For those who enrolled in the Beacon Program at Suffolk County Community College, I have been informed that I should be receiving a spreadsheet from them with your student ID's by mid June. Once the course is complete, I will be able to enter your final grade for the course through Suffolk. I will be posting more information at a later date regarding transcript requests from Suffolk. LIMITS at INFINITY So far, we have been checking what the limit of a function is as a function approaches a specific x-value. Now we are going to find the limits for a function as x goes to either positive or negative infinity. The basic strategy is to determine where the biggest power of x is in a function. If you understand this, you can basically just LOOK at the problem and give the correct answer. There are NOT a lot of calculations that need to be done, so do not OVER-COMPLICATE the process. There is one of 3 possibilities if a limit exists. A) The biggest power of x is in the numerator. The limit will be either plus or minus infinity depending on the sign of the function. B) The biggest power of x is in the denominator. The limit will be 0. C) The highest power of x is the same in both the numerator and the denominator. The limit will be the ratio of the lead coefficients of the highest powers. The sign will be either positive or negative depending on the value of x being plugged in (- x if going to -infinity or +x if going to + infinity) and the signs of the lead power coefficients. This might sound familiar to you. If you think back to when we were graphing rational functions and had to determine if there was a horizontal asymptote (HA), we had to check to see whether the highest power in the function was on top (= no HA), bottom (HA of y=0) or equal on top and bottom (HA of y= ratio of lead coefficients). It is also possible that no limit exists if the function is not approaching a single value as x goes to +/- infinity. Videos: This first video shows the mathematics behind the process. The intro part explains the concept well and I recommend you watch that piece. When it demonstrates finding the limit of a rational function (fraction) and starts dividing everything by the largest power of x in the denominator, this is OVER-COMPLICATING how to solve the problem. Skip to where he gets to the final answer and then go back to the beginning and use my rule of finding the biggest power. You should be able to visually come up with the same answer without all the math that he shows. www.youtube.com/watch?v=lMJ_-rq71l4 Shows the short cuts - getting the answer by inspection. (Where is the biggest power.) www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-15/v/limits-at-positive-and-negative-infinity www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-15/v/more-limits-at-infinity www.youtube.com/watch?v=75xO9xy7TTQ www.youtube.com/watch?v=NmLljBAg82o Assignment due 6/15/20 - page 915: numbers 1 – 14, 20, 22, 24 Assignment due 6/18/20 - Review - two handouts A) Review sheet #8528 - omit problems 16 and 17 B) Review sheet of limits - problems 1-25
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Our last unit this year is limits. In basic terms, you have some function, f(x). It can be anything, from f(x)=x+1 to f(x)= x^2-1 to f(x) = (Sin(Cos(6x-10)))^3. The limit is the y value that the function approaches as x gets closer and closer to some particular value (usually denoted as c for some constant). You can also find limits of a function as the function approaches positive or negative infinity. We will work on those limits next week. For this week, we will work on functions that are approaching a specific numerical limit value (c). Let's say the function is f(x) = x+1 and you want to find the limit of f at x = 5. As x gets closer and closer to 5 (imagine plugging in values of x where x = 1, then x=2, then x=3, then x = 4, x=4.5, x=4.9, x= 4.999, etc.). The y values that result are getting closer and closer to 6. Thus the limit of f(x) = x+1 as x approaches the value of five is 6. Y is getting closer and closer to 6. For the same function, the limit of f(x) = x+1 as x approaches 99 would have a limit of 100. Limits are very useful in calculus and finding results for problems that can not be calculated directly. The above example is a simple one. Calculating a limit will often be more involved then simply plugging in the value and finding what y equals. Limits can be found graphically, numerically and algebraically. The first ways we will try are graphically and numerically. The problems for homework are: page.889: #'s 2, 8, 14, 24 and page. 897 #'s 1 and 2 These are due Monday, 6/8/20. Finding limits algebraically takes a little more work. The easiest way to find a limit algebraically is to plug the limit number into a function. If you get an answer, that value is the limit. But very often, the function is undefined at the plug in limit value. Even though a function is undefined at some value, the limit at that value may or may not exist. In these cases, you will have to do a bit more work to determine what limit if any exists. The problems for homework are: page 897 #'s 4-20 evens, 33. These are due Thursday, 6/11/20. Below are some videos giving examples on limits and calculating the limit as a function approaches a specific constant value. Introduction to limits www.youtube.com/watch?v=ahZ8LLtgu_w Properties of limits (rules for how you can combine the limits of different functions) www.youtube.com/watch?v=US9EXMXqM3I&feature=youtu.be Determining a limit numerically (no algebra. use of calculator) www.youtube.com/watch?v=l7Tcay720vw www.youtube.com/watch?v=KesTHnYwRMg www.youtube.com/watch?v=58u7vaqHg68 Determining a limit graphically. www.youtube.com/watch?v=LdewtuWi7fM www.youtube.com/watch?v=BsgrGFIbMdU&feature=youtu.be www.youtube.com/watch?v=Vi4BiJj-n0g&feature=youtu.be www.youtube.com/watch?v=fky6rtTMOwM&feature=youtu.be www.youtube.com/watch?v=3iZUK15aPE0 www.youtube.com/watch?v=-f_U7Asybsk&feature=youtu.be Finding limits Algebraically: Plugging in limit value: www.youtube.com/watch?v=VLiMfJHZIpk Algebraically when plug-in doesn't work. - Factoring and then plug in limit value www.youtube.com/watch?v=qHfyB0J57qo&feature=youtu.be www.youtube.com/watch?v=gRk24f3SUWQ Finding limits algebraically when plugging in factoring will not work www.youtube.com/watch?v=HnmWte9ZTLE Algebraically by rationalizing when plugging in doesn't work. www.youtube.com/watch?v=ptgGkxhExA0&feature=youtu.be www.youtube.com/watch?v=ouWAhqeAaik&feature=youtu.be Examples of finding different limits algebraically. (Can plug in, factor and cancel, rationalize, one sided limits) www.youtube.com/watch?v=FItPk577Shg
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June 2020
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