Calculus: Research Project
Maximum group size: 2. Each group must do a different project.
Choose a calculus or other math topic from among the list below, research it, and prepare
a presentation on it which you will give to your classmates. This is to be treated like a
mini-lesson where you will assess your classmates and ensure they understand the topic
you have taught.
 [Extending Optimization]: 3-d calculus: Finding local maxima, minima,
and saddle points on surfaces of the form z  f  x, y  . (I have some nifty
free software that allows you to visualize these surfaces . . .)
 [Extending algebraic solutions to Differential Equations]: Using
Integrating Factors to solve differential equations of the form:
dy
 g ( x) y  h( x) (that are not solvable by separation of variables).
dx
 [Extending numerical solutions to Differential Equations]: Using Euler’s
Method to generate solutions for differential equations; how Euler’s
method relates to slope fields.
 [Extending Antidifferentiation techniques]: Antidifferentiating functions
of the form: f ( x)  g ( x) h( x) (i.e products) using the technique of
Integration by Parts;
 [Extending Antidifferentiation techniques]: Antidifferentiating rational
functions using the technique of Partial Fractions.
 [Extending Tangent Lines]: Rather than just using f (a ), f (a ) as the
basis for an approximation to f ( x) at the point x  a (i.e the tangent
line), what about using additional information such as f (a), f (a),...
etc. to find a polynomial approximation? Research McLauren and Taylor
polynomials.
 [Extending Slicing]: Finding lengths of curves; Circumference of circle;
Catenaries.
 [Formalizing Limits]: using the    definition of limit to prove
continuity of basic functions, and to prove rules about continuity (e.g. the
sum and product of two continuous functions are continuous)
Days Remaining
May 8, 12, 15, 19, 20, 22
Presentation Dates
May 28, 29
Math 126 Sample Schedule
(Calculus III)
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Week 9
Week
10



Intro + Section 12.1: 3D coordinate system
Section 12.2: vectors
Section 12.3: dot products, projections



Section 12.4, 12.5: dot products, lines in space
Section 12.5: planes in space
Section 12.6: cylinders and quadric surfaces



Section 10.1, 13.1: vector functions and curves
Section 10.2, 13.2: derivatives and integrals of vector functions
Section 10.3: polar coordinates



Section 13.3: curvature
Exam I
Section 13.3: normal and binormal vectors, normal plane



Section 13.4: velocity, speed, acceleration
Section 14.1: multi-variable functions
Section 14.3, 14.4: partial derivatives, tangent planes



Section 14.4, 14.7: differentials, optimization
Section 14.7: Optimization
Section 15.1, 15.2: double integrals over rectangles



Section 15.3: double integrals over general regions
Section 15.4: double integrals in polar coordinates
Section 15.5: mass and center of mass



Exam II
Taylor Notes Section 1: first-order Taylor polynomials
Taylor Notes Section 2, 3: second- and higher-order Taylor polynomials



NO CLASS: Memorial Day
Taylor Notes Section 3, 4: Taylor polynomials and Taylor series
Taylor Notes Section 4, 5: new series from old


Taylor Notes Section 5: more series
Review for Final Exam
Math 125 (Calculus II) Outline
Week
Outline/Study
Guide
Exam Archive
Topics and Textbook Sections
Outline 1
Antiderivatives, Areas and Distances and the
Definite Integral.
(Sec. 4.9, 5.1, 5.2)
2
Outline 2
The Fundamental Theorem of Calculus,
Indefinite Integrals and Total Change, the
Technique of Substitution.
(Sec. 5.3, 5.4, 5.5)
3
Outline 3
Areas between Curves, Computing Volume:
Washers and Shells.
(Sec. 6.1, 6.2, 6.3)
4
Outline 4
1
MIDTERM #1
Archive
Applications: Work and Average Value of a
Function. Midterm #1
(Sec. 6.4, 6.5)
Outline 5
Techniques of Integration: Integration by Parts,
Trigonometric Integrals and Trigonometric
Substitution.
(Sec. 7.1, 7.2, 7.3)
6
Outline 6
More Techniques: Partial Fractions and
Combining Techniques. Approximation of
Integrals.
(Sec. 7.4, 7.5, 7.7)
7
Outline 7
Improper Integrals and the Length of a Curve.
(Sec. 7.8, 8.1)
8
Outline 8
5
9
Outline 9
MIDTERM #2
Archive
More Applications: Center of Mass. Midterm #2
(Sec. 8.3)
Introduction to Differential Equations:
Separable Equations and Exponential Growth
and Decay.
(Sec. 9.1, 9.3, 3.8, [9.4 optional])
Math 300
Introduction to Mathematical Reasoning
Math 300 is a course emphasizing mathematical arguments and the writing of proofs. The
course gives students the opportunity to learn how to formulate mathematical arguments
in an elementary mathematical setting. It serves as a complement to calculus by
introducing ideas of discrete mathematics. In addition to forming a foundation for more
abstract mathematics it should appeal to students preparing to be teachers or computer
scientists.
The topics covered will include:
1. Logic and Mathematical Language
 Compound statements (``and,'' ``or,'' ``implies'') and their negations.
 The meaning of ``necessary,'' ``sufficient,'' ``if,'' ``only if,'' contrapositive, converse.
 Quantified statements and their negations.
2. Proofs
 Direct, indirect, and contrapositive proofs.
 Existence and uniqueness proofs.
 Disproof by counterexample.
 Proof by induction.
3. Elementary Set Theory
 Basic ways of defining sets.
 Subsets, unions, intersections, set differences.
 How to prove one set is a subset of another, or equal to another.
 Finite Cartesian products.
4. Functions
 Definition of a function between two sets, and what it means for a function to be welldefined.
 What it means for two functions to be equal.
 Injectivity, surjectivity, bijectivity.
 Composition of functions.
 Inverse functions.
5. Cardinality
 What it means for two sets to have the same cardinality.
 What it means for a set to be finite, infinite, countable, uncountable, and how to prove
it.
 Subsets, finite products, and finite unions of finite sets are finite.
 Subsets, finite products, and finite unions of countably infinite sets are at most
countable.
 The rationals are countable, the reals are not.
Probability and Statistics Course Outline