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.