This week in Ap Calc we learned about slope field and separable differential equations. Overall I found myself to understand these two concepts pretty well. Slope fields is basically just substituting in number and solving or their output. Once you have the output you just have to draw a line with that slope on the corresponding coordinates on the graph. This concept is pretty simple the most important thing to remember is which way the lines go for positive and negative slope. This is a useful concept because it can help you see the shape of the graph just by looking at the slopes.
This week in Ap Calc we also learned about separable differential equations. When solving these types of problems the first step is to use basic algebra skills to get y and dy on the same side and dx and x on the same side. Once you have done this you need to integrate both sides and then solve the equation for C. Then you write the final answer is the anti-derivative with the C value plugged in. The most important thing to remember in this type of problem is the +C because if you forget the +C when originally integrating you will have trouble getting to the final answer.
Last week in Ap Calc we focused on finding the area between curves. We focused on finding area between curves in terms of x and in terms y. If one of the functions is clearly above the other function at all values then you take the integral of the top function minus the bottom function in terms of x. For the bounds you have to find the interception point of the graphs. If one of the graphs is not consistently above the other one then you must solve in terms of y In order to find the area you must the integral of the right function minus the left function. Some of the graphs can only be solved if they are split into two parts if they don't have a function above the other or right and left of the other.
Last week we also started learning about the disk and washer method. The washer method is only used if when rotated around the axis there is a hole formed. Otherwise the disk method should be used. I tend to understand the disk method pretty well because its based off of the area function but the washer method confuses me a lot. One of the main problems I have is my struggle to visualize 3D shapes. This has been an obstacle for me in the section because the visualization.
This week in Ap Calculus we focused on the application of integrals. We spent most of the week learning 7.1 which focuses on velocity and acceleration. I found this section to be pretty easy because it was applying the steps we already know how to do with story problems. When approaching velocity problems I found it very helpful to first figure out what equation was given to me and what I was trying to solve for. Once I figured out what equation I had and what equation I needed I found it very easy to perform the steps needed to solve the problem because I either had to antiderive or derive it depending on the problem.
I think 7.1 does a good job of combining what we have already done all together because it uses the skills we learned in chapter 5 with the applications of derivatives with the integral work we learned in chapter six. Due to my prior knowledge with integrals I feel I have a very good understanding of solving word problems involving velocity.
This week in ap calc we also learned how to find the area between two curves this also made a lot of sense based on the addition and subtraction rules of integrals. The most challenging part is figuring out what equation is the upper equation but this is pretty easy if you use your graphing calculator to look at the graphs.
Overall I felt really good about all the things we covered in class this week.
his week is calculus we took the chapter 5 test and u substitution came back to haunt me in chapter 6. Overall I felt pretty good about the chapter 5 test. I have figured out that i have a pretty god understanding of integrals because chapter 5 is the first chapter this whole year that I fully understood what I was doing. Sadly, it was a short lived glory because now were learning how to do definite and indefinite integrals using U substitution. This has been very difficult for me because I did not understand how to use U substitution when we learned it the first time so I'm trying to figure that out along with do the integral part. When I was doing the assignment I started to semi understand how to do it but its still a struggle. I have figured out how to do it when u is very obviously given to me but if it is a trig identity or not clearly stated I get very confused. Even though I still dot completly undertsand U substitution I'm slowly learning it for example this week I released that du is the thing that can be canceled out which is one thing i did not pick on before. I'm hoping if i practice it more I will get way better at it.
When learning the fundamental theorem of calculus I primarily relied on deductive learning. I have always been a deductive learner because just knowing the definition of the theorem does not click for me. I need to see an example of how to apply this theorem before I will understand it. The fundamental theorem of Calculus is important because it combines derivatives with integrals and helps find the area for exact data points which is more useful than just being able to find the derivative. To me the fundamental theorem of calculus means that you can find the integral without a calculator by finding the antiderivative and subtracting F(B)-F(A). The fundamental theorem fits into the context of calculus because it applies derivatives and antiderivatives as well as integrals together. It will also help me with the applications of derivatives because know we can find not only the maximums and minimums but we can find the inbetween. The notation connects to what we have already learned because it applies Dy/Dx and Dx. Overall the fundamental theorem of calculus brings together the chapter four and chapter five stuff.
This week we learned chapter 5 section three which is about definite integrals using antiderivatives. The first thing we learned in section 5.3 was the rules for definite integrals. I found that I understand the rules of integrals pretty well. In an integral at the bottom of the notation is the smaller number also known as A and on top is bigger number know as B. The goal of an integral is to find the area of a section of a graph. To find an integral you can either use Nint on the calculator or do it by hand. We learned how to do the integral by hand by antideriving the function and then subtracting F(B)-F(A). We Also learned how to find the average value. To find the average value you have to do 1/(B-A) multiplied by the integregal. I have found that chapter 5 is way easier than chapter 4. I did not really understand chapter 4 very well so I wasn't very motivated to do the homework because everytime I did it I got really frustrated because I didn’t know how to do it . However with chapter 5 I have been more motivated to do my homework because I actually know how to do it.
This week in ap Calc we learned about derivatives of inverses of trigonometric functions and derivatives of exponential logarithms. Overall I found the trigonometric assignment relatively easy because it was just identifying U and then plugging u into the formula. This was pretty easy because we have identified u when doing the chain rule and implicit differentiation so I'm pretty familiar with identifying U. However I found the exponential logarithms semi difficult simply because I have always struggled with the rules of logarithms. So i'm trying to remember the rules for logarithms from the past years and trying to apply them to derivatives at the same time.
This week we have also been working on the chapter three review assignment to prepare for the test next Wednesday. Overall I have found that I have a pretty good understanding of most of the things covered in chapter 3 individually however I struggle a little but when applying multiple of the concepts together the same time. I have also found that I know how to set up the problem but the problem is so difficult that the simplifying process is messing me up causing me to get the wrong answer. Overall even though at first I struggled a lot in chapter three I think I have a pretty good understanding of all the concepts and should be able to do semi decent on the test on Wednesday.
This week in a Ap Calc we learned about u substitution and implicit differentiation. I also learned a lot about how to bring my success rate up in my homework. I have found if I go back and try to figure why the book did what they did to get the answer they got I’m correcting myself and learning how to do the math correctly and learning what not to do. I’m hoping this new strategist will help me more more successful in this class. Overall the U substitution stuff makes sense however there are times where I get confused on which is Du and Dx. However I believe when I follow my notes and put a lot of thought into the problem I can get it right. As far as implicit differentiation goes, I understand the process of implicit differentiation where you solve for dy/dx however sometimes I struggle identifying where something is a chain rule and a product rule. I believe this is due to the fact that I’m trying to master the u substitution concept leading me to forget to also focus on what rule of derivatives is being used. I’m hope that will more practice I will get better with the combining the two concepts because torwards the end of the assignment I started to make less errors and get more problems right.
This week in ap calc we took the quiz over 3.3 and 3.5. The quiz covered the derivative rules and anti deriving equations. Overall I think the quiz went better then most of the quizzes I have taken in this class thus far. I feel as though I had a good understanding of what I was doing. I also gained a better understanding of what a derivative is when I went through and corrected my 3.1-3.2 quiz. Before I did not understand the connection between tangent lines and derivatives but after revisiting it I now understand it. This week we also learned the chain rule. The chain rule is used when you have a composite function. For a chain rule involving a power you bring the power forward and multiply it by the inside and then you multiply that by the derivative of the inside. The most important part of the chain rule is figuring out which part in the inside part and which part is the outside once you figure that out its a repetitive process that becomes easier the more its performed. Overall I understand the chain rule pretty well however sometimes I get confused when there's a problem involving doing the chain rule inside the chain rule.
This week in ap calc we learned more about derivatives. I honestly don’t really understand what a derivative is. I thought I had a good understanding of derivatives however after taking the section 3.1 - 3.3 quiz I have realized I really don’t understand derivatives at all. I have a good understand of how to find the derivative when given the function however, I don’t understand how to the directive when given the graph of the function. I semi understand bits and pieces of it revolving maximums and minimums and zeros but I don’t overall know how to graph the derivative.
This week we also learned about derivative rules and how to find the anti derivative. Overall I understand a majority of the rules however I got very confused when I attempted to do the anti derive worksheet and I could not find the anti derive material in the book to help me. I also don’t understand how to find the slope of the tangent. I know how to plug in the points but then I get confused and don’t know how to finish the problem. I am hoping that now that Golf is finished and I’m actually in class things will start to make sense.