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14 Cards in this Set

  • Front
  • Back
What are the three steps to prove a function is continuous at a point?
1.) f(c) exists
2.) lim f(x) exists as x→c
3.) lim f(x) = f(c) as x→c
What is the formal definition of a limit / What is the definition with symbols?
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
What do all those symbols mean?
∀ = for all
∃ = there exists
⇒ = implies
⇔ = if and only if
| = such that
Definition of limit as x approaches infinity
L = lim f(x) ⇔ x→∞ or x→-∞
∀ Ɛ>0, ∃ D>0 | x>D ⇒ |f(x) - L|<Ɛ
Definition of an infinite limit as x approaches c
lim f(x) is infinite as x approaches c ⇔
∀ E>0, ∃ δ>0 | |x-c| < δ ⇒ |f(x)|>E
Definition of an infinite limit as x approaches infinity
lim f(x) is infinite as x→∞ or x→-∞ ⇔
∀ E>0, ∃ D>0 | x>D⇒ |f(x)|>E
Definition of the intermediate value theorem
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.
Properties: Reciprocals of zeros + infinity
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 -∞)
Slope of a tangent line at a point...
derivative f(c) = lim as x approaches c of ( f(x) - f(c) ) / (x - c)
Definition of differentiable
Has a derivative
★The Theorem★
If a function is differentiable at x=c, then it is continuous at x=c.
Is the converse of The Theorem true? What are three examples of this.
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
What is local linear approximation?
Using the tangent line to approximate the y-values of a function near the point of tangency.
Difference Quotients: Average Rate of Change
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