Finishing off Euler's Method

Hi everyone, this is Ben and I'll be your scribe for today.

First, we went over the homework that was assigned last night. My images may not show up because blogger is undergoing maintenance today according to the notices, shown here. So I will do my best with my scribe. I will fix it once blogger is back up and running.

Here was the homework that was supposed to be finished:


and



Mr.K put some of the values near the end. So if you are not getting those numbers, you are doing something incorrectly. One reason why may be because of rounding. Rounding is not recommended because it will cause inaccuracies. You may round on your paper, but store the values onto the calculator.

Okay after that, Mr.K introduced us to a new calculator program that does the Euler Method, similar to what we did yesterday, but faster and more accurately. The program also graphs the numerical results. So we looked at the differential function on the homework last night. We saw how we could improve the accuracy of Euler's Method. By making the changes in x smaller and increaing the number of times the method is applied will increase the accuracy of the curve that is formed to the actual graph.

After playing around with the Euler Program. We worked on more differential equation problems. Here is the first one:



The key to these types of questions is to write down what you know. What is given may be explicitly shown while some may require some thinking. For example, in the question above, we are told that the automobile starts from rest. That means that the automobile has zero velocity and has travel zero feet at time zero. We were also told that the acceleration is constant. An initial value is also given. Now that we know this we can solve the problem using anti-differentiation techniques.

Here is the second question:



The goal of this question was to find the year that the country ran out of gas. So to do this, lets see what we were given. First, we were given the amount of gas the country had. We were also given the rate of gas consumption function. Notice that we were give the rate of gas consumption. That means at t=0, no gas was consumed so that means A(0)=0. So to put the goal in algebraic terms we need to find t when A(t)=100. So we anti-differentiate the given function and find c. once that is done, just make that function equal to 100 and solve for t.

I noticed that Mr.K solved the problem using a function similar to an accumulation function. But if you actually evaluate the integral you will get the exact same function to equate to 100.

"There are more ways to skin a cat. But don't really skin a cat, because that is cruel!"
~Mr.K

Just a note and a heads up, Chapter 9.1 was nothing more than applying new terminology to old things. Chapter 9.2, which involved the slope fields, shows how to solve differential equations graphically (using a graph).Chapter 9.3 talks about differential equations numerically, creating a table of values. As predicted, Chapter 9.4 will involve solving differential equations symbolically, using only algebra. (YESSS!!)

That's all we did today. So homework is Chapter 9.3 all questions. The next scribe will be Justus.

Now to continue the Youtube tradition:



Here's a little game my brother told me about. It is so funny, I couldn't stop laughing when I started playing :P

*LINK*