1) Differentiation and integration are considered inverse operations, but they were invented separately and have different geometric interpretations.

a) Give the formal definition of derivative. Using a specific example, compute the derivative using the definition. Show how the derivative can be interpreted algebraically and graphically.

b) Give the formal definition of integral. Using a specific example, compute the definite integral using the definition, or approximate the integral using a Riemann sum. Show how the integral can be interpreted algebraically and graphically.

c) Discuss at least two applications using derivatives. Use an example for each application.

d) Discuss at least two applications using integrals. Use an example for each application.

2) In probability theory, there are two kinds of random variables, discrete and continuous.

a) Explain the difference between discrete and continuous random variables. Give examples of each.

b) Answer the following questions on discrete random variables:

i) Describe the characteristics of a discrete random variable and its probability distribution.

ii) How do you find the probability of an event involving a discrete random variable? Use an example to help illustrate.

iii) What purpose do the mean, median, and standard deviation serve in describing the characteristics of the distribution of a discrete random variable? How do you find the mean, median, and standard deviation? Use the example in part ii) to help illustrate.

iv) The binomial distribution is the most famous of the discrete distributions. Describe the properties of this distribution and how you find the mean, and standard deviation. How do you find the probability of an event? Provide an example.

c) The study of the distribution of continuous random variables is the part of probability theory most closely related to calculus. Associated with each continuous random variable is a function called a probability density function. Answer the following questions on continuous random variables:

i) Define a probability density function. Give an example.

ii) How is the probability density function used to find the probability of an event? Use example from part i) to illustrate.

iii) Give formulas for finding the mean, median, and standard deviation for the distribution of a continuous random variable. Use example from part i) to illustrate.

iv) Many important random continuous phenomena are modeled by a normal distribution. What is the probability density function for the normal distribution?
3) The art of mathematics is creating proofs. Just as a painter has some basic modes of painting, such as oils and watercolors; so the mathematician has some basic modes of proof.

• Proving by direct conditional proof. We assume P with the explicit intention of deducing Q.

• Proving by contrapositive. We assume with the explicit intention of deducing , i.e., using the equivalence .

• Biconditional proof. Proving sentences of the type using the equivalence
.

• Proof by cases. Proving sentences of the type using the equivalence
.

• Proof by contradiction. A proof by contradiction of a sentence P is a proof that assumes and yields a sentence of the type , i.e., using the equivalence
.

a) Show that the pairs of logical statements in the last four bullets are logically equivalent.

b) Give two examples of each type of proof. You may select proofs from Calculus, Linear Algebra, Abstract Algebra, Geometry, Number Theory, Introduction to Real Analysis, and Foundations of Mathematics.There should be a variety of examples, i.e., the proofs should not be similar in form, and should include a variety of different topics. (Note: For these examples, you need not modify the proof, but be sure to cite the source.)

i) Direct conditional proof.

ii) Conditional proof using contrapositive.

iii) Biconditional proof.

iv) Proof by cases.

v) Proof by contradiction.

4) The concept of infinity has been studied in Calculus, Modern Geometry, and Foundations of Mathematics. Discuss the uses of infinity in these courses as indicated below.

a) In Calculus, infinity has been used in some limits, improper integrals, sequences, and series. Give examples of the different ways that infinity is used.

b) In Geometry, points of infinity or ideal points were discussed in hyperbolic geometry.

c) In Foundations of Mathematics, infinity played a major role in sets and cardinality. Give examples of sets of each infinite cardinality.

5) A function is one of the key concepts in many areas of mathematics.

a) Define a function, a 1-1 function, an onto function, an inverse of a function. Provide examples for each as you define them.

b) Functions were also studied in various mathematics courses. In these courses, the functions had additional properties. Discuss these functions, including their additional properties, and provide examples. Note: Some of these functions may have different names in different courses.

i) Linear Algebra.

ii) Abstract Algebra.

iii) Modern Geometry.

6) Algebraic systems relate sets of elements with binary operations.

a) Describe the algebraic systems setting up a hierarchy beginning with groups, then abelian groups, rings, integral domains, and fields. Explain how each additional more restrictive system is an extension of a previous system with additional properties. Give examples for each of these five algebraic systems. Be sure to include examples of groups that are not abelian. Also, provide examples of rings that are not integral domains, and examples of integral domains that are not fields.

b) Consider the set with operations matrix addition and multiplication. Discuss the properties of an algebraic system that has. Is a group and/or an abelian group? Is a ring and/or an integral domain? Show, by examples, which properties of a field this algebraic system does not have.

c) Define a vector space. Is a vector space a group and/or an abelian group with respect to vector addition? Is a vector space a ring, integral domain, and/or field with respect to addition and scalar multiplication? Pay particular attention to the definitions of the arithmetic operations for each of the algebraic systems.