Calculus: Key Concepts (9/5/12)

1
Calculus Key Concepts (9/5/12)

• What are the 3 or 4 key concepts of calculus?
• What do they mean?

2
Limits

• What does it mean to say thatlimx ? a f (x) L
  ?
• Some limits are obvious.Example limx ? 4 1/x
  ?
• But some are not!Example limx ? 0 1/x
  ?Example limh ? 0 ((x h)2 x2) / h ?
• And other types of limits...?

3
Derivatives

• What do we mean by the derivative of a function f
  (x) at a point (a, f (a))?
• Whats the (analytic) definition?Note Its a
  limit!!
• The derivative of a function at a given point is
  a number. But if we free up the point, we then
  get the derivative function f ?(x).

4
Recall Some Derivative Facts

• What is the derivative function of each of these
  functions?
• xr (r any fixed real number)
• a x (a any fixed positive number)
• loga(x)
• sin(t)
• cos(t)
• tan(t)
• And what about all the derivative rules?Example
  Whats the derivative function of f (x) x
  e(x2)?

5
Antiderivatives

• Given a function f (x), what is an antiderivative
  of f ?
• Why do we say an, not the?
• Is computing antiderivatives mechanical process
  like computing derivatives?
• Do all elementary functions have formulas for
  their antiderivatives?

6
Recall (?) Some Antiderivative Facts

• ?x r dx (provided r ? ?)
• ?1/x dx
• ?a x dx
• ?loga(x) dx
• ?sin(t) dt
• ?cos(t) dt
• ?tan(t) dt
• ?e(x2) dx

7
Definite Integrals

• What does it mean to integrate a function over
  some part of its domain?
• That is, given a function f (x) defined on an
  interval a, b, what doesmean?
• How can we compute this number?

8
Assignment for Friday

• Obtain the book.
• Carefully review all of the ideas we discussed
  today.
• Remember to always bring your text and your
  clicker (oh, and your brain!) to class.