1.3, 2.4, 4.1, 4.2 Rates of change, second derivative, the local maxima and minima and inflection point

Main points:
• Rates of change
○ Average rate of change of y between t=a and t=b: delta(y)/delta(t)=[f(b)-f(a)]/(b-a)
○ The units of average rate of change of a function are units of y per unit of t
○ Increasing function: the values of f(x) increase as x increases
○ Decreasing function: the values of f(x) decrease as x increases
○ Concavity
§ Concave up=bends upwards as we move left to right
§ Concave down=bends downwards as we move left to right
○ Average velocity=change in distance/change in time
• Second derivative
○ f''
○ If y=f(x) then second derivative can be written as (d2y)/(dx2)
○ f''>0 on an interval --> f' is increasing --> graph of f is concave up
○ f''<0 on an interval --> f' is decreasing --> graph of f is concave down
• Local maxima and minima
○ f has a local minimum at p if f(p) is less than or equal to the values of f for points near p
○ f has a local maximum at p if f(p) is greater than or equal to the value of f for points near p
○ "local" because it is only near p
○ Critical point
§ Point p in the domain of f where f'(p)=0 or f'(p) is undefined
§ Point (p,f(p)) on the graph of f
○ Critical value
§ Value, f(p), of the function at a critical point
○ First and second derivative test
• Inflection point
○ A point at which the graph of a function f changes concavity
○ At the inflection point, f'' is zero or undefined (not always!)

Challenges:

Second derivative was pretty straightforward, the ones I had more problems was local maxima and minima and inflection point. Nevertheless I found myself still struggling with more complex forms of equations that contained e, ln, chain rule, product rule etc. I think it would be lot of help to go at least through one example of finding the local maximum, minimum and point of inflection.

Reflection:


Example about investment was interesting and I would like to see more problems like it. Otherwise I still felt that there was a lack of examples and problems related to my are of interest which is economics. This time I spent a bit more time studying the chapter and went through all of the problems, some for the second time too and it really helped to understand the topics. I don't know why I don’t find the time to do so every single time.