Calc 1, Lec 24B: Optimization (Mathematica), Geometric Optimization, Intro to Differential Equations

Use Mathematica to do optimization in Calculus. Differential equations are also introduced.

(0:00) Plugging the functions into Mathematica and using Solve to find critical points (make sure you note the double equal signs == when using Solve). Evaluate functions and get numerical approximations. Make graphs and use Manipulate to make animations (in the case, to see how the graph changes as the volume of the cylinder increases).
(9:10) Geometric Optimization Example 1: Find the point on the graph of y = (x - 1)^2 which is closest to the origin.
(18:11) Geometric Optimization Example 2: Find the x value between 0 and 20 which maximizes the area of a rectangle whose vertices are at (0,0), (x,0), (x,f(x)), and (0,f(x)), where f(x) = x^2/3 - 50x + 1000.
(23:02) Introduction to Differential Equations in the case of exponential growth and decay. A function of the form y = C*e^(k*t) has the property that its derivative is proportional to the function itself (with proportionality constant k). This makes intuitive sense: the more rabbits there are, for instance, the greater the rate at which baby rabbits will be born (it's a basic fact of life).