Math 1220-002
Camacho
Fall 2012
Calculus Review
Calculus Review
Limits
Rules
1.
2.
3.
4.
5.
6.
7.
Techniques for Evaluating Limits
1.
Factor numerator and denominator and cancel when possible
2.
If taking limit to infinity, divide numerator and denominator by highest order term.
Continuity: A function,
1.
2.
3.
, is continuous at
exists.
exists.
.
if...
Math 1220-002
Camacho
Fall 2012
Calculus Review
Derivatives
Definition:
If the limit does not exist, then the function is not differentiable. All of the above notations for
derivatives are interchangeable.
Rules:
1.
2.
3.
4.
5.
(power rule)
6.
(product rule)
7.
(quotient rule)
8.
9.
10.
Higher Order Derivatives:
1.
2.
3.
4.
(chain rule)
Math 1220-002
Camacho
Fall 2012
Calculus Review
Implicit Differentiation:
1.
Differentiate both sides with respect to , keeping in mind the chain rule when
differentiating an expression involving (since is itself a function of ).
(e.g. given
, differentiate to get
)
2.
Distribute when necessary and move all terms having a
(e.g.
)
3.
Factor
(e.g.
to one side.
and solve (the derivative will likely depend both on
and so
)
Integration
Indefinite Integrals (Anti-Differentiation):
1.
2.
3.
4.
5.
6.
7.
8.
Fundamental Theorems of Calculus:
Here we are assuming that
is an anti-derivative of
.
and ).
Math 1220-002
Camacho
Fall 2012
Calculus Review
Method of Substitution (Indefinite Integrals):
1.
2.
Identify inner function and set equal to u.
(e.g. for
the inner function
Determine du by using the formula
(e.g. if
then
so
).
)
3.
Substitute u and du into the integral so that it is completely in terms of u.
(e.g.
)
4.
Anti-differentiate in terms of u. (e.g.
5.
Substitute x into the solution using the definition of u.
(e.g.
)
)
Practice Problems
Limits:
Section 1.3: #1 - 12 (evaluating limits)
Section 1.3: #25 - 34 (using limit theorems)
Derivatives:
Section 2.3: #1 - 20
Section 2.4: #1 - 18
Section 2.5: #1 - 20
Section 2.6: #1 - 14
Section 2.7: #1 - 12
(finding derivatives using rules of differentiation)
(differentiating trigonometric functions)
(differentiating using the chain rule)
(higher order derivatives)
(implicit differentiation)
Integrals:
Section 4.3: #9 – 16
Section 4.4: #1 - 14
Section 4.4: #15 - 26
Section 4.4: #35 - 52
(properties of definite integrals)
(evaluating definite integrals)
(finding anti-derivatives using the substitution method)
(evaluating definite integrals using the substitution method)