# Syllabus for Calculus

**Calculus, Part II with probability and matrices. (4h. 1
c.u.)
**Functions of several variables, partial derivatives, multiple integrals,
differential equations;

introduction to linear algebra and matrices with applications to linear programming and Markov

processes. Elements of probability and statistics. Applications to social and biological sciences.

Use of symbolic manipulation and graphics software in Calculus. Note: This course uses Maple.

Texts:

[C] Thomas/Finney, Calculus, 9th (or Alternate) Edition

[P] Probability and Matrices, Custom Edition Containing
Material Taken From:

Probability and Statistics, 3rd Edition, Morris H. DeGroot and Mark J. Schervish

and

[F] Finite Mathematics, 7th Edition, Margaret Lial,
Raymond N. Greenwell and

Nathan T. Ritchey

Maple/Calculus Lab Manual for Math 104/114/115

**Syllabus:**

Chapter |
Section & Topic |
Core Problems |

12. [C] Multivariable Functions
and Partial Derivatives. |
||

12.1 | Functions of Several Variables. | 5, 8, 13, 14, 15, 16, 17, 18, 19, 25, 45 |

12.2 | Limits and Continuity. | 11, 13, 16, 35 |

12.3 | Partial Derivatives. | 5, 19, 30, 47, 57, 63, 65 |

12.4 | Differentiability, Linearization, and Differentials |
5, 11, 20, 23, 24, 34a |

12.5 | The Chain Rule. | 3, 8, 17, 40, 41 |

12.7 | Directional Derivatives, Gradient Vectors and Tangent Planes |
2, 3, 17, 18, 27, 31, 43, 55 |

12.8 | Extreme Values and Saddle Points. | 6, 11, 17, 29, 36, 39, 42, 53 |

12.9 | Lagrange Multipliers. | 5, 14, 21, 23, 32, 46 |

13 [C] Multiple Integrals |
||

13.1 | Double Integrals | 5, 8, 15, 21, 23, 26, 29, 33, 45, 53, 54, 68 |

1 [P] Introduction to Probability |
||

1.5 | The Definition of Probability | 7, 8 |

1.6 | Finite Sample Spaces | 1-4 |

1.7 | Counting Methods | 1, 3, 4, 5, 7, 8 |

1.8 | Combinatorial Methods | 2, 3, 4, 7, 17, 18 |

1.9 | Multinomial Coefficients | 1, 3, 6, 8 |

1.10 | The Probability of a Union of Events | 1, 4, 5, 6, 9 |

2 [P] Conditional Probability |
||

2.1 | The Definition of Conditional Probability | 6, 9, 10 |

2.2 | Independent Events | 7, 8, 9, 10, 12, 16, 19 |

2.3 | Bayes' Theorem | 1, 2, 10, 11, 12 |

3 [P] Random Variables |
||

3.1 | Random Variables and Discrete Distribution | 2, 3, 4, 8 |

3.2 | Continuous Distribution | 2-8 |

3.3 | The Distribution Function | 3-8 |

3.4 | Bivariate Distribution | see below |

4 [P] Expectation |
||

4.1 | The Expectation of a Random Variable | 1, 2, 4, 5, 9 |

4.2 | Properties of Expectations | 6, 7, 8, 10 |

4.3 | Variance | 1, 2, 3, 6, 7, 9 |

5 [P] Special Distributions |
||

5.2 | The Bernoulli and Binomial Distribution | 3-7, 10 |

5.4 | The Poisson Distribution | 2, 3, 6 |

5.6 | The Normal Distribution | 3, 7, 9, 10 |

5.9 | The Gamma Distribution | 8, 9, 13, 14 |

1 [F] Least Squares Fit |
||

1.3 | The Least Squares Line | 8, 10, 11, 17 |

2 [F] Linear Equations and
Matrices |
||

2.1 | Solution of Linear Systems by the Echelon Method |
23, 25, 29 |

2.2 | Solution of Linear Systems by the Gauss-Jordan Method |
17, 24, 27, 28, 33, 39, 40 |

2.3 | Addition and Subtraction of Matrices | 27, 29 |

2.4 | Multiplication of Matrices | 15, 24, 30, 31 |

2.5 | Matrix Inverses | 11, 13, 15, 17, 19, 22, 23 |

2.6 | Input-Output Models | 1, 5, 17 |

2 [P] Markov Chains |
||

2.4 | Markov Chains | 2, 3, 11, 12, Also find limiting distributions |

3 [F] Linear Programming I |
||

3.1 | Graphing Linear Inequalities | 21, 23, 25, 29 |

3.2 | Solving Linear Programming Problems Graphically |
7, 9, 13 |

3.3 | Applications of Linear Programming | 17, 21 |

4 [F] Linear Programming II |
||

4.1 | Slack Variables and the Pivot | 5, 7, 21 |

4.2 | Maximization Problems | 7, 9 |

4.3 | Minimization Problems; Duality | 5, 7, 13 |

SAMPLE EXAM QUESTIONS also form a part of the core.

The core problems indicate the kind of basic problems you
will need to be able to solve by hand.

They also provide a guide to the basic level of difficulty to be expected on the
final exam.

Note: All sections of Math 115 have a COMMON FINAL EXAM