Theorums Flashcards
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.
Extreme Value Theorem
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
Intermediate Value Theorem
a point near which the function values oscillate too much for the function to have a limit
Oscillating Discontinuity
a point of discontinuity where one or both of the one-sided limits are infinite
Infinite Discontinuity
a point of discontinuity where the one-sided limits exist, but have different values
Jump Discontinuity
a discontinuity c of f for which f(c) can be redefined so that the limit as x approaches c of f(x)=f(c)
Removable Discontinuity
if f is integrable on [a,b] its average value on [a,b] is
Definition of Average (mean) Value
a point where the graph of a function has a tangent line and where the concavity changes
Definition of a Point of Inflection
the graph of a differentiable function y=f(x) is a) concave up on an open interval I if y’ is increasing on I b) concave down on an open interval I if y’ is decreasing on I
Definition of Concavity
a point in the interior of the domain of a function f at which f’=0 or f’ does not exist is a critical point of f
Definition of a Critical Point
if f’(c)=0 and f’‘(c)<0 then f has a local maximum at x=c if f’(c)=0 and f’‘(c)>0 then f has a local minimum at x=c
Second Derivative Test for Local Extrema
if a function f has a local maximum value or a local minimum value at an interior point c of its domain and if f’ exists at c then f’(c)=0
Local Extreme Value Theorem
Let f be a function with domain D. Then f(x) is the a) absolute maximum value on D if and only if f(x)<=f(c) for all x in D b) absolute minimum value on D if and only if f(x)<=f(c) for all x in D
Definition of Absolute Extreme Values
the derivative of velocity with respect to time
Definition of Acceleration
the derivative of the position function s=f(t) with respect to time
Definition of Instantaneous Velocity
the absolute value of velocity
Definition of Speed
if and only if it is continuous at every point of the interval
a function f is continuous at point c
the slope of a curve y=f(x) at the point (a,f(a)) is f’(a) provided f is differentiable at a
Definition of the slope of a curve at a point
the derivative of the function f with respect to x is the function f’ whose value at x is
Definition of a Derivative
xzAa function y=f(x) is differentiable on a closed interval [a,b] if it has a derivative at every interior point of the interval and if h exists as h approaches 0 from the left or the right
One-Sided Derivative
F(x) becomes arbitrarily close to a particular, finite value L as x becomes arbitrarily close to some value a.
Definition of a limit
If f(x) is continuous at c then the limit is lim x–>c f(x) = f(c)
Direct Substitution
If f(x) < g(x) < h(x) and the lim x–>c f(x) = lim x–>c h(x) = L then lim x–>c = L
Sandwich/Squeeze Theorem
The line y=b is a ____ if the graph of f(x) is either lim x–> 00 f(x) = b or lim x–> -00 f(x) = b
Horizontal Asymptote
The line x=a is a ____ of f(x) if either lim x–> a- f(x) = +/- 00 or lim x–> a+ f(x) = +/- 00
Vertical asymptote
(f(b) - f(a))/ b-a
Average rate of change
lim h–> 0 f(x+h) -f(x) / h aka derivative
Instantaneous rate of change
A line perpendicular to the tangent line at the point of tangency
Normal line
Slope of tangent line
derivative
lim x–>a f(x) - f(a)/ x-a
Alternate definition of the derivative
f is continue at x=c and the derivative on the left and right of the c are equal
Derivative Piecewise
No line/point/jump discontiuity
Continuous
Non corners/cusps/vertical tangents/discontinuities
Differnitability
If a function is differentiable then it is continuous
Differentiability Implies Continuity
The process of using the tangent line to estimate the value of a funtions
Linearization
When a limit results in indeterminate form (0/0) by direct substitution, then one is able to take ether derivative of the numerator and the derivative separately
L’Hoptial’s rule
