AP Calc AB

Unit 3

Theorems:

Mean Value Theorem (MVT):

  • f is increasing whenever x2>x1 implies that f(x2)>f(x1); larger inputs result in larger outputs

  • suppose x2>x1 and f’(x)>0. the MVT says that f’(c )=(f(x2)-f(x1))/(x2-x1); suppose the inputs x2 is larger than the input x1 and the derivative of the interval is positive. the MVT states that there is a c value within [x1, x2] such that f’(c )=(f(x2)-f(x1))/(x2-x1). so if f’(x) is known to be positive on the interval [x1, x2], then f’(c ) must be positive because c is within the interval, and the denominator is known to be positive, then the numerator has to be positive so f(x2)>f(x1)

5.3 AP Classroom DV 1

  • functions are said to be increasing or decreasing within a domain

  • increasing function: bigger inputs result in bigger outputs and smaller inputs result in smaller outputs

  • decreasing function: bigger inputs result in smaller outputs and smaller inputs result in larger outputs

  • when a derivative is positive, a function is increasing: positive derivative means ROC with respect to the input of the function is positive; positive slope men’s increasing function values

  • when a derivative is negative, a function is decreasing

  • the derivative is positive when the y coordinate on the derivative graph is positive or above the x-axis

  • f(x) is increasing when f’(x) > 0

  • f(x) is increasing on [a,b] because f’(x)>0 there (or possibly f’(x)≥0 there)

  • f(x) is decreasing when f’(x)<0

  • f(x) is decreasing on [a,b] because f’(x)<0 there (or possibly f’(x)≤0 there)