Math 1220-002
Camacho
Definition:
Rules:
1.
2.
3.
4.
and
5.
and
Fall 2012
Final Exam Review

Understand and use the exponential growth formula:
 Use separation of variables to solve differential equations. Especially the following: o Exponential Growth Equation o Newtown’s Law of Cooling
 Use the integrating factor technique to solve linear first order differential equations: o
 Solve for the integration constant, C , using an initial condition.
 Extract appropriate information from a word problem to write down and solve an appropriate differential equation and answer any questions about the model.

Math 1220-002
Camacho
1.
2.
and and
3.
and
Fall 2012
Final Exam Review
Also keep in mind that….
1.
2.
3.
1.
Familiarity: Recognize the integrand as the derivative of a familiar function.
2.
Simplification: Algebraically simplify the integrand as much as possible before attempting to integrate.
3.
U-Substitution: Let u equal some function of x. Then calculate du in terms of dx.
Finally, substitute so that the integral is in terms of u. Solve the integral in terms of u.
Then substitute x into the solution.
4.
Combination: Some problems require some combination of the three techniques above.
 o Know how to set u and dv appropriately to find a solution.

Tabular Integration o Understand the process. o Know how to set u and dv appropriately to find a solution o Know when to stop integrating and differentiating.

Math 1220-002
Camacho
Fall 2012
Final Exam Review
You should be able to solve integrals having the following terms by using the corresponding substitutions




It will be helpful to know the following trigonometric identities:
 Pythagorean Identities: o o o
 Power Reduction Identities o o

Product to Sum Identities o o o

Math 1220-002
Camacho
 Distinct Linear Factors:

Repeated Linear Factors:
Fall 2012
Final Exam Review

Quadratic Factors:

L’Hopital’s Rule: provided that either…. o o
 Types of Indeterminate Forms:
OR o or o or o or or
Apply L’Hopital’s Rule directly.
Convert to fraction and Apply L’Hopital’s Rule
Take natural log, simplify, and apply L’Hopital’s Rule


 If has an asymptote at , where is in the interval then

Math 1220-002
Camacho
Fall 2012
Final Exam Review
 represents the infinite sequence of numbers
 The limit of the sequence is defined as o The sequence converges if exists and is finite
. o The sequence diverges if does not exist or is infinite.
, where

An infinite series is the sum of all terms in an infinite sequence
 The th partial sum in a sequence is the sum of the first terms.

A geometric series has the form . o If , then o If , then diverges o
 The term test says that if , then the series diverges.

The integral test says that converges if and only if converges.

A p-series , , converges when and diverges when .

The comparison test says that if , then o If converges, then converges o If diverges, then diverges

Math 1220-002
Camacho
Fall 2012
Final Exam Review
 The limit comparison test says that if o o
and
and
both converge OR
both diverge
 The ratio test says that if , then o If then converges o If then diverges o If then the test is inconclusive.
where , then either
 The alternating series of the sequence is defined as .
 An alternating series converges if
 A series is absolutely convergent if converges

A series is conditionally convergent if converges but does not.
 The absolute ratio test says that if , then o If the series converges absolutely o If the series diverges o If the test is inconclusive.
 An power series is an infinite series having the form

The convergence interval of a power series is the interval of values for which the power series converges. The interval is found by using the absolute ratio test and determining the values for which the limiting ratio is less than 1.

Operations on a power series can be formed term-wise. To differentiate a power series, one can simply differentiate each individual term and add the results together. One can anti-differentiate similarly. One can also substitute an expression for each value of x to come up with a new power series.

Math 1220-002
Camacho
Fall 2012
Final Exam Review
 The Taylor series of a function, , centered at , is a power series having the form

The Maclaurin Series of is a Taylor series centered at .

A function can be closely approximated by the first terms of its Taylor polynomial when is close to .

The Remainder (error) from using the first terms of a Taylor Series, centered at , to approximate a function, , is given by where is some number between and . This can’t be used to find the exact error, but it can be used to approximate an upper bound for the error.

Parametric Functions have the form where

Parametric functions are graphed in such a way that the x-coordinate of a point and ycoordinate of a point change independently over time. All possible curves in 2D can be graphed using a parametric function, including those that violate the vertical line test and those that intersect themselves.
 There are three important quantities we can calculate about a curve that is described parametrically.

Math 1220-002
Camacho o Slope of tangent line = o Speed of Object = o Length of Curve =
Fall 2012
Final Exam Review

It is possible to convert a parametric function to a Cartesian function by eliminating the parameter. We normally do one of the following to accomplish this. o Use the or equation to solve for . Plug this definition of into the other equation to obtain an equation that only depends on and . If possible, isolate
or onto one side of the equation. o When and are involved, solve for these trig functions in terms of and . Then use the fact that .

Polar Coordinates describe the location of a point using its distance from the origin, , and its angle relative to the positive -axis, . One can convert between polar and
Cartesian coordinates using the following formulas: o and o and

Polar Functions describe the radial distance, , in terms of . They have the form
.

To find the area of a non-uniform sector, we apply the following formula
 To find the area between to non-uniform sectors, we use the formula

Math 1220-002
Camacho
Fall 2012
Final Exam Review

It can sometimes be trick to figure out exactly what the limits of the integral should be.
This normally involves drawing a picture of the functions first and seeing where they intersect.

Math 1220-002
Camacho
Logarithmic & Exponential Functions:
Section 6.1: #3 – 14, 15 – 26
Section 6.3: #3 – 10, 11 – 22, 37 – 44
Section 6.4: #17 – 26, 27 – 33
Differential Equations:
Section 6.5: #1 – 4, 5 – 24, 39 – 45
Section 6.6: #1 – 14
Inverse Trig Functions:
Section 6.8: #39 – 54, 55 – 72
Integration Techniques
Section 7.1: #1 – 54
Section 7.2: #1 – 36, 37 – 48
Section 7.4: #1 – 16, 17 – 26
Partial Fractions:
Section 7.5: #1 – 40
Section 7.6: #1 – 12
Indeterminate Forms:
Section 8.1: #1 – 20
Section 8.2: #1 – 44
Improper Integrals:
Section 8.3: #1 – 24
Section 8.4: #1 – 32
Fall 2012
Final Exam Review

Math 1220-002
Camacho
Infinite Series
Section 9.1: #1 – 20, 21 – 30
Section 9.2: #1 – 14, 15 – 24
Section 9.3: #1 – 12, 13 – 22, 23 – 26, 27 – 32
Section 9.4: #1 – 4, 5 – 10, 11 – 34
Section 9.5: # 1 – 6, 7 – 12, 13 – 30
Power Series
Section 9.6: #1 – 8, 9 – 28
Section 9.7: #1 – 10
Taylor Series
Section 9.8: #1 – 18, 19 – 24
Section 9.9: #1 – 8, 9 – 14, 37 – 42
Parametric Functions
Section 10.4: #1 – 20, 21 – 30, 31 – 34, 35 – 46
Polar Coordinates and Functions
Section 10.5: #9, 10, 11 – 16, 17 – 22
Section 10.6: #1 – 38
Section 10.7: #1 – 32
Fall 2012
Final Exam Review