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-test
for 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.