If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k
60
New cards
Global Definition of a Derivative
61
New cards
Alternative Definition of a Derivative
f '(x) is the limit of the following difference quotient as x approaches c
62
New cards
∫-sin(x)
63
New cards
∫sec²(x)
64
New cards
∫-csc²(x)
65
New cards
∫sec(x)tan(x)
66
New cards
Extreme Value Theorem
If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.
67
New cards
Rolle's Theorem
Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).
68
New cards
Mean Value Theorem
The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.
69
New cards
First Derivative Test for local extrema
70
New cards
Point of inflection at x=k
71
New cards
Combo Test for local extrema
If f'(c) = 0 and f"(c)
72
New cards
Horizontal Asymptote
73
New cards
L'Hopital's Rule
74
New cards
x+c
75
New cards
sin(x)+C
76
New cards
-cos(x)+C
77
New cards
tan(x)+C
78
New cards
-cot(x)+C
79
New cards
sec(x)+C
80
New cards
-csc(x)+C
81
New cards
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
82
New cards
Fundamental Theorem of Calculus #2
83
New cards
Mean Value Theorem for integrals or the average value of a functions