Reviews for Tests
Most tests and quizzes will consist of short answer questions, calculations, and other "typical" math problems. Some questions are similar to homework, while others ask you to synthesize the text and class discussion. Expect one short essay question. You should look over your class notes, your returned homework, and the homework answer sheet. The following questions provide a guide to the important ideas of each chapter.
Analytic Geometry
- What is mathematics? What do mathematicians study? Is math a science or an art?
- What different kinds of numbers are there? Give examples.
- What is analytic geometry? What are its "parts"? Why is it a great idea? Give some examples of how analytic geometry approaches problems. Compare and contrast to other approaches.
- What is algebra about? What is it an abstraction of? What is algebra used for?
- How do you solve a quadratic equation?
- What is geometry about? Who were the first people to do geometry?
- What are some of the terms that Euclid defined? What are his "Common Notions" and what role do they play? What are his Postulates and what role do they play?
- State some theorems (propositions) that are found in Euclid's Elements. How do you prove them?
- Which is easier: algebra without geometry, or geometry without algebra? Explain.
- What is a line? How do you find the equation of a line? What information do you need to find this equation?
- What is a circle? How do you find the equation of a circle? What information do you need to find this equation?
- What are some other useful formulas from Cartesian geometry?
Modeling
- What is a mathematical model? What purposes does it serve?
- What families of models did we study? Draw graphs characteristic of each kind of model. What type of parameters a and b are associated with each graph?
- How do you decide which model is most likely to fit a data set?
- How do you find a model to fit a set of data? Describe the role of the "new data" and how it is derived from the original data for each kind of model.
- How do you measure the goodness of the fit?
- What kind of real world situations can mathematics model well? Which models go with which situations most frequently? What kind of situations might mathematics have a harder time with?
- Who in history was instrumental in developing the mathematical view of the world that scientists use today? What, briefly, did they do, and when did they live?
- Write a paragraph about mathematical modeling: what is it, why is it a great idea? Give an example to illustrate your points.
Calculus
- What kind of questions does calculus answer? Why are simpler methods not sufficient?
- Where besides math is calculus used?
- Compare and contrast the value of a function f at x = a with the limit of the function at a.
- Summarize the "calculus approach" without resorting to short-hand formulas. In particular, write an English description of how you find the instantaneous velocity of an object from the position function. Also, how do you find the area under the graph of a function and over an interval on the x-axis?
- What four different kinds of problems are united under the calculus umbrella?
- State the Fundamental Theorem of Calculus in some form.
- What does the symbol Δ represent? How is it used to indicate change and rate of change?
- What units are typically used for rate of change?
- Compare the ideas of average rate of change and instantaneous rate of change.
- What is a derivative? an antiderivative?
- Using tables of derivatives or antiderivatives, how do you do the following?
- find the instantaneous velocity of a position function?
- find the position function from the velocity function?
- find the slope of a line tangent to a graph? find the equation of that line?
- find the area under the graph of a function?
- How do you estimate answers to the above when the tables cannot be used?
- What does the program SUM do, and for what kinds of problems is it helpful?
- How old is calculus? Who first articulated the methods of calculus? When?
- Give at least one interesting fact about the history of calculus or its discoverer that is not found in our textbook.
Statistics
- What kinds of questions are addressed by probability and statistics?
- What is a probability? How are probabilities calculated?
- What is a statistic? How many did we study? How do you calculate them?
- What is a statistical distribution? How is a distribution described?
- What is a Bernoulli trial? Give some examples.
- What kind of experiments probably result in normal distributions? Give some examples.
- In a normal distribution, what is the relationship between an x-value and a z-value?
- Given a population with some normal attribute, how do you determine either what fraction or how many of the population are within a certain range for the attribute?
- What is the Z-test? When can it be used? How is it performed? What kinds of answers does it produce? How accurate is it?
Proof
- What is a mathematical proof? What purposes does it serve?
- What is an axiomatic system? Give two examples we have studied in this course.
- Explain the terms primitive term, axiom, theorem.
- Explain the terms and, or, not, negation, converse, contrapositive, implies, all, some. How are these ideas and the rules of logic used to create new statements from old ones?
- How do you negate a statement? That is, how do you turn a false statement into a true one and vice-versa? How can you simplify negated statements?
- Give some examples of true mathematical statements.
- Give an example of a true statement that is not addressed in the field of mathematics.
- How is mathematical truth different from truth in other fields?
- Give an example of one theorem and its proof.
- What rules of arithmetic and algebra did we use in proofs? In particular, what is the Zero Principle?
- How do you prove that something exists? How do you prove that something is unique? How do you prove that a given set of numbers is the entire solution set to an equation? How do you prove that an algebraic equation is always true? How do you prove that an algebraic equation is not always true?
- What is a natural number? What are the precise definitions of even number and odd number?
- What philosophical issues must mathematics address about proof?
- How do mathematicians judge whether a proof is correct?
Famous Theorems
- State several famous theorems that we studied.
Final Exam
- Know Euclid's proof methods for Propositions I.1, I.3, I.9, and I.10.
- Explain how analytic geometry contributes to each of the other areas we studied: modeling, calculus, statistics, and proof.
- Were there hints of the methods of calculus in Greek geometry? Explain. How did calculus improve upon ancient Greek geometry?
- How does calculus help in statistics?
- What are the elements of an axiomatic system? Would the ancient Greek mathematicians have recognized this approach to math?
- What theorem from this course do you think is especially important or beautiful? Support your choice.
- What is mathematics?
Class, College, and Life
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