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14 Cards in this Set
- Front
- Back
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What are the three steps to prove a function is continuous at a point?
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1.) f(c) exists
2.) lim f(x) exists as x→c 3.) lim f(x) = f(c) as x→c |
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What is the formal definition of a limit / What is the definition with symbols?
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L is the limit of f(x) as x approaches c if and only if for any positive number epsilon, there exists a positive number delta such that if x is within delta units of c, but not equal to c, then f(x) is within epsilon units of L.
------------------------------------------- L=lim f(x) as x→c ⇔ ∀ Ɛ>0 ∃ δ>0 | 0<|x-c|<δ ⇒ |f(x)-L|<Ɛ |
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What do all those symbols mean?
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∀ = for all
∃ = there exists ⇒ = implies ⇔ = if and only if | = such that |
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Definition of limit as x approaches infinity
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L = lim f(x) ⇔ x→∞ or x→-∞
∀ Ɛ>0, ∃ D>0 | x>D ⇒ |f(x) - L|<Ɛ |
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Definition of an infinite limit as x approaches c
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lim f(x) is infinite as x approaches c ⇔
∀ E>0, ∃ δ>0 | |x-c| < δ ⇒ |f(x)|>E |
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Definition of an infinite limit as x approaches infinity
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lim f(x) is infinite as x→∞ or x→-∞ ⇔
∀ E>0, ∃ D>0 | x>D⇒ |f(x)|>E |
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Definition of the intermediate value theorem
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If function f is continuous for all x in the closed interval [a,b] and y is a number between f(a) and f(b), then there is a number x = c in (a,b) for which f(c) = y.
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Properties: Reciprocals of zeros + infinity
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1.) if f(x) = 1/g(x) and lim g(x) = 0 as x→c
then lim f(x) is infinite as x→c 2.) if f(x) = 1/g(x) and lim g(x) is infinite as x→c then lim f(x) equals 0 as x→c (also applies to as x→∞ or -∞) |
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Slope of a tangent line at a point...
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derivative f(c) = lim as x approaches c of ( f(x) - f(c) ) / (x - c)
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Definition of differentiable
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Has a derivative
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★The Theorem★
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If a function is differentiable at x=c, then it is continuous at x=c.
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Is the converse of The Theorem true? What are three examples of this.
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No. Vertical tangency, cusps, and corners all show the converse of The Theorem to be false as they are continuous functions at x=c but are not differentiable at x=c
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What is local linear approximation?
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Using the tangent line to approximate the y-values of a function near the point of tangency.
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Difference Quotients: Average Rate of Change
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Forward: ( f(x+h) - f(x) ) / h
Backward: ( f(x) - f(x-h) )/h Symmetric: ( f(x+h) - f(x-h) ) / 2h h = tolerance a.k.a Δx |
