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 6/12/20 - page 915: numbers 1 – 14, 20, 22, 24 Assignment due 6/17/20 - Review - two handouts A) Review sheet #8528 - omit problems 16 and 17 B) Review sheet of limits - problems 1-25
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 Wednesday, 6/3/20. 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 on Monday, 6/8/20. 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
Congratulations everybody. We have two units left to go. The first topic is partial fraction decomposition. Since it is not a large unit, it will not have its own assessment. We will finish it this week. Next week we will start our last unit, which is limits. The assessment for limits will include material on partial fractions. If you combine two fractions through addition or subtraction, you find the common denominator then add or subtract the adjusted numerators to get your answer. For example, x /3 + (x-1) / 4 = (4x / 12) + 3(x-1) / 12 = (4x + 3x-3) /12 = (7x-3) / 12. You can think of partial fractions as doing the opposite. Instead of being asked to add fractions together, you are being given the final sum of (7x-3) / 12 and asked to find the fractions that were added to get that value, in this case x/3 and (x-1)/4. There is a process to do this. It often involves solving a system of equations with multiple variables. This is where knowing how to find determinants on your calculator and using Cramer's Rule becomes useful, especially when solving systems with many variables. Homework - Due Friday - 5/29/20 page 720 - #'s 2, 12, 14, 16, 18, 22, 24, 26, 28, 32, 36 Videos: Setting up the partial fraction decomposition - (determines how many linear and quadratic factors you will need to find). www.youtube.com/watch?v=N1lx9yn3SLI&feature=youtu.be Example one with 2 linear factors www.youtube.com/watch?v=WoVdOcuSI0I&feature=youtu.be Example two with 2 linear factors www.youtube.com/watch?v=RFTAVAWHgN0&feature=youtu.be Example with repeated linear factors www.youtube.com/watch?v=6DdwGw_5dvk&feature=youtu.be Example with linear and quadratic factors www.youtube.com/watch?v=prtx4o1wbaQ&feature=youtu.be Example with repeated quadratic factors www.youtube.com/watch?v=Dupeou-FDnI&feature=youtu.be Another example www.youtube.com/watch?v=04RSkBwVzK0&feature=youtu.be
Attached is the link to an assessment on system of equations and matrices.
Good Luck! docs.google.com/forms/d/e/1FAIpQLSdM-XUecydJQF1UrdyAGChf_v6gVQFD3msb9hwciEWwyvUq5Q/viewform Attached is a review sheet and answer key for systems of equations and matrices. You should finish the assignment by Thursday, May 21. The assessment will be on Friday, May 22. There will be a link on the next post to the assessment. Email me with any questions. Stay healthy!
Now that we know how to calculate a determinant, we can use it to solve various problems. Determinants of matrices can be used to solve systems of equations using Cramer's Rule. We can use determinants to solve various problems such as finding the area of a triangle. Below are the notes and videos demonstrating how to use these techniques as well as the answer key for the homework. Homework assignment due Tuesday, May 19. pages.713 – 14: numbers 30, 34, 38, 40, 42, 48, 52 Videos Cramer's Rule using determinants www.youtube.com/watch?v=ItxF3IjC_uw Example of solving a 2 x 2 system of equations using Cramer's Rule www.youtube.com/watch?v=Z0i_LTUKHVA&feature=youtu.be Example of solving a 3x 3 system of equations using Cramer's Rule. www.youtube.com/watch?v=NhSsdEfJIDo&feature=youtu.be Finding the area of a triangle using determinants www.youtube.com/watch?v=zWI9QeQf2JI
A determinant is a number associated with a square matrix that has certain properties which allows us to 1) determine if a matrix has an inverse (If the determinant is 0, the matrix has no inverse) 2) solve system of equations in matrix form. (using Cramer's Rule). We can find the determinant of any square matrix using Minors and Cofactors. There is a simple formula for finding the determinant of a 2 x 2 matrix. There is a very handy shortcut for finding the determinant of a 3 x 3 matrix called the diagonal method. Most prefer this method much more than using minors and cofactors. You can also use your calculator to find the determinant of a matrix as well. Homework assignment due Friday - 5/15/20 Page 713 numbers 2-20 evens Videos Finding the determinant of a 2 x 2 matrix, a 3 x 3 using diagonals, 3 x 3 using minors and cofactors www.youtube.com/watch?v=OI07C1HsOuc Examples Determinant of a 2 x 2 www.youtube.com/watch?v=JTeAPYturls&feature=youtu.be Determinant of a 3 x 3 using diagonals www.youtube.com/watch?v=Ibm0jZrxolo&feature=youtu.be www.youtube.com/watch?v=YsPMbFeoAno&feature=youtu.be Determinant of a 3 x 3 using minors and cofactors www.youtube.com/watch?v=umXgFjqPfsk&feature=youtu.be www.youtube.com/watch?v=sa90Tx9RuSg&feature=youtu.be Finding determinants using the calculator www.youtube.com/watch?v=5gMK5v_Sc64
In the basic mathematical operations of addition/ subtraction, multiplication/division we have identity elements and inverses. In addition and subtraction, the identity element is 0. If you add 0 to any number, you get the same number back again. x + identity = x (0 is the identity element in addition) The identity element in multiplication is 1. If you multiply anything by 1, you get your original element. 5 x identity = 5 (1 is the identity element in multiplication) The same thing is true with matrices. Any square matrix (same number of rows and columns) will have an identity matrix where you have 1's along the diagonal from top left to bottom right and zeros everywhere else. If you multiply a matrix by its identity matrix, you get the original matrix back. See the notes and videos for specific examples. Just like basic mathematical operations, matrices also have inverse matrices. In basic addition, the inverse element is the number you add to get the identity element of 0. Ex: 5 + inverse = 0(identity). So the inverse of 5 would be -5. -2 + inverse = 0(identity). So the inverse of -2 is +2. In multiplication, the inverse element for a number is its reciprocal. 5 x inverse = 1(identity). So the inverse of 5 for multiplication is 1/5. -1/2 x inverse = 1(identity). So the inverse for -1/2 is -2. -(1/2) x(-2) = 1(identity element). Matrices, if they are square matrices (same number of rows and columns), also have inverse matrices. If you multiply a matrix by its inverse, the result is the identity matrix. In math terms, Given a matrix A and its inverse matrix A^-1 then A x A^-1 = I (where I is the identity matrix) We can compute the inverse matrix of a given matrix. There are a few ways to do this. For a 2 x 2 matrix, there is a simple formula that can be used. For any square matrix whether 2 x 2 or larger, you can find the inverse by using Gauss-Jordan elimination on a matrix while simultaneously applying the same steps to the identity matrix. The assignments for this next section will involve finding inverse matrices of a given matrix. Homework due Wednesday, 5/13/20 pages 697-98: Numbers 4, 6, 8, 10, 12, 16 Videos Finding and identifying identity matrices www.youtube.com/watch?v=hPAS6H6xFa0 Formula for finding the inverse of a 2 x 2 matrix www.youtube.com/watch?v=oXV66LG5xUA Finding inverses using the calculator www.youtube.com/watch?v=_fFj4NbLcTU Finding inverse matrices using an augmented matrix (Gauss - Jordan) for sizes larger than 2 x 2 matrices www.youtube.com/watch?v=KBYvP6YG58g Example finding the inverse of a 3 x 3 matrix using Gauss - Jordan www.youtube.com/watch?v=NXCz9dw5Ce0&feature=youtu.be www.youtube.com/watch?v=6KEmzBwCBzI&feature=youtu.be
Up to this point we have been representing equations as systems of matrices. We have done operations on single rows in a matrix. We can also do operation on entire matrices themselves. The basic operations include addition, subtraction, multiplication. We can do matrix equations to solve for a matrix given several matrices. This next section deals with those operations. For the most part, it involves the basic mathematical operations of addition, subtraction, multiplication and division and knowing when you can use them. The videos give examples and more notes are included in files at the bottom of the page along with your assignment for this section. The problems to be done are on pages 684 -685: #'s 2, 4, 6, 8, 10, 12, 30, 32, 38, 40 This assignment is due Monday, May 11. Videos: Matrix operations - addition, subtraction, and scalar multiplication www.youtube.com/watch?v=iNty4CSFIpU Examples of addition and subtraction www.youtube.com/watch?v=cTKjm5fgqKM&feature=youtu.be Example of scalar multiplication www.youtube.com/watch?v=kDCka2iqE1Y&feature=youtu.be Combining various operations www.youtube.com/watch?v=SYzQPWMUhV8&feature=youtu.be Matrix multiplication www.youtube.com/watch?v=6Hmzu-WKCjc Examples www.youtube.com/watch?v=RqA4HLtJBBs&feature=youtu.be www.youtube.com/watch?v=l--THwUU73g&feature=youtu.be www.youtube.com/watch?v=zAjUyPe-4mI&feature=youtu.be Using the TI-84 calculator www.youtube.com/watch?v=C7Dc414qmlk
A matrix (plural matrices) is a format to represent data. A matrix represents data with rows and columns of numbers. Rows are horizontal and columns are vertical. A matrix can be used to represent a system of equations. Example: 2X + 3Y = 25 4X - 7Y = 22 can be represented in matrix form as _____ _____ | 2 3 25 | The elements of a matrix are enclosed in the | 4 -7 22 | [ ] symbols. ____ ____ Note that the rows represents each of the equations. The columns represent the various variables and the constants in the equation. X's are the first column. Y's are the second column. The constants are the 3rd column. In this first lesson, we will discuss syntax and notation of matrices, representing systems of equations using matrices, and reducing matrices/solving systems of equations using the method of Gauss -Jordan elimination / row-reduction. Homework: Text page 673: #'s 2, 4, 6, 16, 18, 20 Due: Tuesday 5/5/20 Homework: Text page 674: #'s 30, 34, 38, 42 Due: Wednesday 5/6/20. (More practice on matrices.) Videos: Basic notation and description of the order or size of a matrix. www.youtube.com/watch?v=ilFJYjfKYjk&feature=youtu.be Changing an equation into matrix format and then reducing the matrix to row-echelon form using row/reduction or Gauss Jordan elimination. www.youtube.com/watch?v=BWBckWPjfpw Solving a system of equations (2 x 2) using an augmented matrix www.youtube.com/watch?v=V8mb5BFmJO0&feature=youtu.be Solving a system of equations with no solutions using matrices www.youtube.com/watch?v=NgXXKmQHFDg&feature=youtu.be Solving a system of equations with infinite solutions using matrices www.youtube.com/watch?v=9_wGz6zcZ6s&feature=youtu.be Solving a system of three equations using a system of matrices www.youtube.com/watch?v=WiVeiVIu_SM&feature=youtu.be Solving a system of equations using matrices and REDUCED row-echelon form. www.youtube.com/watch?v=-bPPDq0Y8s4 Using the TI-84 matrix menu to enter a Matrix and show row operations on a Matrix www.youtube.com/watch?v=ABY5rYu3Rmc&feature=youtu.be Using the TI-84 to show Matrix operations on the home screen of the calculator www.youtube.com/watch?v=syx9LY7qMgo&feature=youtu.be There are more videos at the page www.mathispower4u.com/alg-2.php if you wish to see more examples. On the page, press the control key and the f key to open up a search box and type matrix to see the list of videos.
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