Derivatives of periodic functions / Partial derivative

3.5, 9.3, 9.4

1) Main Points

Chapter 3.5 continues our exploration of derivatives, this time finding the derivative of periodic functions. The book noticed that the rate of change for a sine function was also in a periodic but reversed manner. They graphed the points and found that the derivative function of a sine function was a cosine function. They proved it by finding the derivative of a certain point on the sine graph and comparing it to the cosine graph. They did this also for the derivative of cosine and found that it was -sin(x). 9.3 and 9.4 introduce the concept of derivatives to multivariable functions. We know that in a multivariable function if you make one variable a constant the function changes in accordance with the other variable, so it is possible to find the rate of change of the function when keeping a variable constant. This is called the partial derivative and if (y is fixed) is shown by: the limit as h approaches zero {f(a+h,b) - f(a,b)}/(h). If x is fixed then its: lim (h app. 0) [f(a+h,b)-f(a,b)]/(h). You can use this model to find the partial derivative of a table or a contour diagram. Partial derivatives can also be computed algebraically by using our knowledge of derivatives. If you consider one variable in a two variable equation a constant you can solve for the derivative in that situation. The book wrote some shortcuts for these as Second-order partial derivatives.

2)Challenges

I am confused about what the table of "Second-Order" partial derivates is about. Another challenge that I foresee is when solving for partial derivatives in multivariable functions you treat the fixed variable as a constant which means that you just multiply the function by it and it doesn't just disappear or become zero.


3)Reflections

Knowing the derivatives for periodic functions seems like it will have plenty of real world applications. So much of the real world changes periodically like day light, tides, population (sometimes), temperature/weather, using the derivative will help us predict those changes. Multivariable functions also have many real world applications and it will be useful to know the rate of change for each part of the equation.