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Calc Ch. 7 and 8.2

Dr. Shelly, AP Calculus BC Chapter 7 and 8.2

Euler's Method for Solving Differential Equations 1. Solve the differential equation for dy in terms of x, y, and dx, 2. Substitute values of x, y, and dx to calculate a value of dy, 3. Find the approximate new value of y by adding dy to the old value of y, 4. Repeat procedure to find next dy at next x
Logistic Differential Equation dy/dx=ky*(M-y)/M or dy/dx = k/M*(y)(y-M) where M is the maximum sustainable value of y as x increases & k is a proportionality constant
Logistic function y = M/(1+ae^(-kx)), the solution of the logistic differential equation, where the constant a is determined by the initial condition
critical point occurs at x = c if and only if f(c) is defined and f '(c) is either zero or is undefined
point of inflection or inflection point The point (c, f(c)) if and only if f "(x) changes sign at x = c
cusp (c, f(c)) if and only if f ' is discontinuous at x =c
plateau point (c, f(c)) if and only if f '(c)=0, but f '(x) does not change sign at x = c
local maximum If f ' (x) goes from positive to negative at x = c, and f is continuous at x = c
local minimum If f '(x) goes from negative to positive at x = c, and f is continuous at x = c
concave up at x = c If f "(c) is positive
Concave down at x = c If f "(c) is negative
The second derivative test If f '(c)=) and f "(c) is positive, then f(c) is a local minimum. If f '(c)=0 and f "(c) is negative, then f(c) is a local maximum. If f '(c) & f "(c)=0, then it is not distinguishable
Created by: sissiloo