Calc 1, Lec 15A: Differentiability, Derivatives of Exponentials, Compound Interest & Cnts Int Rate

(0:00) Lecture plan.
(0:52) Review differentiability and continuity (differentiability implies continuity, so discontinuity implies non-differentiability).
(2:41) Derivatives of exponential functions. Explore with a symbolic approach and Mathematica approximations. Find out what makes the number e so special, including by looking at graphs.
(14:05) Compound interest application. Describe the general situation and then look at an example with r = 0.04, n = 4, and P = 1000. Find and interpret the function values and derivative values at t = 0 and t = doubling time. Discuss Leibniz notation with a vertical line to indicate evaluation.
(22:38) Relationship to the continuous interest rate (continuous growth rate). Determine how to write everything in terms of base e (the base of the natural logarithm ln(x)). Discuss the geometric meaning of the continuous interest rate (continuous growth rate).