3.5, 9.3 & 9.4 Derivatives of periodic functions / Partial derivatives

Main points:

• Sine and cosine function are periodic-->their derivatives must be periodic also
• For x in radians
○ (d/dx)(sin x)=cos x
○ (d/dx)(cos x)=--sin x
• If z is a differentiable functions of t, then
○ (d/dt)(sin z)=cos z(dz/dt)
○ (d/dt)(cos z)=-sin z(dz/dt)
• If k is a constant, then
○ (d/dt)(sin kt)=k cos kt
○ (d/dt)(cos kt)=-k sin kt
• Partial derivative of R with respect to x at (a,b) is the derivative of f with y constant
○ Fx(a,b)=rate of change of f with y fixed at b, at the point (a,b)
• Partial derivative of f with respect to y at (a,b) is the derivative of f with x constant
○ Fy(a,b)=rate fo change of f with z fixed at a, at the point (a,b)
• Local linearity
○ Change in f ≈ rate of change in x-direction * delta x+ rate of change in y-direction * delta y
○ So, delta f ≈ fx*delta x + fy*delta y
• The mixed partial derivatives are equal
○ If fxy and fyx are continuous at (a,b)
§ Fxy(a,b)=fyx(a,b)

Challenges:

The derivatives of periodic functions where ok, as I have done them in the past. I still need to revise them in order to be comfortable with them. Partial derivatives are something what I haven't done in the past, and my first impression was just general confusion. Also the notation of partial derivative of R was something completely new to me and I would like to know where the notation comes from. I would like to see more emphasis put on the partial derivatives in tomorrow's class as they were just lot more complex compared to sine and cosine functions and their derivatives.

Reflection:

I found the airline example interesting in partial derivatives section as it did relate to my field of interest. The problem whether to increase labor or machinery is something what I am very likely going to encounter in my principles of economics class next semester. The topics seemed very relevant to my area of interest (economics) so I am looking forward to really gain deep understanding of partial derivatives.