Term

Definition
limx>a p(x) exists and is equal to k if and only if the limit equls k and aproaches from the left and right. 


Term
Theoretical Definintion of a Limit 

Definition
if f is a function on an open interval containing (c) and (L) is a real number, then the limitx>c=L means that for each E>0, there exists at least one d>0 such that if xc<d, then f(x)L<E. 


Term

Definition
if h(x)≤g(x) for all x in a n open interval containing c, except at c itself, and if lim>c h(x)=L=limx>cg(x), then limx>f(x) exists and =L. 


Term

Definition
a function that has 1 yvalue for each xvalue in the open interval and doesn't jump fom one value to another without taking on every value in between. 


Term

Definition
a function is continuous at x=c if:
1. f(c) exists
2. limx>c f(c) exists
3. if lim x>c f(c)=f(c) 


Term
(Discontinuity) Removable/Hole 

Definition
can make function continuous by either adding or moving a point. 


Term
(Discontinuity) Non removable


Definition
1. Jump any funtion where one sided limts exist but don't equal each other.
2. Infinite Discontinuity (VA) limit @ one or both sidesm
= ±∞.
3. oscillating limit DNE 


Term
Intermediate Value Theorem 

Definition
If f is continuous on the clsed interval [a,b] and K is a number fetween f(a) and f(b) then there is at least one number c in [a,b] such that f(c)=K. 


Term

Definition
x=a is a VA if f is either limx>a f(x)=±∞
OR
lim x>a+ if f(x)=±∞ 


Term

Definition
y=b is a horizontal asymptote for f if either lim>infinity from the left or right and still equals f(x)=b. 


Term

Definition
f(x) is indicated f'(x) where the derivative of the function f(x)=Δx>0 (f(xΔx)f(x))/Δx. 


Term

Definition
Represents the slope of the tangent line. 


Term

Definition
ability to take derivative at a point.
Except:
1. any discontinuity
2. Vertical Tangent
3. Corner/Cusp 


Term
Logarithmic Differentiation 

Definition
a method of finding derivatives that changes (y=) functions into ln, so we can use ln properties. 


Term

Definition
1. Absolute extrema the highest (absolute max) and lowest (absolute min) values on a graph. 


Term

Definition
If f is continuous on closed interval [a,b], then f has both an absolute max and min value. 


Term

Definition
points higher (relative max) or points lower (relative min) than the points on either side. 


Term

Definition
numbers in the domain of a function where f'(x)=0 or where f'(x) DNE
1. at max or min, we have a horizontal tangent line
2. f'(x) DNE @ the end points b/c derivatives are limits. 


Term

Definition
Let f be continuous on [a,b] and differentiable on (a,b). If f(a)=f(b), then there is at least 1 number c in the interval (a,b) such that f'(c)=0. 


Term

Definition
if f(x) is continuous and differentiable on (a,b), then there is at least one c, in (a,b) such that f'(c)= f(b)f(a)/ba 


Term

Definition
y=f'(c)(xc)+f(c)
method that uses tangent line approximation to estimate a function at a given point. 


Term

Definition
1. find derivative
2. find critical numbers (solve for 0)
3. create test table and plug in intervals
shows max and min for function 


Term

Definition
1. Domain
2. find ppoi
3. Test the ppoi in chart 


Term

Definition
(Δx/3)[f(x)+4f(x)+2f(x)...f(x)] 


Term

Definition


Term

Definition
Let F(x) be defined on [a,b] and let Δ be an arbitrary partition of [a,b]. The c_{i} is any point in the i^{th} subinterval, [ε f(c_{i})Δx] 


Term

Definition
Δ>0 then the summation f(ci)Δx_{i} is defined and exists on[a,b], then F is integrable on [a,b] and the true area is found. 


Term

Definition
For a definite integral to be interpreted as area, then f must be continuous and nonnegative. 


Term

Definition
if f is continuous on [a,b] and if F is any antiderivative of f on [a,b], then ∫f(x)dx=F(b)F(a) 


Term

Definition
if F is continuous on [a,b], then all x in [a,b] d/dx[∫f(t)dt]=f(x) 


Term
Average Value for Integral Area 

Definition
If f is integrable on [a,b] then av(f) = (1/ba)on the integral f(x)dx 


Term

Definition
If f is continuous on [a,b], then at some points c in [a,b] f(c) =(1/ba) on the integral f(x)dx


