Dr. Shelly, AP Calculus BC Chapter 7 and 8.2
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| 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
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| 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
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| Logistic function | y = M/(1+ae^(-kx)), the solution of the logistic differential equation, where the constant a is determined by the initial condition
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| critical point | occurs at x = c if and only if f(c) is defined and f '(c) is either zero or is undefined
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| point of inflection or inflection point | The point (c, f(c)) if and only if f "(x) changes sign at x = c
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| cusp | (c, f(c)) if and only if f ' is discontinuous at x =c
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| plateau point | (c, f(c)) if and only if f '(c)=0, but f '(x) does not change sign at x = c
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| local maximum | If f ' (x) goes from positive to negative at x = c, and f is continuous at x = c
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| local minimum | If f '(x) goes from negative to positive at x = c, and f is continuous at x = c
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| concave up at x = c | If f "(c) is positive
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| Concave down at x = c | If f "(c) is negative
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| 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
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