Instructions for the Final Exam

The final exam will be given in two parts. The first part (which will count as 80% of the total

score on the final exam) will be given in class on Wednesday, April 12 between 12:30 and 2:30.

The second part (20%) will be take-home and it will cover material from the programming

language Maple. No collaboration is allowed on either part of the exam. Cheating will be penalized

by at best giving a lower letter for the course grade.

For the in-class part of the exam you are allowed with 3 sheets of paper (6 pages) of formulas

from the course, in addition to the tables for the normal, t-distribution, chi-square distributions.

Also, you may use calculators, but you have to show your work for every step in your solution

(including integration, substitution, use of identitites, etc.).

A comprehensive list of subjects on which you may be tested:

Statistics

-- Averages, Variance, Covariance, Correlation coefficient for random variables

-- T- distribution - definition, the 100 * u percentile of a t-distribution

-- Confidence interval for the mean of a normal distribution (computing the sem = standard

error of the mean by using the sample standard deviation).

-- Chi-square distribution - definition, the 100 * u percentile of a chi-square distribution

-- Confidence interval for the variance of a distribution

-- Hypothesis Testing ; definition of H_0 and H_1; One tailed test and the two tailed tests

(when the variance is known and when the variance is not known) for the mean of a normal

distribution (p-values and their significance).

-- Linear Regression; Total SS, Reg SS, Res SS. The F distribution. The F-test for simple

linear. The t-test for simple linear regression. The p-value of the test.

-- Multiple Regression. The F-test for assessing if some variables are good predictors for

the dependent variable ; the p-value of the test. The t-test (or the F-test) for assessing if

one variable is significant when all the other variables are significant in predicting the

dependent variable; the p-value(s) of the test.

-- Multiple Logistic Regression. The logit transformation. The odds ratio OR and a

confidence interval for OR. Hypothesis testing in multiple logistic regression.

-- Rank correlation. Ranking procedure. The Spearman rank-correlation coefficient and

the t-testfor Spearman rank correlation (the p-value of the test).

Differential Equations

-- Separable, Linear first order DE. Exact Equations. The substitution method.

-- Euler, Bernoulli, Riccati Equations

-- Modeling with DE: logistic model, connected tanks, interacting populations (predator-prey,

competing, cooperating).

-- Linear Second order DE with constant coefficients. The method of undetermined coefficients

and the variation of parameters.

-- Existence and uniqueness issues for linear and nonlinear DEs. Obtaining global existence

from local existence.

-- Slope fields. Phase line analysis and stability of critical points for DEs. Bifurcation.

-- Systems of DE . Finding solutions with the eigenvalue-eigenvector method. Phase plane analysis

(nullclines, linear trajectories) and stability of the critical points.

-- Almost linear systems of DEs. Linearization. The Hartman-Grobman theorem. Nonlinear

Trajectories.

-- The Undamped Pendulum and the conservation of energy. Period, frequency, amplitude.

Matrix Theory

-- Solving a linear system of equations with row reduction

-- Linear transformations. The matrix of a linear transformation

-- Finding eigenvectors and eigenvalues. Generalized eigenvectors (the case of

duplication for eigenvalues)

Maple

-- define functions, plot graphs

-- solve differential equations, plot slope fields, solution curves

-- solve systems of differential equations, phase portraits, trajectories

-- compute eigenvectors and eigenvalues

-- Euler's method and the improved Euler's method for differential equations and for

systems of differential equations. Plot the solution and the approximations in the same plot.