Math 1210-007
Final Exam Review
Fall 2010
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.
at
2.
Asymptote
3.
Infinitely fast oscillations
(e.g.
)
at
)
(e.g.
One sided limits
1.
(limit approaching from the right side)
2.
(limit approaching from the left side)
Continuity: A function is continuous at x=c if...
1.
2.
3.
exists.
exists.
.
at
)
Math 1210-007
Final Exam Review
Fall 2010
Derivatives
Rules:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Higher Order Derivatives:
1.
2.
3.
4.
Implicit Differentiation:
1.
Differentiate both sides with respect to , keeping in mind the chain rule when
differentiating an expression involving (since y is itself a function of x).
2.
Solve for
in terms of
and .
Math 1210-007
Final Exam Review
Fall 2010
Applications of the Derivative
Related Rates:
1.
Determine quantities two interest (e.g. r = radius, A = area)
2.
Determine the relation between the quantities (e.g.
3.
Differentiate the relation with respect to time in order to determine how the rates
of the two quantities are related (e.g.
)
4.
Plug in the values that are given for the quantities, and the rate of change of one
of the quantities, and solve the equation in step 3 for the unknown rate.
Sometimes only one of the two quantity's value is given, in which case you use
the original relation (before differentiating) to determine the value of the other
quantity.
)
Critical Points: There are three types of critical points...
1.
Stationary points are the values of x where
(i.e. numerator is 0).
2.
Singular points are values of x where
3.
End points are the largest and smallest values over which the function
being considered. (i.e. on the interval
the end points are
and
is undefined (i.e. denominator is 0).
is
)
Extreme Values: Maximum and Minimum values of a function
1.
The extreme values of a function MUST occur at a critical point
2.
To find the extreme values of a function first find the critical points, then evaluate
the function at all known critical points to determine which is the maximum value
and which is the minimum value.
Monotonicity & Concavity:
1.
2.
3.
4.
5.
means
means
means
means
means
is increasing
is decreasing
has a horizontal tangent line
is concave up
is concave down
Math 1210-007
Final Exam Review
Fall 2010
Optimization Problems:
1.
Determine all quantities of interest (e.g. h = height, w = width, A = area)
2.
Determine the relationships between quantities (e.g.
3.
Identify the objective function (e.g.
4.
Use the relationships to write the objective function in terms of only one variable
(e.g.
, therefore
).
5.
Determine the maximum/minimum value of the objective function by first finding
all critical points and then plugging the critical points into the objective function
to determine which yields the optimal value. In some problems you are looking
for the maximum value and in others you will be looking for the minimum value.
Integration
Indefinite Integrals (Anti-Differentiation):
1.
2.
3.
4.
5.
6.
7.
Fundamental Theorems of Calculus:
1.
1st Fundamental Theorem:
2.
2nd Fundamental Theorem:
,
)
)
Math 1210-007
Final Exam Review
Fall 2010
Method of Substitution (Indefinite Integrals):
1.
2.
Determine the correct definition for u in terms of x.
(e.g. for
the correct definition of u is
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.
)
)
Method of Substitution (Definite Integrals):
1.
2.
3.
Determine the correct definition for u in terms of x.
(e.g. for
the correct definition of u is
Determine du by using the formula
(e.g. if
then
so
).
)
Determine the new limits of integration given the definition of u.
(e.g. when
,
and when
,
limits of integration are from
to
)
. So the new
4.
Substitute u and du into the integral so that it is completely in terms of u. Also,
substitute the new limits of integration with respect to u.
(e.g.
)
5.
Evaluate the definite integral in terms of u. Note that there it is not necessary to
substitute back in for x when evaluating a definite integral.
(e.g.
)
Math 1210-007
Final Exam Review
Fall 2010
Physics
Two-Dimensional Motion:
= horizontal acceleration
= horizontal velocity
= horizontal position
= vertical acceleration
= vertical velocity
= vertical position
= horizontal acceleration
= initial horizontal velocity
= initial horizontal position
= vertical acceleration
= initial vertical velocity
= initial vertical position
When acceleration is constant you can use the following formulas:
Horizontal Motion:
Vertical Motion:
Right is the positive direction, Left is the negative direction.
Up is the positive direction, Down is the negative direction.
Vectors:
1.
2.
3.
4.
magnitude of the vector
direction of the vector
horizontal component of the vector
vertical component of the vector
Important Laws of Physics:
1.
2.
3.
4.
Newton's Second Law:
(i.e. the net force is equal to the sum of all forces on the object)
When gravity is the only force acting on an object then you may assume
a.
b.
Make sure that all of the units in your problem are consistent.
Math 1210-007
Final Exam Review
Fall 2010
Applications of Integration
Area:
1.
Vertical Slices:
2.
Horizontal Slices:
(here
is the height of the slice at x)
(here
is the length of the slice at y)
Method of Disks:
1.
This method is used when the object is a solid of revolution without a hollow center.
Furthermore, your slices should be perpendicular to the axis of rotation.
2.
Vertical Slices:
3.
Horizontal Slices:
(
= radius at x).
(
= radius at y).
Method of Washers:
1.
This method is used when the object is a solid of revolution with a hollow center.
Furthermore, your slices should be perpendicular to the axis of rotation.
2.
Vertic. Slices
(
= inner radius,
= outer radius)
3.
Horiz. Slices
(
= inner radius,
= outer radius)
Method of Shells:
1.
This method is used when the object is a solid of revolution. Your slices should be
parallel to the axis of rotation.
2.
Vertical Shells:
3.
Horizontal Shells:
(
= radius and
(
= radius and
height).
height).
Math 1210-007
Final Exam Review
Fall 2010
Arc Length:
1.
Parametric:
2.
Standard:
Work:
1.
2.
(
Spring:
= force applied at x)
(x = distance the spring is stretched beyond its natural length)
Math 1210-007
Final Exam Review
Fall 2010
Practice Problems
Limits:
Section 1.3: #1 - 12
Section 1.3: #25 - 34
Section 1.3: #41 - 44
Section 1.6: #1 - 16
(evaluating limits)
(using limit theorems)
(left and right side limits)
(identifying continuous functions)
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)
Applications of Differentiation:
Section 2.8: #1 - 9
Section 3.1: #5 - 26
Section 3.2: #1 - 10
Section 3.2: #11 - 18
(related rates problems)
(finding critical points and extreme values)
(finding where a function increases or decreases)
(finding where a function concave up or concave down)
Integration:
Section 3.4: #10 - 16
Section 3.8: #1 - 36
Section 4.4: #1 - 14
Section 4.4: #15 - 26
Section 4.4: #35 - 52
(optimization problems)
(finding anti-derivatives)
(evaluating definite integrals)
(finding anti-derivatives using the substitution method)
(evaluating definite integrals using the substitution method)
Physics:
1D Practice:
2D Practice:
http://www.math.utah.edu/~camacho/files/pp_11_8.pdf
http://www.math.utah.edu/~camacho/files/pp_11_9.pdf
Applications of Integration:
Section 5.1: #1 - 10
Section 5.2: #5 - 16
Section 5.2: #18 - 26
Section 5.3: #1 - 12
Section 5.4: #1 - 10
Section 5.5: #1 - 8
(calculating area of a region)
(evaluating volume using the method of discs)
(evaluating volume using the method of washers)
(evaluating volume using the method of shells)
(evaluating arc length of a curve)
(evaluating work done on a spring)