1.3 Rates of Change 2.4 The Second Derivative 4.1 Local Maxima and Minima & 4.2 Inflection Points

Main Points:
The average rate of change is defined as the change in y divided by the change in x. The average rate of change for a linear function is its slope. A function is linear if the rate of change is constant on all intervals. A function is increasing if the values of y increases with x, the opposite is true if y decreases as x increases. A function is concave up if it bends upward from left to right, the opposite is true if the function bends downward from left to right. If the second derivative of a function is greater than 0 on an interval, then f ' is increasing, so the graph is concave up. If the second derivative is less than 0, then the opposite is true. The second derivative can be thought as the rate of change of the slope. A critical point is one where f '(x) = 0. This means that the tangent line has a slope of 0 at this point (indicating a possible minimum or maximum). You can test for a local maxima or minima by using the first derivative test. If f changes from increasing to decreasing, then the critical point is a maxima, if the function changes from decreasing to increasing, then the critical point is a minima. If the second derivative is positive at the critical point, then it is a minima. If the second derivative is negative, then the critical point is a maxima. The point at which a function changes concavity is called an inflection point. A point of inflection is where the second derivative equals zero. The sign of the second derivative changes from one side of an inflection point to the other.

Challenges:
The only thing that was challenging was trying to wrap my head around the idea of a rate of change for a rate of change. I understand what it means, but it's a little hard to visiualize at first. One can think of an exponential function when trying to understand this.

Reflections:
This material seems like it could have many practical applications in the real world, especially finding the maximum or minimum of a function. I'm wondering how finding an inflection point of a function could be usefull. Second derivatives could be usefull for finding information about exponential functions that represent real world problems.