It's all about Euler.

Today in class we spoke mostly of a man named Euler. A famous mathematician that made many discoveries in the field of calculus. He was responsible for the number "e", the symbol of pi and the use of "i" to represent an imaginary number. Euler had an identity and a method.



Euler also worked with many brilliant people, one such person is Goldbach, who stated that any even number larger than 2 can be written with the sum of any 2 prime numbers, if you feel like being famous some day, find an exception to this statement. Euler was also a father and he claimed to have made many mathematical discoveries while holding a baby in his hands, and as his other children played around his feet, now that is some fatherly love. Mr. K thought this was quite the favorable trait.

Now that we know a bit about Euler we can move on to his method. Which was the main focus in today's class. If we have a derivative that tells the slope of any line and you want to find a specific function, and not just a family of functions. You would need a slope and point, then you can get the equation of the line from the slope field. Calculate the equation of the line by moving a known point on one of the slope field lines by 0.1 units, find the slope at that point of the line then move the point 0.1 units on the new line and find a slope line at that new point, and repeat. Using the tangent lines gives a close approximation of the equation of the line. A great example is found on slide 2.


You can decrease the amount of error by using smaller intervals, such as 0.001 units instead of 0.01, but this is a very tedious process when done by hand. This method generates a table of values.

Another example we used was using local linear approximation to find 1.001¹⁰⁰.


Find the derivative of the function by replacing 1.001 with x to make it easier. Then replace x with a number close to 1.001 such as 1, this will be the x-value. By solving the original equation the y-value given would be 1. Solving the derivative equation would give the slope, which is 100. Inputting this into the equation of a line and then replacing 1 with 1.001 to find a close approximation of the original value of 1.001¹⁰⁰ which would be 1.1

Our homework is found on slides 6. We are given the initial point, the differential equation, the amount of steps between the given interval and the change in the x-value. The first few steps are done for us. All the equations we need to use are given at the top, and we find the slope by inputting the points in the differential equation. The change of x remains constant. Have fun.


and slide 7 is also homework.

Also, great job to Jamie for her performance in the talent show! Good luck on your homework and the next scribe will be Benofschool.