Calculus Final Review Test 1 (Unit 4-5)

• Epsilon-Delta/Formal Definition of a Limit

  If for every ε > 0, there exists a δ > 0 such that if 0 < | x-a | < δ , then |f(x) - L| < ε
• Steps on proving limits

  First find the delta value, then do the proof of the limit
• lim as x approaches a f(x) = L

  f(x) is defined on an open interval that contains "a" except possibly the value of "a" itself.
• In your proofs, make sure to add

  Choose E > 0, Let D= the delta that you solved for, Assume | x -a | < Delta
• Continuity

  A function is continuous if it isn't broken up
• To be continuous at a point x=g

  lim of f(x) as x approaches g- = L and lim of f(x) as x approaches g+= M are equal to each other & f(g)= L, which has to equal the same number/has to exist
• Removable Discontinuity

  There is a "hole" in the graph
• Jump Discontinuity

  There is a "jump" in the graph
• Intermediate Value Theorem

  If f if continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c) = k
• Includes the point (point is filled in)

  Backets [ ]
• Does not include the point (point if not filled in)

  Parenthesis ()
• Delta is the distance from a number *remember to subtract!
• Cos(3pi)

  -1