# Shared Flashcard Set

## Details

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

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|
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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