Term
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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. |
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| Theoretical Definintion of a Limit |
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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. |
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Term
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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. |
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Term
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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. |
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Term
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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) |
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Term
| (Discontinuity) Removable/Hole |
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Definition
| can make function continuous by either adding or moving a point. |
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Term
(Discontinuity) Non removable
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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 |
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Term
| Intermediate Value Theorem |
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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. |
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Term
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Definition
x=a is a VA if f is either limx->a- f(x)=±∞
OR
lim x->a+ if f(x)=±∞ |
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Term
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Definition
| y=b is a horizontal asymptote for f if either lim->infinity from the left or right and still equals f(x)=b. |
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Term
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Definition
| f(x) is indicated f'(x) where the derivative of the function f(x)=Δx->0 (f(x-Δx)-f(x))/Δx. |
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Term
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Definition
| Represents the slope of the tangent line. |
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Term
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Definition
ability to take derivative at a point.
Except:
1. any discontinuity
2. Vertical Tangent
3. Corner/Cusp |
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Term
| Logarithmic Differentiation |
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Definition
| a method of finding derivatives that changes (y=) functions into ln, so we can use ln properties. |
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Term
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Definition
| 1. Absolute extrema- the highest (absolute max) and lowest (absolute min) values on a graph. |
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Term
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Definition
| If f is continuous on closed interval [a,b], then f has both an absolute max and min value. |
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Term
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Definition
| points higher (relative max) or points lower (relative min) than the points on either side. |
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Term
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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. |
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Term
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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. |
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Term
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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 |
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Term
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Definition
y=f'(c)(x-c)+f(c)
method that uses tangent line approximation to estimate a function at a given point. |
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Term
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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 |
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Term
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Definition
1. Domain
2. find ppoi
3. Test the ppoi in chart |
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Term
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Definition
| (Δx/3)[f(x)+4f(x)+2f(x)...f(x)] |
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Term
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Definition
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Term
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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] |
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Term
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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. |
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Term
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Definition
| For a definite integral to be interpreted as area, then f must be continuous and non-negative. |
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Term
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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) |
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Term
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Definition
| if F is continuous on [a,b], then all x in [a,b] d/dx[∫f(t)dt]=f(x) |
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Term
| Average Value for Integral Area |
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Definition
| If f is integrable on [a,b] then av(f) = (1/b-a)on the integral f(x)dx |
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Term
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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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