Math 1310-001
Camacho
Spring 2013
Final Exam Review
Limits
Rules
1.
2.
3.
4.
5.
6.
7.
The limit does not exist at a point where the function has a...
1.
Jump (e.g.
2.
Asymptote
L’Hopital’s Rule
If
Then
at
(e.g.
)
at
)
Math 1310-001
Camacho
Spring 2013
Final Exam Review
Derivatives
Definition:
provided that the limit exists. Otherwise the function is not differentiable at that value of .
Rules:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Math 1310-001
Camacho
Spring 2013
Final Exam 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 ).
2.
Move all terms having a
3.
Factor
to one side (it may be necessary to distribute first)
and solve (the expression will likely depend both on
and ).
Implicit Differentiation can be used for…

Differentiating Inverse Trig Functions

Differentiating Functions in which the base and exponents are both variable.
Applications of the Derivative
Related Rates:
0.
Understand the problem and draw a diagram the situation.
1.
Identify and name the quantities of interest. In most cases, the rate of one quantity will
be given, and the rate of another quantity is being asked for.
2.
Use information in problem to write a relation between the quantities.
5.
Differentiate the relation with respect to time to get a relation between the rates of the
quantities.
6.
Substitute the known values of each variable to find the numerical value of the
unknown rate. Sometimes the quantity of only one variable is given while both are
needed. To find the value of the other quantity use the relation between the two
quantities, substitute the known quantity and solve for the unknown.
Optimization Problems:
0.
Understand the problem and draw a diagram.
1.
Identify the quantities of interest
Math 1310-001
Camacho
Spring 2013
Final Exam Review
2.
Use information in problem to determine the objective function and constraints.
3.
Use constraints to write the objective function in terms of only one variable.
4.
Maximize/Minimize the objective function within the appropriate domain.
5.
Explain the results in a way that answers the original question
Integrals
Indefinite Integrals:
1.
2.
3.
4.
5.
Limit Definition of Definite Integral:
Fundamental Theorems of Calculus:
Math 1310-001
Camacho
Spring 2013
Final Exam Review
Techniques of Integration:
1.
Recognition and Lookup Tables
2.
Algebraic Simplification, Trigonometric Identities, Partial Fraction Decomposition
3.
Method of Substitution
4.
Integration by Parts / Tabular Integration
5.
Trigonometric Substitution
Properties of Definite Integrals:
1.
2.
3.
Approximation Techniques:
1.
Riemann Sums
2.
Midpoint Rule
3.
Trapezoidal Rule
4.
Simpson’s Rule
Math 1310-001
Camacho
Spring 2013
Final Exam Review
Practice Problems
Limits:
Section 2.3: #2 – 7, 9 – 24, 29 – 32
Section 2.5: #5 – 10, 15 – 36
(using limit laws)
(limits involving infinity)
Derivatives:
Section 2.6: #5 – 26
Section 2.7: #19 – 32, 35 – 38
Section 3.1: #3 – 26, 27 – 32
Section 3.2: #3 – 24, 25 – 27
Section 3.3: #1 – 22
Section 3.4: #7 – 36, 37 – 40, 41 – 44
Section 3.5: #3 – 16, 17 – 26
Section 3.7: #2 – 20, 21 – 27
(slope of tangent and instantaneous velocity)
(derivative of a function)
(rules of differentiation)
(product and quotient rules)
(trigonometric derivatives)
(chain rule)
(implicit differentiation)
(derivatives of logarithms)
Application of Derivatives:
Section 4.1: #1 – 30
Section 4.5: #5 – 46, 27 – 32
Section 4.6: #2 – 30
Section 4.8: #1 – 16, 19 – 36
(related rates)
(l’Hopital’s rule)
(optimization problems)
(anti-derivatives)
Integrals
Section 5.2: #2 – 4, 17 – 20, 21 – 25
Section 5.3: #1 – 30
Section 5.4: #7 – 17
Section 5.5: #1 – 6, 7 – 36, 41 – 57
Section 5.6: #3 – 24
Section 5.7: #1 – 6
Section 5.7: #11 – 18
Section 5.7: #21 – 32
Section 5.8: #1 – 22
Section 5.9: #7 – 16
(Riemann sums)
(evaluating definite integrals)
(fundamental theorem of calculus)
(substitution method)
(integration by parts)
(simplifying using trig identities)
(trigonometric substitution)
(partial fraction decomposition)
(using lookup tables)
(trapezoidal, midpoint, and Simpson’s rules)