Sections 1.3, 2.4, 4.1, and 4.2

Main Points: Section 1.3 was already read a few weeks ago, but it was a review of the first derivative to prepare us for the second derivative. Section 2.4 introduces the second derivative. Its notation is either f " (x) or d^2y/dx^2. It is the rate of a change of a rate of change. The second derivative gives information about the first derivative, such as if it is increasing or decreasing. It also tells whether the original funciton is concave up or down. Section 4.1 talks about local maxes and mins. If f(p) is less than or equal to valuesf for points near p, then the function has a local minimum at p. Also, f has a local maximum at p if f(p) is greater than or equal to values of f for points near p. Also, a critical point is a point, p, where f ' (p) = o or undefined. A critical value is the value of f(p) where the critical point is. If a function that is continuous on an interval has a local max or min at p, then p is a critical point or enpoint on the interval. Two ways to test for the maxes and mins of a function are the first and second derivative tests. The first derivative test says that if f changes from negative to positive at a point, then f has a local minimum at p. Also, if f changes from positive to negative, it has a local maximum at p. The second derivative test states that if p is a critical point of f and f ' (p)=0, and if f is concave up at p, then f has a local min at p. Also, if f is concave down at p, then f has a local max at p. However, not ever critical point is a max or min. In section 4.2, the text introduces inflection points. Inflection points are points where the graph of a function change concavity. This happens at points where f '' (x) = 0 or undefined. However, not all points where f '' (x) = 0 is an inflection point.

Challenges: The theory behind maxes and mins can be a bit confusing, especially the difference critical values and critical points.

Reflections: My teacher in high school calc spent a lot of time on maxes and mins, along with inflection points. Although I am pretty familiar with these concepts, I still don't understand their importance in solving problems in real life or how they are actually applied to real world situations.