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Calculus Theorems and DFN
Major definitions of calc 1
32
Mathematics
Undergraduate 1
12/02/2010

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Cards

Term
Limit Definion
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 |x-c|<d, then |f(x)-L|<E.
Term
Squeeze Theorem
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
Continuity
Definition
a function that has 1 y-value for each x-value in the open interval and doesn't jump fom one value to another without taking on every value in between.
Term
Continuity at a Point
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
Vertical Asymptote
Definition

x=a is a VA if f is either limx->a- f(x)=±∞

OR

lim x->a+ if f(x)=±∞

Term
Horizontal Asymptotes
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
Derivative
Definition
f(x) is indicated f'(x) where the derivative of the function f(x)=Δx->0 (f(x-Δx)-f(x))/Δx.
Term
Derivative
Definition
Represents the slope of the tangent line.
Term
Differentiability
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
Extrema
Definition
1. Absolute extrema- the highest (absolute max) and lowest (absolute min) values on a graph.
Term
Extrema Value Theorem
Definition
If f is continuous on closed interval [a,b], then f has both an absolute max and min value.
Term
Relative Extrema
Definition
points higher (relative max) or points lower (relative min) than the points on either side.
Term
critical numbers
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
Rolles Theorem
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
Mean Value Theorem
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)/b-a
Term
Differentials
Definition

y=f'(c)(x-c)+f(c)

method that uses tangent line approximation to estimate a function at a given point.

Term
1st Derivative Test
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
2nd Derivative Test
Definition

1. Domain

2. find ppoi

3. Test the ppoi in chart

Term
Simpon's Rule
Definition
(Δx/3)[f(x)+4f(x)+2f(x)...f(x)]
Term
Trapezoidal Rule
Definition
a=.5h(b1+b2)
Term
Riemann Sums
Definition
Let F(x) be defined on [a,b] and let Δ be an arbitrary partition of [a,b]. The ci is any point in the ith subinterval, [ε f(ci)Δx]
Term
Definite Integrals
Definition
||Δ||->0  then the summation f(ci)Δxi is defined and exists on[a,b], then F is integrable on [a,b] and the true area is found.
Term
Theorem with no name
Definition
For a definite integral to be interpreted as area, then f must be continuous and non-negative.
Term
Evaluation part of FTOC
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
FTOC Part 2
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/b-a)on the integral f(x)dx
Term
Mean Value Theorem
Definition

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

 

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