This week in class we learned to compare the graphs of the first derivatives and second derivatives. The tools we use to find anti derivatives in the previous functions are used heavily in this new section. By comparing these graphs we were able to understand the shape of the original function. The methods we learned come in handy when we are not given full information on the original equation. However, this week I struggled with the material. I had a hard time understanding how to solve the problems without the use of the calculator. Much of this confusion had me while I was working on the assignments. I fairly understood the notes in class, however when I started the assignment it was more difficult and the format was different. I understood well how to find requirements in graphs when using the calculator where it would should concavity. I need to get better at using skills without the use of a calculator. Next week, I am guessing I will learn more strategies and comparison between derivatives in graphs that will be good practice without the use of a calculator. I was also confused on all the separate rules in 4.3. I understood the concepts but had a hard time remembering how to solve a problem with using support for the rules. But with help from these two websites I was able to understand the concept a bit better. http://www.math.hmc.edu/calculus/tutorials/secondderiv/ and http://mathworld.wolfram.com/FirstDerivativeTest.html. What I found with the second derivative test is when x=c in a particular graph the concavity would be concave down in the original function due to the graph of the second derivative of c being negative where c is the critical point. The second derivatives can be handy in order to solve the zeros to find the inflection points and whether it is concave up or down. Overall, although I struggled this week I would say I tried my best in participating in what I could do group discussions and homework. This section defiantly was a challenge!
This week, we jumped into the beginning of chapter 4. Chapter 4 consists of using the same processes of derivatives we have used in the past, just additional information. In 4.1 and 4.2 it discussed the properties of extreme values and the mean value theorem. Much of my graphing skills came in handy for this section to interpret the solutions. When solving solutions I learned that much of it was just common sense, like the mean value theorem which states that a point (a,b) is continuous for f(x)=y and differential at every point (a,b). If this statement is true the mean value then states that at least one point c in (a,b) must be the rate of change formula, f(c)= f(b)-f(a) / b-a . At first, I had a hard time understanding what the complex definition meant. However, once I put it to work I began to grasp it better. To show that a function satisfies the mean value theorem the steps to solve it goes like this: You first must find the derivative of the original function. You then need to find the average rate of change by plugging it into the equation with the b and a values. To check you take the derivative equal to the rate of change value to find x. The x of that solution becomes the c value. To find the y coordinate for the c value, plug c back in to the regular function for x to get y. Section 4.2 I understood fairly well. It was the same process of taking anti-derivatives but instead of just leaving the answer as c, we solved for that value. I found this to be easy because it was the same method as what we had our test over in chapter 3 except to find c you plug in the x and y values. My participation I felt was good this week. I have gotten a lot out of the CCC of the homework, and when I am confused on a question or concept I try to go over it or ask for help. Next week, I am as summing we will continue looking at chapter 4 and going further in depth. I ran across some questions on the activity that became a little confusing.
These websites/video are helpful for more information on the MVT: http://www.youtube.com/watch?v=xYOrYLq3fE0 http://tutorial.math.lamar.edu/Classes/CalcI/MeanValueTheorem.aspx This week we finished up the sections in chapter 3. We learned how to imply inverses in dervative trig functions and how to take derivatives involving exponentials and logarthims. These new sections involve using the same dervative methods like before except it involves new equation proofs in order to solve. However the chain rules, product rules, and quotient rules, still apply of taking the dervative of (d/dx). These are important concepts to learn in order to continue learning rules to tak dervatives of more complex functions. This week was a bit easier than last week in dealing with dervatives. Taking the inverses were easier to grasp because I have already had lots of practice in taking dervatives of regular functions. Because of this, it was simple just to use the same methods but including the inverse equation and rules. Taking derviatives of exponentials and logarthiums were also easier to understand because you just had to memorize the certian definitions involved in certain situations in order to solve. The most difficult part of this week was answering more complex problems that delt with all the dervative rules used together. It becomes difficult to sort out which rules and defintions must be used to solve certain problems. Therefore really looking at the equation to the problem and understanding specifically what it is displaying will help sort out what techniques must be used in order to solve the dervative or inverse. For example, when I got my quiz back I was a little disspointed in my score. This was because I thought I understood the concept of dervatives and the rules that delt with them. However, I failed to notice that in some of the problems I had not used the product rule along with the chain rule. With solving for inverses of trig functions you must remember to also use the co inverses also. These co functions are easier to sort out once you memorize or remember the orginal ones of sin, tan, sec. To find csc, cot, and cos you simply take the negative of its oppisite. So in other means the inverse of sin= - cos inverse. Inverse of tan= - cot inverse. Inverse of sec = - csc inverse. Knowing these rules will allow you to take the dervative of inverse trig fucntions by applying the x value given and plugging it into the x value in the equation and following the chain, product, ect rules with dervatives with it to simplify. My participation again this week I thought went well. I particapted in group questions and continued to work on problems I did not understand to where I didabnd asked for help when necessary on assignments. Next week I believe we will continue applying all these techniques and mothods with dervatives with us to carry over into story problems and real life situations. Therefore it will be important for me to continue to do well in understanding all of these dervative rules.
http://tutorial.math.lamar.edu/Classes/CalcI/DiffInvTrigFcns.aspx http://www.sosmath.com/calculus/diff/der08/der08.html This week was very complex. I learned a lot of new calculas topics. These topics delt with finding dervatives using the chain rule with anti dervatives and implict differntiation. Along with learning these new skills, I also had to review and use basic algebra skills in order to solve and simplify the techniques used to solve the dervatives for these complex problems. What we did last week, focusing on solving dervatives and anti-deravatives really carried over into the skills of this week. Technically last week were the building blocks that lead up to this week. Next week, I am precicting that we will continue using the concepts and techniques we used this week and last week to continue solving dervatives of different kinds of functions. This week was probably one of the more difficult weeks for me. I found it difficult to keep everything straight and in line when solving for the dervatives. One little mistake could really mess up the whole problem! The chain rule, was the concept I understood the most. All the chain rule is are functions inside of functions. In order to solve these, you just split the dervatives apart into pieces. So you would take f of g (x). Into the dervative of f of the dervative of g times the dervative of g of x. This concept at first was a little difficult. However as we continued to practice and use it with more complex problems I got better at it. The implict differnitation was the concept I had the hardest time understanding. There were lots of steps involved using algebra skills that I tended to get mixed up and throw off my whole equation and answer. However these webstites really helped with my understanding of it better: https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/implicitdiffdirectory/
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=2&ved=0CC8QFjAB&url=http%3A%2F%2Fwww.sosmath.com%2Fcalculus%2Fdiff%2Fder05%2Fder05.html&ei=aWdqUsCfMaOTyQGwhIGYBw&usg=AFQjCNEpDIwIogwargfFttNIODnYgbX_Dw. For next week I will need to work on becoming stornger with using my algebra skills to solve complicated complex problems. Overall, my participation was well this week. I particapated and contributed to class disscussions and questions at the beginning of class. This week we continued to use derivatives but with trig functions. The same method is used from previous weeks to find the slope at a tangent but becomes more complex. For example, I thought using derivatives with trig functions were easy. It is the same process except using memorizing the co function compliments that go with each trig value. Another method we used this week was using the chain rule. You use the chain rule when we solve for composite functions. This means there is a function inside a function. I also thought this method was pretty simple. It is simply like a chain, you just keep taking the derivative of the function. However, to find the anti derivative with the chain rule I found to be a little more difficult. The comparison when you took a polynomial anti derivative from last week was a lot more simple. When taking an anti derivative of a composite function you must replace the inside function with a u. With this you must find du/dx is to find what du is to substitute it back into the equation. The video lesson on this was unique, a different way of teaching. I felt I was able to grab the concept just the same as if I was in class. The one plus side of using this kind of teaching is if I was falling behind I could always go back and rewind the lesson. I found these two websites that also described the relationship of using the chain rule with composite functions. This website also gave extra practice examples and their solutions http://sydney.edu.au/stuserv/documents/maths_learning_centre/compositefunctionrule.pdf. http://www.pinkmonkey.com/studyguides/subjects/calc/chap4/c0404801.asp. During the first day of the chain rule I was refreshed on simplifying answers more thoroughly. Picture one shows this concept. In the future I want to work on simplifying complex equations more easier and of natural habit. This I believe will just take some practice of refreshing old algebra skills. I would say my participation this week went crucially well. I participated in group discussions, reasoning, and asked questions when necessary. Next week I am assuming that we will continue using the chain rule and process of derivatives with more complex questions, then taking the chapter test over all of chapter 3.
This week I learned the concepts of derivatives better. I got a better understanding of its function in definition that it is the slope of a tangent. I learned many techniques in finding the derivative in relation to its parent function or new equation. Using the variety of formulas to solve for the derivative allows for a better understanding of the relationship it has with the parent function. The lab we did in class helped clarify the relationship between the two. It also was good practice in reapplying the definition of a derivative. However in section 3.2 we learned when there is no derivative. Examples of these would be when there is use of corners, cusps, vertical tangents, or discontinuity. All of these concepts were applied in the section quiz earlier this week. I understood these concepts well but lacked the strength of comprehending it with the questions that were being asked on the quiz. In the future, I plan to become better at not only knowing the material but being able to clearly comprehend to answer unexpected questions that could be on an assessment. I found the second half of this week to be a lot easier. However, now by going over the quiz and continuing to reapply the similar concepts in new lessons has made me really understand it even better. I found the shortcut method and technique rules a lot easier to do. All it really is, is algebra skills to solve. Almost like a puzzle, you just plug the right numbers into the correct places. However, learning the long way first allowed me to actually process and comprehend what I was actually doing when I would be solving for the derivative. However knowing the shortcut techniques would have came in handy for the previous quiz in checking my answers. I believe these tools will help me later in future calculus problems when solving for real life applications. Therefore for the chapter test I plan on taking advantage of knowing both methods in order to check my answers thoroughly. One thing I found to be kind of interesting this week was the higher order of derivatives. I did not know that derivatives had these kind of unique relationships. If f (x) prime is differential so are the following x primes on going. How it is written is similar with the blank (power used) of y with respect to x. An interesting fact to go along with this is if the function is raised to certain power then it will take that power plus one more step for the derivative to accumulate to zero. So for example the prime value of y would be 4 to equal zero when the power is raised to 2 in the first prime function. For more information I visited http://www.owlnet.rice.edu/~fjones/chap3.pdf and http://www.encyclopediaofmath.org/index.php/Differential_calculus. Overall I again thought I participated well this week in discussion and during labs. I frequently asked questions and help when needed. Next week I assume we will be continuing and finishing up with derivatives and their function with calculus mathematics.
Even though we had a shortened week, this was not the case for a short amount of material learned. Actually in my opinion, this week was the most difficult in means of assignments and content. Last week, we learned about finding the slope on a zoomed up point, which we found out was the derivative. This week, we took those concepts and expanded it into finding the derivative algebraically while finding the equation of a tangent line in order to understand line relationships better. At first finding the derivative was hard. However, once I got the practice at it, it became very simple. To find the derivative you plug f(x) to solve for f(x)'. To compare this you can graph both the original and derivative function on graph paper to see the result and the tangent to the point where the line intersects. I found it most difficult to find derivatives with fractions, because it felt more complex. However, these websites were a good source for help http://www.math.brown.edu/UTRA/derivtips.html http://www.howtoace.com/HTACFiles/node21.html. For instance when we worked on the exercise assignment on Friday (when you were gone) it was good practice. Therefore now, I feel very confident in finding the derivatives of a function. The tip I learned when finding derivatives that came in handy for me was to keep everything organized and neat. If you rush through a problem it is very easy to mess something up when cancelling out variables which will totally throw off your whole equation in the end. Therefore to keep it neat, it is best to use the four step process we learned in class. The picture below is an example of this four step process in depth. When I became good at this, it started to become a routine and I began combining the steps together. The topic I had the hardest understanding and still continue to struggle on is section 2.4 of what we learned on Tuesday. I was not in class that day, so I was not there for the lesson. I have asked for peer help and I get some problems dealing with section 2.4 but I still have grey areas with it. However, with practice and continuing to ask questions when needed I am sure these grey areas will be filled. Overall I would say I participated well again this week. I got help when needed and helped others who needed help also. Again, I find it very beneficial to do the group work on the white boards and then discussing them at the end. Next week, I hope to focus on getting my grade up in the class and going further in depth with derivatives.
This week we finished up the limits and continuity sections with the chapter test on Tuesday. I appreciate you giving us an extra day to go over the review before the test. I felt that the activity allowed me to participate in really understanding the concepts of limit and continuity problems. On Wednesday we started a new lab that involved the introduction of derivatives. Mathematicians use derivatives frequently in calculus. From our discussion on Wednesday in class, I learned that knowing the limits of a function will help measure the derivative of the function. Therefore of what we learned in the second chapter will come in great handy for this new chapter involving derivatives. This is because you can not use the value 0 but instead you can use numbers very close to the number zero to find the values. Right now I do not know a well enough understanding of what exactly a derivative is except the fact that it is considered a slope of a point at a certain time. Next week I am assuming I will learn more about derivatives and knowing what their purpose serves. When I looked up what a derivative was the explanation that was given to me was the measure of a function in relationship to its change in input. This would make sense in why it would involve knowing the slope when you zoom up on a point like we did in the lab. There are special formulas involving derivatives I am assuming I will learn in the future that I found on http://mathworld.wolfram.com/Derivative.html. This website helped me understand the meaning of derivatives better. This week I felt I participated well in class by working in my group during the lab and in discussions. I felt I understood the concepts in chapter 2 well and I struggled the most with derivatives. This is to be expected though because I only worked with derivatives for one day in which we didn't even get to finish the lab (since I was gone on Friday). However, my goals for next week are to understand and learn more about derivatives and their relationships within a function.
This week I continued to learn more about limits. By learning how to solve limits in forms of continuity and infinity allows you to interpret values with holes, assympotes, ect. These sections were a review of what we learned last year in pre calc. My knowledge expanded in learning how to solve limits involving infinity. I also got a better understanding in knowing how to solve limits without the use of a calculator. Learning how to solve these problems algebraically is an important skill that will come into handy with future sections when I have to find limits and understand graphs better for more complex problems. Telling whether functions were continuous or not was an easier subject for me. I understood the difference in a removable discontinuity, jumps, and infinite. Graphs that involve assymptotes will be a continuous function. How so? This is because the definition of a continuous function is that every point in the DOMAIN has to be continuous. Therefore the assymptote is not in the domain. However, this does not mean it is a function. For more information in continuous functions I checked out other methods at http://mathworld.wolfram.com/ContinuousFunction.html. The thing I struggled most this week was probably the quiz we had over limits. I thought I had a good understanding of limits, but I did not do so well on the quiz. I made errors, that I realized I had made after I took the test. In the future I need to get better at thinking what the question is really asking in conceptional thinking. In problem 69 in section 2.3 I had a hard time trying to figure out what the question was asking of me. However, by participating in peer discussions I grew to understand how simple the problem really was. All I had to remember was that you can take a log of something X to the n power and make it n log x. From this technique it made the problem into a basic algebra to solve. I believe I participated well this week, which allowed me to understand the concepts better. For next week I really want to do well on the test and if we have another quiz to make up for the previous two quizzes.
This week after we took our review quiz we learned the basics on limits. The limits lab was a good review that helped me rewind the concepts of limits from last year. I was able to grasp the concept on limits better without the relying heavily on my calculator. I found the rules of limits to be very simple, almost like common sense. These limit rules like the constant, identify, sum/difference, product, quotient, constant function, and power rules were similar to basic algebra rules. Therefore basic algebra skills can be very helpful when finding certain limits. This method comes in handy when the denomonator happens to be zero in a function where you would not be able to directly substitute in to find the limit. The expression when the limit of (sinX)/(X) as X approaches 0 equals 1 will be very helpful I believe in future sections to find limits more easily when working with cos, sin, tan, ect kinds of functions to simplify in finding limits. The problems I best understood this week were the direct substitution ways to find limits instead of the algebraic. I found it easier just to plug the values in rather than making a mistake with my algebra skills. However, I need to become better with using the algebraic method for finding limits because not only will help me get an answer write on the test but is a good way to double check my answers. One and two sided limits state that when f(x) has a limit as x approaches c if the right and left hand limits both are equal (in symbols.) Therefore for a function to be continuous it must be said that when f(x) x=x(0) is defined at x0. From understanding the relationships of limits to be either one or two sided of where the values approach from you can really get a true understanding of the graph itself. To go even more in depth with different kinds of functions including continuous you can go visit http://www.mathamazement.com/Lessons/Pre-Calculus/11_Introduction-to-Calculus/limits-of-functions.html. I also found this website http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties.aspx helpful with the limits section.
|