral with summation (Riemann formula) 1-8 ; quiz demain.
01/o9–pg. 251–71, 73, 76 (you use this formula every day in my class when the bell rings), 78, 79, 80, 84. Assignment tomorrow, similar to hwk/cwk assignment probs..there will be a calculator prob. Review derivatives and anti derivatives on pg. 244.
01/8–pg. 249: 23-41, odd; 45-48*, 50, 52, 62, 64, 67-69 (*–> use www.mathgraphs.com to print a copy). Expect a quiz on this material Wed. Je crois qu c’est assez pour aujourd’hui,ne penses-pas tois, Zunair?)
12/13–pg. 234: 33, 37;; pg. 322: 45-48; 71, 78
12/11–Very similar to last test: MC (Know your theorems: IVT, MVT, EVT), FR + interpreting a graph (concavity, incrasing/dec)…justify your answers as you justify your existence. You will need a scientific calculator! 39 points…
12/8–pg. 233–1-6, 8, 12
12/7–Same as 12/6…Test on Tuesday, 12/12
12/6–pg. 217: 24, 25 (cool prob), 27 (similar to prob done by Suraj), 33 (present you are sending me for the holidays 😊), 45
12/5–pg. 6³–2, 5, 16,12 (Hassan Malik, “the soccer player”, not the other Malik), 1 8 (Nameera), 20
12/1–Read 3.7..you may also look up “Optimization” in Khan Academy, Paul’s Notes or in your workbook
11/30–Test 30 MC + 1 FR–Review the conditions (hypothesis, the “if” ) for IVT, MVT, and EVT , first/second derivatives, concavity, POIs. You will have to interpret several graphs and establish relationships between f(x), f'(x) and f”(x)..max/min, etc. (all the stuff we have been doing lately). Test should take the whole period. READ QUESTIONS CAREFULLY (ZUNAIR, ANDREW RUIZ, ET AL –OF COURSE, SOME OF YOU DON’T READ AT ALL!) . Test is wort 39 points, i strongly recommend you do the FR first as it is easy compared with the MC and it is worth 9 points (do the math, about 23%). Points will be added to this test.
11/29–Handout: 47, 48, 49, 51,53. Textbook: pg. 209–32, 38, 41
11/28-Overall, disappointed with the quiz results; obviously you are not reading the book much to your detriment. There were a few unexpected pleasant surprises (Justin Hugger),however, we shall move on; Test on Thursday (will tell you more about it tomorrow). HW: Read 3.6, do pg. 208 1-6, 15 (follow guidelines on pg. 202)
11/27–It’s tough coming back after a week..since I had nobody to roast, I roasted a turkey…lame, but accurate, so I must make up for lost time.
Now, for your hwk…handout, 24-26, 53, 36 (compliments of Hassan), and 40 (we need to do at least one trig fn, easy, compared to #39–Don’t Rachel Thevenot troll you, it is f(t)= t + cos t (she thought it was “i + cos t”…too much gymnastics may interfere with cerebral processes right, Patrick?).
Ryne, please remind me about prob 62 form 11/15 (good test prob).
11/17–Due Mon, 11/27: From hand out given on Thursday, 11/16: 17, 22, 23, 28, 29, 39 (review f ‘ (x); f ” (x), min/max, inc/dec, POI, concavity, how do first and second derivatives affect a function’s behavior and vice versa. Check Khan Academy and Paul’s notes from Lamar University. We will also go over any pending probs form 11/15 (see below)).
11/15–pg. 189: 24, 38, 54, 56, 58, 62, 69
11/14–pg. 17²– 18-20, 26, 36. For the problems listed, find extrema (use first derivative to identify critical points (f'(x) = 0 or f'(x) dne, then test f’ (x) at the x values to see if f'(x) changes from + to – indicating a rel max or from – to + indicating a rel min. ALTERNATIVELY, you could use the Second Derivative test (see pg. 188) ). Also, find POI by using the second derivative and test for concavity (If f ”(x) changes from + to – , concavity changes from concave up to concave down; if f ”(x) changes from – to +, concavity changes from concave down to concave up). Remember: f”(x) > 0—> concave up; f ” (x) < 0, concave down). Don’t forget, once you find the x values for either inc/dec or for POI, you must substitute the x values in the given function to obtain the extrema and/or POI (points means (x,y) coordinates.
11/13– pg. 183 #62. Also, read 3.2 and come prepared to discuss concavity, inflection points and second derivative test, . Time for a quiz, Andrew, on 3.1 and 3.2? Sounds good.
11/10– Increasing and Decreasing functions
KNOW DEFINITION ON PG. 174. Remember that a function increases or decreases over an x- interval (just like our weight –My weight either increased, decreased, or stayed constant over the last month). We need to find the INTERVALS of inc/dec…so, follow guidelines on pg. 175.
FIRST DERIVATIVE TEST, PG. 176 (really a test for inc/dec..focus on change. What happens with f(x) = 1/x?
Follow examples in book. HWK: pg. 181: 4,9,10,19, 27, 29, 32, 34, 58 (compliments of the marine engineer)
11/9–pg. 172: 25, 27, 28( print graphs for the last two),33, 35, 37, 39, 40, 43, 44, 49
11/6–Read 3.1 Do: 2-10, all; 12-36, even.
11/3–Test on Monday: 4 related-rate probs, all done in class and/or assigned for hwk (only #s have been changed); review your derivative rules. Some questions are similar to questions from the last test. Bring a scientific calculator, although you may leave your answers in exact form (preferred, particularly if π is involved…don’t forget, three places behind decimal (that’s why I prefer exact answers.)). You may leave your answer as follows: (5π+ √47/(23² -(1/4)) provided that it simplifies to the correct answer (acceptable on AP exam).
11/2–We will finish with Related Rates–TEST ON MONDAY!!
10/31–Just finished grading C2–have been spooked! Happy Halloween? Really? HWK: Handout given in class this morning (review of related rates) + probs assigned on 10/27 that we haven’t discussed yet..Remember: It is your responsibility to ask questions–probs from workbooks are welcome.
10/30–My trick or trick to you (Yes, I’ll be grading papers on 10/31)–Test tomorrow. What is included? It is easier to respond as to what is not included: related rates. So: everything we have done since Ch 1 (your AP exam will be comprehensive), with an emphasis on CR and implicit differentiation, Johnny Knotobrite, a FR question and a a couple of deja vu probs with a Hernandez twist. N calculator.
10/27–pg. 151: 28, 32, 39, 43 Need help with related rates? (Yes!) Come during lunch Monday or during sixth period.
10/26–pg. 149–24, 27, 31 (deja vu–see below, 10/25) In honor of the World Series (Dodgers vs Astros (Mr. T’ ‘s team..easy to know I should root for…you find out whatever Mr. T is for and you take the opposite side)), #33 and 34 (Frankly, I’d like Houston to win for sympathetic reasons (Harvey), solidarity ). Neel and Suraj (and anyone else who would like to try)– 25, 35, 36 (“shadow” probs have appeared on the AP exam in the last three years…time to throw shade (don’t encourage me.).
10/25–HWH finish FRQ 6, 2002B (two ships traveling at right angles from each other). I would like to discuss this prob before our Major Assignment tomorrow. So, if there is anyone who has advisory at C1 (Jacob Edwards?), please come towards the end and put prob on board so that we’ll be ready to discuss it or if you have band or PE, the class and I will appreciate it. Ditto for anyone in C3 (Kristina started the prob at the end of class today, but we did not finish going over prob). Test tomorrow: Chain rule, second derivative, continuity, differentiability, horizontal/vertical tangents, implicit differentiation, easy related rate prob, motion (position, velocity, acceleration, jerk–don’t take the last one personally), instant vs average velocity, def of derivative..combination of MC and short answer).
10/24–Both classes: pg. 149–13, 16, 18, 19, 20, 22…also: C3–FR questions (handout, AP 2002)
10/23–Two hwks to go over tomorrow (10/18 and 10/20); In +: C2/C3 –do prob 3 (“Warm Up to Related Rates Worksheet”, cone prob); C3 (Megan & Co), prob. 1
Related Rates Worksheet ( “Sheila walks to Lake Menomim…)
10/20–HWK: Read 7.7, When do we use L’ Hôpital? When direct substitution produces an indeterminate form (the form most commonly tested on the AP exam is “0/0”). Pay attention to the theorems on pg. 531, particularly 7.4 (explained by our friend Neel and our “friend” Zunair). L’ Hôpital was originally (years ago) a topic covered in Cal AB, then it was moved to BC, a move that always mystified me; as of last year, the topic was kicked back to AB, which is where it belongs (tested on last year’s exam). I prefer to work a limit problem by using direct substitution or one of the methods learned in Ch. 1. However, I find L’ Hôpital convenient. In order to invoke (use) L’ Hôpital, you must ascertain that direct substitution gives you an indeterminate form and then state ,”by L’ Hôpital….” . Now, for hwk (i know, Zunair, less is more, but I am taking lessons from Manal), pg. 537: 5-20 (omit 18, unless you know the d (ln x)/dx (Hassan M knows).
10/18–How do we find the derivative of an absolute value function at the points where it is differentiable? Refer to prob 104, pg. 136 (proven in class this AM), Now do pg. 136, 105-108. Tomorrow, L’ Hôpital rule (Neel and Suraj, are you ready?)
10/17–Major assignment tomorrow: implicit differentiation, tangent and normal lines, limits, potpourri..
10/16–Handout–Since you did so well on the participation assignment this morning, perhaps you would like another for tomorrow? I’m in.
10/13-Minor assignment on “Implicit Differentiation” Monday, 10/17.
10/10 HWK–pg. 142: 24, 26, 28, 35, 43, 48 due 10/12. Good luck to the juniors on their PSAT. See you on 10/12 (with the test graded); Minor assignment on Friday. Chain rule? Trig derivatives? Unit Circle?
10/9–Je ne vais pas ajouter plus des questions a votre examen pour demain, meme si Justin Hugger veut que j’ajoutte la regle de chaine. Test is wort 31 points (will have a bonus point on the board) and I will be adding extra points if you have any (we know Nicole and Zunair have at least one each for being absent). HWK for Thursday, 10/12 (no class 10/11..I will miss you..NOT!). pg. 142: 24, 26, 28, 35, 43, 48. Hope to have the tests graded by Th.
10/6–Test on Tuesday. whether we have school on Monday or not. . HW for Monday (Yes, I think (hope) we’ll be in school Monday pg. 142: 1-15. odd). Now, if we don’t have school on Mon and Tue, then we’ll have the test on Thursday, but I don’t think this option will materialize. What do yo do on a rainy weekend? Math hwk and catch up with your resume (Nameera), brag sheets, college essays..I’ll be writing LOR and catching up with other cal teachers, my BFFs..maybe this hurricane will blow some calculus your way (study limits, continuity, IVT, tangent lines, normal lines, etc…test could have been given to you two weeks ago..no chain rule or derivatives rules on this one).
10/5–Will return test from Tue 10/13 tomorrow..maybe a participation or minor assignment? Read section on “Implicit Differentiation” and pay attention to the examples..ask yourself, “what’s this stuff”, “when and why do we use it”?
10/4–Handout from Stewart book (Zunair is checking SLADER for answers since he is apprehensive about my collecting homework): 29, 32, 34 ,43, 45, 51, 53-57
10/3–Test tomorrow, Ch. 2 , up to but not including Chain Rule (we’ll get that later, Justin..no estan listos todavia).
10/2–Bruno may be a careful driver, but he is a distracted deriver! Chain Rule ..we’ll be on this topic for two days (at least); the problems in the book can be labor intense due to the algebra (we call this “cleaning up the problem”) and may not be representative of what you may see in the AP exam. So Hassan, how far do we need to go? OK, if you can factor and simplify, do it; if not, then why bother? Common Factor, simplifying a fraction (reducing), combine like terms…be careful with negative signs (like we are careful with negative people..don’t worry about “zero” people since they don’t count). Still with me? OK, here is your homework:
pg. 133: 21, 24, 27, 30, 33 (you need practice with the quotient rule); 37, 38( chain rule is not necessary; may use power rule), 43, 47, 53, 56, 58, 61, 64…NO TEST TOMORROW! (Thank your advocate, Lisa Pham), but expect a participation grade (5-6 questions) on topics already covered (No chain rule and no limits, Allie S.)
9/28–We will go over the hwk assignment from 9/26 that were not addressed this AM before the quiz tomorrow. The quiz tomorrow will focus on quotient and product rules (2.3)
9/26–Thank you, Zunair, for reminding me about posting the hwk assignment (+1?): pg. 124–72. 76. 83-88, 93-98..Maybe a participation grade tomorrow or a minor assignment…pret la regle de la chaine? la regla de la cadena? the chain rule?
9/25–Allie A and Greer K posting hwk @ 1: 20 PM (a rarity): pg. 124; 43, 46, 48, 47 (redux, using the derivatives we learned today), 54, 59, 61, 78 (Hassan M’s question..well, one of the many),
9/22-pg. 116 #106; Practice the quotient rule: pg. 124: 11, 12, 18,26, 27, 30,31,33,37,38, 47,57–Don’t forget–practice all the rules (we will also go over the hwk assignment given on 9/21)…will Monday 9/25 be a good day to collect the hwk and give a participation grade? With something to think about, this is Mr. Hernandez signing off wishing you an enjoyable weekend.
9/21–pg. 124: 5, 6, 17, 31, 39, 40, 50, 52, 53 as well as pg. 115: 93 , 97, 98 –We will have a minor assignment tomorrow–focus on power rule, definition of derivative at a point and probs similar to hwk from 9/20 and 9/19).
9/20–pg. 115: 91, 95, 96
9/19–pg. 114: 75, 89, 90 101, 107, 111
9/18–We learned formulas for the derivatives of : a constant fn, a polynomial (power) fn., sine and cosine fns, the sum/difference of two continuous fns. Tomorrow we will learn the product and quotient rules (the derivatives of the product and quotient of two functions). Remember that for the derivatives to exist the functions have to be continuous on their domain. We also learned how to graph the derivative of a function by using the nDer command (Found under “Math 8”—NDer( y, x, x); NDer(y,x,#) gives us the numerical derivative, a value (please note “#” means number, not hashtag in this case..the “#” has been around forever, long before the current hashtag interpretation..there was a world before you discovered it). HWK: pg. 113, 3-48,(multiples of 3), 57, 62, 64, 66. Hope to return your papers tomorrow.
9/15–Read 2.2 Neel (C2) and Suraj (C3) will devlop the proof for us on Mon (proof relies on the binomial theorem).
9/14–Test tomorrow (major assignment). MM (and others, I’m sure) would like to know what’s on the test? The test is a combination of FR and MC questions; I expect you to show the work for ALL questions, including the MC. Topics? Limits, continuity, domain and range, Sandwich Theorem, limits at infinity, vertical and horizontal asymptotes, instantaneous rate of change (IROC, for future reference), review whether a function is odd or even (odd: if f(-x)= –f(x); even, f(x)=f(-x) or in plain English, “odd, if substituting “–x for x” gives the opposite (negative) of f(x); even if substituting “-x” for “x”, returns original function) y=x³ and y=sin(x) are odd fns; y=x² and y=cos(x) are even fns (in fact, all trig fns EXCEPT cos(x) and sec (x) are odd fns). BONUS PROB UPON REQUEST.
9/13–Remind me to do #32, pg. 102; also, 39-44, 46, 66, 70, 85
9/12–37,38. Gracias, Bruno, your visit enlightened me.
9/11–Using the definition of derivative lim Δx–>0 ((f(x + Δx) – f(x))/Δx) (slope of tangent line, instant rate of change) HWK: pg. 102–8, 23, 31, 34 ( Spoiler alert: messy algebra may be involved)
9/8–Test on Monday–You will need your graphing calculator for some problems (make sure you have batteries (Sophie)). What’s on the test? IVT, Average values, limits at infinity as well as infinite limits, you will have to illustrate certain situations dealing with continuity, VA, HA (no slant, Hassan, not this time). I am keeping the sandwich in the fridge.
9/5–Pls see below
9/1–On Tuesday we will finish limits at infinity, discuss dominance, slant asymptotes and look at exponential and log functions. Minor assignment on this topic on Wed (Greer feels a sense of relief). HWK: (same as 8/31 which some of you did not do due to attending the game (MM?) plus: 33, 35, 37, 38 (use hint). Remind me to investigate the special limit lim x—>∞ (1 + (1/x))^ (x) (Neel, please develop the algebraic proof).
8/31– Magnetize your brain: pg. 199: 25-32 (algunos problemas interesantes–Squeeze Theorem comes into play..don’t forget about the Squeeze Theorem for it will not forget about you).
8/29–Tests from Mon will be returned tomorrow (abbreviated classes, less time to commiserate). Today we looked at “limits at infinity) or lim x–>∞, aka, “end behavior”, which gives us a horizontal (HA) or slant asymptote (you will not see the latter very often). We specifically looked at rational functions ( Q(x) = P(x)/R(x)), where the numerator and denominator are algebraic expressions (polynomials). In general, divide both num and denom by the highest degree (exponent) in the expression, then see what happens as x–>∞. Short cuts discussed in class:
- if both num and denom have same degree, then lim x–> ∞ = ratio of leading coefficients = y
- If degree of numerator < deg of denom, then lim x –> ∞ = 0 and HA y= 0
- If deg of num > deg of denom, then lim x –> ∞ is ∞ and we may write, “DNE”. In this case we obtain a slant asymptote and perform long division to get the equation of the asymptote
The above is summarized on pg. 195 HWK: pg. 199 15, 17; 19-24 (all).32
8/28–There is nothing better than to snuggle with a calculus book and work calculus probs while it is raining outside..along those lines, read 3.5, ‘”Limits at Infinity” pp. 192-198.
8/24–Infinite limits occur at vertical asymptotes. Remember to always check your domain. Hwk: 24-28; 32, 42-48, 58,60, 62
8/23–pg. 77: 41, 45, 46, 48, 49, 52,62.63,77, 83,86
8/21–Some of you had a total eclipsed; some, partial and some none at all..HW: Review 1.4 (one-sided limit, continuity and IVT (fn must be continuous!! is a solar eclipse continuous? Yes, while it lasts, particularly in its totality path) pg. 77, 15-20, 69-72
8/18–Hope the eclipse on Monday, won’t eclipse your recent grade (we will be finding limits algebraically, much to David Hutton’s (aka “David Lampo”) delight.
HWK: Read 1.4–do 1-14, pg. 76
8/17–Same as /16
8/16–Please remind me to collect fees.. We will go over probs 67-78 (assigned yesterday, not addressed today); pg. 66: 79-82, 84; 101-104 (compliments of Mr. Prentice). Quiz demain? Peut etre…
8/15–pg. 66: 56 (use conjugate), 58, 59-62 (care to comment Suraj? Andrew? Alex? do these probs look familiar? Umm..l’ avenir nous attends). in +: 67-78
8/14–pg. 55: 9-18 (graphic interpretation); pg. 65: 27-34 ( time to review the trig values of special angles..unit circle, ASTC), 41-44; 49-52
12/14–Our imaginary “bro” Zunairline suggested the following probs based on the Law of Cosines (pp;. 352-53, old book: 5, 8, 13, 15, 17)
12/13–pg. 439: 12,15, 18,20,38, 40
12/12–Photocopied pg. 4, 5, 10-15
12/11–pg. 2^9–39, 44, 49, 58,58, 61,80 (how do 80 and 58 compare?)
12/7–pg. 511: 31-38 (Use DeMoivre’s theorem, pg. 507; follow examples 5 and 6)
12/6–So, what do we do on a cold, rainy, miserable evening–stay home, order pizza from Reginelli’s and do math homework: pg. 511–19- 28( Use formula on pg. 505
12/5–pg. 511: 1-1, all (see examples 1 and 2, pp. 503-504). Quiz tomorrow–review polar graphs.pg
12/1–pg. 501: 34, 17, 18, 29, 40
11/30–pg. 501: 13, 32 (not a cardioid; dimpled limacon, does not go through zero (Why?)..remind me to show its rectangular version–check for symmetry, if in doubt, graph y= 6 – 5 cos Θ , then reflect over x axis to get an idea what the polar will look like (r,Θ)) 34 . Flowers tomorrow?
11/29–pg. 492 (in your text), 35-49, odd (graph them in your calculator)
11/28–pg. 20² (photocopied)1-11 (odd; columns “a” and “c” only).. remember : If given rectangular —> polar, use r = (x² + y²) ^ (1/2); Θ = tan ¯¹ (y/x), also remember that points don’t move, so if (x,y) is in Q2, its polar version should be in Q2 as well (the angle “rules” in polar coordinates).
If given Polar (r, Θ) —-> x = r cos Θ; y= r sin Θ. Tomorrow, we’ll be having fun graphing polar equations (remind me to give you a handout).
11/27–pg. 492: 1-4 (follow ex. 3, pg. 488); 27-30 (“a” only; follow ex. 4)
11/17–Read 6.4 in your textbook or google “Polar Coordinates” (Khan Academy)
11/15–Photocopied pg. 384 33-40 (prove identities by using appropriate double angle formulas, work with more complicated side)
11/14–Chris went fishing for trig problems and he found the following which he would like to share with the whole class: pg. 383, Written exercises 1-10; pg. 384: 13-18; 20, 22
11/13–Tangent of Sum of two angles; angles formed by the intersection of two lines–textbook pg. 425: 2, 7, 8, 15, 16, 21, 22, 28, 29, 53
11/10–Photocopied page, old textbook: 1-6 (condense); 7-10 (prove by expanding and using formulas); 14-26, even (use formulas to expand); 28, (check for quadrants; if in doubt draw a Δ); 33-38, identities (expand, combine and simplify as needed). Corresponding pages in your book for reference: pp. 421-424: exs 1-4.
11/9–Test on equations and identities
11/8–pg. 411: 57-62 PLUS probs assigned on 11/3 (see below)
11/7–I was not absent because I had gone fishing (a rumor started in social media….fake news!)..I was indisposed. HW: Photocopied pg. 326 (old book): + the following probs: 17-22
11/6–pg. 326: 1-16, all
11/3–pg. 411: 51-56
11/2–pg. 460–16, 26, 28, 34, 48
10/31–pg. 321: 13-24; pg. 322: 31-36
10/30–pg. 411 (27-32); pg. 418: 5-10, 11-20..going fishing….
10/27–pg. 410: 9-26, all Use Pythagorean, co-functions, and even/odd relationships as outlined in your text, section 5.1. Google “trig identities” and see how probs are worked. Don’t forget algebraic techniques: common factor, common denominator, etc.
10/26–Reading assignment: Read 5.1, pay attention to the example probs as well as material highlighted in blue rectangles. Do pg. 410. 1-4 (exercises 5.1)
10/25–pg. 5π/3 (photocopied, old book)–31–36 (interesting probs; you may want to do graph them first–these probs don’t have to do with “period” as the probs we have been doing; instead, approach them analytically). Minor assignment tomorrow, include angle of inclination).
10/24–pg. 306: 12, 13 (top); 22-24–MINOR ASSIGNMENT ON TH, 10/26
10/23–pg. 306: 23-26; make sure you know how to do probs assigned on 10/20 and 10/19
10/20–See below, probs. 25-30
10/19–Assigned probs form Trigonmetry, (Hornsby et al; pg. 273; 7-16)
10/18–photocopied pg. 306 (19-24). I suggest graphing these probs first (radian mode, Zoom 7, 0 ≤ x < 2π) to find the solutions graphically). Remember that the main concern with these problems is the period.
10/17- Finish problems on handout (5-7)
10/16–Handout, 1-4 (use graphing calculator as needed)
10/13–I’ll be checking the hwk assignment given on 10/9 (see below) for a participation grade. If you haven’t done it, please do it.
10/10–Good luck on your PSAT demain! On Thursday we will have a retest (make up) in class. Your test grade from 10/9 will not be dropped, however.
10/09–Grading your papers, I was hoping that this test would elevate your grades but you seem to be having problems with angle of elevation (Nick, Chris) and angle of depression (which describes my present state). HW: pg. 358 (3+5=8, love # combinations: 58, 60, 61, 68, 72-76 (my blood pressure is going ↑).
10/6–See 10/5. We worked fined tuned our graphing tools this AM and you should be ready for the test next time we meet, hopefully Monday..we are at the mercy of Nate..remember: if you play now, you pay later..so if we have days off, we will have to make them up…disruptive. Reginelli’s is closed this weekend (2 PM Sat), so, time to study. Maybe this hurricane will blow some trig your way and make it stay.
10/5–Possible hurricane tomorrow..uhm, I mean “test”..including applications of right triangles and a bit of graphing similar to what we have done in the past two days + some all stuff (it is always good to review what we have learned). No hwk per se as we have not finished going over all the graphing. Recommend you bring a scientific calculator.
10/4–Old book, pg. 313, whole page; pg. 395 #25
10/2–Same as 9/28
9/28–Photocopied page, old book (pg. 305; 1-16). Additionally, pg. 336: 61, 62, 394: 8,15,16,22
9/27–pg. 357 (three consecutive odd #s)–29-40; 43-48..Read 4.4 and come ready for questions. David asked,”What happens if “B” is negative”? In your graphing utility graph y = sin (x); y= sin (-x) AND y= cos (x); y= cos (-x) . What do you notice? Why? Why does your book mention abs (B), but other books don’t? Ponder on that. Test on Friday, 9/29.
9/22–Test (Major Assignment on Monday, 9/25). HWK: Study Guide (photocopied pg. 293, old text..good review guide)
9/21– HW: old book, photocopied pg. 289; 5-8 (bottom); pg. 290: 11-12. We will go over the hwk before the minor assignment tomorrow.
9/20–pg. 385: 1-12 (all); 23-32, 33-35 (expect a hwk quiz tomorrow)
9/19–pg. 385; 1-12, except 4, 5, 8, 11
9/18–Read 4.7–Inverse trig functions..we will be on this topic for the remainder of the week. Bring calculator. When doing an inverse trig function problem, the answer is an angle in a restricted domain. Why? Because the trig functions fail the horizontal line test due to periodicity, we must restrict the domain as we did in our discussion this morning ( y = x² has an inverse for x ≥ 0) in order to make them one-to-one (a “one-to-one” function is a function that passes the vertical and horizontal line tests, i.e, both the fn and its inverse are functions).
9/15–We covered the graphs of the six trig functions. Review the graphs carefully. You should know the domain, range, period, continuity, whether the functions are bounded or not (only sine and cosine are), increase/decrease intervals, even/odd, asymptotes–there is a nice summary at the bottom of pg. 364 (I also gave you a hand out this morning which did not include y= cot (x)).
The minor assignment scheduled for Friday will be given on Mon, 9/18. HWK: Do cot x on the last pg of the handout and pg. 365 (how many days in a year?), top, 5-10.
9/14–Minor assignment tomorrow: review reference angles, unit circle, values of special angles. We will go over the hwk assigned 9/13 before the quiz.
9/13–Judging by your hwk quiz today, you need to study the trig values of special angles ☹️. HWK for tomorrow: pg. 286: 13-18, photocopied.
9/12–Expect a hwk quiz tomorrow on trig values of special angles (we’ll be on this diet for a while). HWK: assigned problems from photographed (not twitted) hwk, (sorry, Hyobin, twitter queen).
9/11–Today we derived the trigonometric function values of special angles (30, 45, and 60º or π/6, π/4, π/3) and their multiples by using right triangle and cofunction relationships. It is important that you familiarize yourselves with these values as we will encounter special angles often in this course, calculus and physics. HW: old book (please try not to use a calculator, although you may consult the table..I don’t memorize, but draw a right triangle as needed. For a 30-60-90, always associate the shorter leg with 1/2). Remember: in Q1, all values are +; in Q2, Q3, and Q4, two trig values are >0 and the other 4 trig values are <0. Where is x>0? where is x<0? y>0; y<0 (ASTC kicks in as a mnemonic device). Associate y with sin Θ; x, cos Θ
HWK: phtocopied pg 280, 11-18.
9/8–pg. 347: 37-42
9/6-pg. 348: 43-48 (use reciprocal relationships–see handout). Please bring hand out tomorrow.
9/5–pg. 347 (3+4=7..we keep getting all these arithmetic relationships), 1-20, all
9/1–Don’t forget: Reginelli’s will donate 10% of your food bill Sat evening (9/2) towards Hurricane Harvey release efforts. As an extra bonus, you’ll get to see Bennett and Hyobin (Clearview location) and have a Dream State or Uptowner on foccacia on my behalf (my favorite). Also, on my behalf: HW ( photocopied pg. 279 (2+7 = 9)): Written exercises (A, bottom of page), 1-4 all. We will have a minor assignment on Wed 9/6; enjoy your long weekend.
8/31–Phtocopied pg. 274; 33-40
8/30–Phtocopied pg. 272(old book): 1-19, odd; do 5 and 6 as well
8/29–pg. 273: 21-29
8/28–Read 4.2..relate right triangle trigonometry to the unit circle.
Look for trig values of special angles.
8/24–Length of an arc (s= Θr, if Θ is in radians; (Θ/360)* (2πr), if Θ is in degrees). Area of a sector (a sector of a circle is the region of the circle bounded by two radii and an arc of the circle, think “slice of pizza”) If Θ is in radians: (Θ/2)* r² or (sr/2)..Which formula to use? Well, it depends on information given in the problem. Remember: anytime you are using radians, think Π. Homework: pg. 265, 3-10 (photocopied, old book).
8/23-photocopied pg. 262: 17-23, odd
8/21– For the hwk, use the formula Θ = s/r (definition of radian measure; formula may be manipulated as needed REMEMBER: Θ MUST BE IN RADIANS .) , pg. 325–25-33 (odd); 35-38
8/18–1-24 (all; you may use a calculator, although not needed (1-8). For probs 9-16, you want the degrees to cancel, so multiply by (π/180º); for 17-24, multiply by 180º/π, cancelling the π. For #23, “2” means ” 2 radians” (remember that π radians ≅ 3.14 radians–think “unit circle”).
8/17–pg. 95: 25-28; 35-40
8/2(8)–pg. 95: 9-16 (state domain and identify the function); 47-54 (remember the tests: If f(x)= f(-x), even, symm wrt y-axis; if f(-x)= -f(x), odd, symm wrt origin)
8/15–pg. 116; QR–1-10 (for help refer to sections indicated). Re-read 4.1. We will probably have a quiz on domain on Th.
8/14– We will continue our review of functions learned in Algebra II. Meanwhile, read 4.1, pay attention to new terms: radian measure and conversion from degrees to radians (we will explain further tomorrow). This is a reading homework assignment. Also; do pg. 94: “Quick Review” (QR) : 5-8 (find the domain–good quiz questions)