2.2 & 2.3 The derivative function and its interpretation

Main points:
• Sign of derivative indicates whether the function is increasing or decreasing
○ f'>0 ,then f is increasing over that interval
○ f'<0 ,then f is decreasing over that interval
○ f'=0 ,then f is constant over that interval
• Leibniz's notation for derivative is f'(x)=dy/dx
○ Disadvantage: awkward to specify the x-value at which a derivative is evaluated
• Units of derivative: units of dependent variable / units of independent variable
• Velocity= ds/dt and derivative of velocity (dv/dt) is acceleration
• Local linear approximation: dy≈f'(x)dx (for dx near 0)

Challenges:
I am finding it frustrating in a way to find estimations of the instantaneous rate of change, as I would prefer an exact value. I am also having hard time remembering that the derivative is the instantaneous rate of change so it actually implies how much y changes in respect to x. For example with the copper mine example (example 3), it shows the cost of extracting the next ton of copper. I didn’t understand the concept of local linear approximation at all, and it is something what I would like to hear tomorrow in the class.

Reflection:

I found the examples interesting, especially business related such as the copper mine, sugar production and cost of building extra square feet of house. I feel that they are real world examples that I might be pondering myself too at some point. I am always been also interested in the great mathematicians, so I found it interesting to be reading about Leibniz. It's the fact that they often have very unusual backgrounds and maybe I'm hoping to kind of have a chance to identify their thought processes as they made their major discoveries.