Minima, maxima, and critical points. Rates of change. Optimization. Rates of change. L'Hopital's rule. Mean value theorem.
Minima, maxima and critical points
- Testing critical points for local extrema
- Identifying minima and maxima for x^3 - 12x - 5
- Concavity, concave upwards and concave downwards intervals
- Inflection points
Graphing using derivatives
Graphing functions using derivatives.
Another example graphing with derivatives
Using the first and second derivatives to identify critical points and inflection points and to graph the function.
Minima, maxima, inflection points and critical points
Can calculus be used to figure out when a function takes on a local or global maximum value? Absolutely. Not only that, but derivatives and second derivatives can also help us understand the shape of the function (whether they are concave upward or downward). If you have a basic conceptual understanding of derivatives, then you can start applying that knowledge here to identify critical points, extrema, inflections points and even to graph functions.
Optimization with calculus
Using calculus to solve optimization problems
Rates of change
Solving rate-of-change problems using calculus
Mean Value Theorem
Intuition behind the Mean Value Theorem
Mean value theorem
If over the last hour on the highway, you averaged 60 miles per hour, then you must have been going exactly 60 miles per hour at some point. This is the gist of the mean value theorem (which generalizes the idea for any continuous, differentiable function).
Limits have done their part helping to find derivatives. Now, under the guidance of l'Hôpital's rule, derivatives are looking to show their gratitude by helping to find limits. Ever try to evaluate a function at a point and get 0/0 or infinity/infinity? Well, that's a big clue that l'Hopital's rule can help you find the limit of the function at that point.