We started a new activity that involved creating gifs. Supposedly this would help us with a new concept we would be learning. Our group was quite oblivious to what the point of making these gifs would be; all we wanted to do was re-create the gif example that we were given. When that was achieved, we were able to go through an explanation of why the equations we used were able to give us the same graph. The equation we started off with was f(x)=1/2x^2 (we were using desmos graphing calculator on the computer to create these graphs) and we had to make a line that would trace the slope of the line. It was quite confusing because we were not familiar with desmos, we knew what we wanted it to do but was not sure how to exactly make it do it. Some trial and error was involved. Finally we used the point slope equation y=(y2-y1)/(x2-x1)(x-x1)+y1 to get the equation and learned how to find derivatives of a curve also being able to recreate the first gif. We typed in f(x)=.5x^2 for the original function then we put in a slider for the function as (sf(x)), when moving the slider back and forth, it would hit all the points on that function. Then we put in another equation y=sx and picked the point of (0.4, 0.8) for desmos to keep that point consistent to find the tangent line of the original equation we typed in. Next, we plugged in (0.4,0.8) to the slope equation to get m= f(s)-0.8/s-0.4. This would allow us to find the tangent line at any point with the equation y=(y2-y1)/(x2-x1)(x-x1)+y1, we entered m= f(s)-0.8/s-0.4(x-0.4)+.08 to finally, finally, get desmos to trace and find the tangent line at any point. For the next gif we changed the different points with different variables. With the information already typed in we took out m= f(s)-0.8/s-0.4(x-0.4)+.08 and added (f(s)-f(b)/s-b)(x-b)+f(b) and put in another slider (bf(b)) to trace the original equation. This shows two points on the equation with the tangent line therefore being able to find the derivative of the function when the distance between the two points got smaller. We changed the original f(x) equation to create the third gif that traced and found the tangent lines of the equation f(x)=.5x+sinx. I recorded Mr. Cresswell’s lesson on these different equations and on how and why they work. This will be helpful for the assignment and be able to refresh me on the topic learned. It was really interesting to use desmos to learn the concepts of tangent lines and derivatives.
Click here for link to gifs.
We started a new activity that involved creating gifs. Supposedly this would help us with a new concept we would be learning. Our group was quite oblivious to what the point of making these gifs would be; all we wanted to do was re-create the gif example that we were given. When that was achieved, we were able to go through an explanation of why the equations we used were able to give us the same graph. The equation we started off with was f(x)=1/2x^2 (we were using desmos graphing calculator on the computer to create these graphs) and we had to make a line that would trace the slope of the line. It was quite confusing because we were not familiar with desmos, we knew what we wanted it to do but was not sure how to exactly make it do it. Some trial and error was involved. Finally we used the point slope equation y=(y2-y1)/(x2-x1)(x-x1)+y1 to get the equation and learned how to find derivatives of a curve also being able to recreate the first gif. We typed in f(x)=.5x^2 for the original function then we put in a slider for the function as (sf(x)), when moving the slider back and forth, it would hit all the points on that function. Then we put in another equation y=sx and picked the point of (0.4, 0.8) for desmos to keep that point consistent to find the tangent line of the original equation we typed in. Next, we plugged in (0.4,0.8) to the slope equation to get m= f(s)-0.8/s-0.4. This would allow us to find the tangent line at any point with the equation y=(y2-y1)/(x2-x1)(x-x1)+y1, we entered m= f(s)-0.8/s-0.4(x-0.4)+.08 to finally, finally, get desmos to trace and find the tangent line at any point. For the next gif we changed the different points with different variables. With the information already typed in we took out m= f(s)-0.8/s-0.4(x-0.4)+.08 and added (f(s)-f(b)/s-b)(x-b)+f(b) and put in another slider (bf(b)) to trace the original equation. This shows two points on the equation with the tangent line therefore being able to find the derivative of the function when the distance between the two points got smaller. We changed the original f(x) equation to create the third gif that traced and found the tangent lines of the equation f(x)=.5x+sinx. I recorded Mr. Cresswell’s lesson on these different equations and on how and why they work. This will be helpful for the assignment and be able to refresh me on the topic learned. It was really interesting to use desmos to learn the concepts of tangent lines and derivatives.
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The concept I had trouble with was finding the hole. I knew that crossing out similar expressions was finding the x of the hole but I did not know how to find the y. I learned that plugging the x back in the what was leftover would give you the y. Above is another example showing I have mastered this concept. I had confused myself on what was concave down and what was concave up. I also was not thinking straight when it asked what point was neither increasing or decreasing. When I learned the right answer I was dumbfounded. So a above are two examples showing I have mastered both concepts. Next time I have to think the opposites for concave up and concave down. The maximum or minimum are also points of neither increasing or decreasing. Sometimes you feel like you're on top of the world, then the next thing you know you're crashing and falling into a pit of fiery evil. That's how I felt when I was doing the homework over the section of limits; I was understanding the concepts and felt confident in my work. Then the quiz happened and I got wrecked. Literally, it was like a different language, and it felt like I forgot how to do everything. I was like "What's wrong with me?!" I should know how to do this, yet I couldn't understand what it was asking. I will definitely do a mastery to actually master the concepts. The last quiz I didn't do too bad but I feel like I should do the mastery for it so that I can keep my grades up and keep the concepts fresh for the next week. Thank god for the cresswellcalc website blog because I like going back to it to find out what we did. My alarm did not go off on Thursday and I missed the introduction day about discontinuity so now I feel kind of lost because I just can't understand the concept of what it looks like, and how to identify it (just like I have a hard time learning science about atoms and what not because I do not know what it looks like) I know we do a lot of discussion to find out why and how the math works yet I wish I could see it in a way I could understand and comprehend it. Salt and pepper graphs... During the activity packet about discontinuity, we learned that f(x)=the lower level of x. That was new. The problem was "a function with domain all reals that is discontinuous at every integer and continuous everywhere else." I had no idea that the little brackets with only the bottom ledges present meant that 4.999 would be 4. Didn't know that existed or that that was even a possible option! So that was cool. I like learning these new concepts and the pace is pretty good overall. Also I am very thankful for the mastery opportunities, they will help me so much throughout the year, thank you!
What a wonderful Friday! We have been working on limits, not limits of physical strength but limits of the value y when x approaches a value. Limits are awesome, especially when I can use algebra to figure out the limit. It's great when you use common sense to figure out a problem instead of using the calculator. Also, did you know that sin(x)/x=1?! That's a great little piece of information. I feel much better about the pace of this class because I was afraid that the rest of the year was going to be like the first packet that we received, just a whole bunch of stuff all up in my business, yet now we have time to look at notes and discuss the concepts. It's cool that we don't have to turn in the homework until the chapter is over. It was also satisfying to use the quote "The limit does not exist" -Mean Girls You probably got a lot of Mean Girls references when you first introduced limits… Anyways, It’s interesting to learn how and why math works the way it works, not just knowing that it does indeed work. It kind of makes me feel like a scientist going through steps of a lab in order to find the correct solution, hence the “Limit Lab”. I’m not sure how the topic of limits we are learning right now will lead or connect to the next topic, so I am just going to focus on this topic until I feel comfortable with being able to teach the class how I got my answers. When we got the notes and was able to practice it a couple times I felt better about my position in the class because I didn’t feel useless when other classmates had questions. I was actually able to help a friend in another hour figure out the limit by using algebra in a way she could not think of. I do need to still work on refreshing old knowledge in order to apply it to the math I am doing today in order to efficiently solve a problem. Overall I feel much better about this class than I did last week!
Sometimes in life you will come across obstacles that frustrate you but challenges you to become a stronger person. The obstacle this week is the Pre-Requisites packet. Who knew that a couple pieces of paper stapled together could create so much stress. I need to put in a lot more effort into reviewing the past concepts if I want to be successful in this class. One page that I was able to work on confidently was the Similar Triangles page. I really enjoy algebra and working with problems where I need to figure out where one side of the equation equals the other side of the equation. So setting up the ratios and cross multiplying was pretty fun! |