# MA0002 Mathematical methods B

If you have any questions, please don't hesitate to send me an email (Martin.Wanvik@math.ntnu.no) or talk to me after the lectures. The lectures will be in norwegian, but the main textbook is in english. The problem sets will also be in english (mostly chosen from the main textbook).

## Information about the course

A lot of what you need to know about this course can be found on this page. This includes a timetable (when and where the lectures/problem sessions are held) and also practical information about the exam (the date, time, duration and what aids you're permitted to use).

## Main textbook

- Calculus for Biology and Medicine, 3. utgave av Claudia Neuhauser

## Preliminary syllabus

~~(The notes are presently only available in norwegian, but that will change).~~

- Chapter 7: Section 7.1, 7.2 and 7.3.2
- Chapter 8: Section 8.1 and 8.2.1, and note about linear first-order differential equations
- Chapter 9: All, and note about determinants of matrices larger than 2x2
~~(presently only in norwegian, but an english version will be made available fairly soon)~~. - Chapter 10: 10.1-10.6 (not including 10.6.3)
- Chapter 11: 11.1-11.3 (except 11.3.2), 11.4.1 and the following note.

## Old exam problems

Can be found here. The page is in norwegian, but that shouldn't pose too much of a problem. It (usually) includes both the problems and suggested solutions ("løsningsforslag"). "Vår" and "høst" means "spring" and "autumn", respectively.

## Problem sets (exercises)

- There will be given a total of
**12 problem sets**, one for each week (starting in week 4). - In order to be allowed to take the exam, you need to hand in an get
**at least 8**of them approved by your teaching assistant. - The problem sets will mostly be taken from the main textbook, and will concern material that have been lectured in the previous week.
- You may cooperate with other students in solving the problems, but each student is required to hand in a separate solution.

** The problem sets are to be handed in by Monday, at 12:00 the week after the problem sessions.**

## Problem sessions - time and place

The problem sessions are completely voluntary - you don't have to show up if you don't want to. But it is an excellent place to work on the problem sets, as there will be a teaching assistant present at all times to help you if you get stuck somewhere.

Send an email to Martin.Wanvik@math.ntnu.no to be assigned to a group and a teaching assistant (please include information about which timeslots are suitable for you - see below for a list of possible choices). If you have an NTNU-issued email address (i.e. one which ends in @stud.ntnu.no), please include it.

A list of all registered students and which group they've been assigned to can be found here.

## Number of approved problem sets

To find out how many problem sets you have gotten approved, enter your username on this page. For this to work, you need to be registered with a NTNU-issued email address; if you don't have one, you should ask your teaching assistant directly instead (se above for email).

## Problem sets

*2nd ed.* refers to the second edition of the textbook, in case that you are using that instead of
the third edition.

Problem set | Week | Problems | Suggested solutions | |
---|---|---|---|---|

12 | 17 | 11.3 | 1,7,9 (2nd. ed: 1,5,7) | |

Exam 2008H (see here) | Problem 1 | |||

Exam 2002V (see here) | Problem 2,5 | |||

(see note) | Write down the general solution of the system \( d\mathbf{x}/dt = A\mathbf{x} \) where \( A = \begin{bmatrix} 2 & 9/2 \\ -1/2 & 2 \end{bmatrix} \) | |||

11 | 16 | 11.1 | 1,7,9,11,13,25,28,29,31,37,45,67 (2nd ed: 1,7,9,11,13,23,26,57. Problems 29,31,37,45 are given below) | |

11.1.29,31,37 | Problem statement identical to 11.1.27 (2nd ed) with \( A = \begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix}\), \( A = \begin{bmatrix} -2 & 2 \\ 2 & 1 \end{bmatrix} \) and \( A = \begin{bmatrix} -3 & -1 \\ 1 & -6 \end{bmatrix} \) respectively. | |||

11.1.45 | Problem statement identical to 11.1.37 (2nd ed.) with \( A = \begin{bmatrix} -2 & 4 \\ -2 & -2 \end{bmatrix} \). | |||

10 | 15 | 10.6 | 3,8,11,21,31,33,34,41,45,56 (2nd ed.: same) | |

9 | 12 | Note: The deadline for this problem set is monday, 8th of april at 12:00 | ||

10.4 | 1,7,19,25,35,45 (2nd ed.: 1,3,13,19,29,39) | |||

10.5 | 1,5,11,19,31,41 (2nd ed.: same) | |||

8 | 11 | 10.2 | 1,11,15,17,23,27,28 (2nd ed.: 1,5,9,11,23,27,28) | |

10.3 | 1,5,13,41,45 (2nd ed.: same) | |||

7 | 10 | More problems than usual this week, but most of them are fairly straigthforward and shouldn't require too much work. | ||

9.4 | 5,9,11,25,37,39,47,51,63,65 (2nd ed: same) | |||

10.1 | 7,19,20,21,25 (2nd ed: 7 given below, 11,12,13,17) | |||

10.1.7 | Evaluate the functions \( f_1(x,y) = 2x - 3y^2 \) and \( f_2(y,x) = 2x -3y^2 \) at \( (-1,2) \). | |||

6 | 9 | 9.2 | 71,75 (2nd ed.: 59,63) | |

9.3 | 49,55,59,61,63,65,69,71,75 (2nd ed: 61 is given below, the numbering is otherwise identical) | |||

9.3.61 | Find the eigenvalues \( \lambda_1 \) and \( \lambda_2 \) for \( A = \begin{bmatrix} a & 0 \\ c & b \end{bmatrix} \). | |||

5 | 8 | 9.2 | 51,52,63,65 (2nd ed.: 43,44,51,53) | |

9.3 | 1,13,29,41,49,55 (2nd ed.: same) | |||

(See note) | Calculate \( \begin{vmatrix} -1 & 0 & 2 \\ -1 & -2 & 3 \\ 0 & 2 & -1 \end{vmatrix} \) and \( \begin{vmatrix} 1 & -1 & 0 & 1 \\ -1 & 2 & 0 & -1 \\ 2 & -3 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{vmatrix} \). | |||

4 | 7 | 9.2 | 1,15,16,21,22,29,31,43,45,47,69,70 (2nd ed.: 1,13,15,16,23,25,35,37,39,57,58 + problem 9.2.16 given below) | |

9.2.16 | Assume that \( A \) is a \( 2\times 2\)-matrix. Find conditions on the entries of \( A \) such that \( A + A' = \mathbf{0} \). | |||

3 | 6 | 8.2 | 1,6 (2nd ed.: same) | |

(See note) | 1. Solve \( x \frac{dy}{dx} + y = e^x, \quad x > 0 \). | |||

2. Solve \( \frac{dy}{dx} - xy = 0 \) using a) the method described in the note and b) using separation of variables. | ||||

9.1 | 1,5, 25, 29,31 (2nd ed.: 15,19 and 21 replaces 25, 29 and 31, respectively. | |||

2 | 5 | 7.3 | 5,13,23,28,45 | |

Write down the partial fraction decomposition for the function \( x/(x^2 + 4x + 4) \). (Hint: first check if the denominator can be factored). | ||||

8.1 | 1,10,15,21,29,47 | |||

1 | 4 | 7.1 | 1,8,17,32,33,39 | solutions |

7.2 | 1,9,12,36,49,59 |

## Lecture plan

Date | Theme | Syllabus |
---|---|---|

2013-04-26 | Repetition: diff. equations (separable/linear first-order), parts of ch. 10 | 8.1,8.2.1, note, 10.5-10.6 |

2013-04-24 | Finished 11.4.1, the harmonic oscillator, techniques of integration (repetition) | 11.4.1, 11.2.2, 7.1-7.2 |

2013-04-19 | Applications/Examples | 11.4.1 |

2013-04-17 | Examples, non-linear systems (Alexander Lundervold was my substitute) | 11.2-11.3 |

2013-04-12 | Solution in the case of complex eigenvalues (finish), examples | 11.1-11.2, note |

2013-04-10 | Equilibria and stability. Solution in the case of complex eigenvalues | 11.1, note |

2013-04-05 | Systems of differential equations (first-order linear homogeneous autonomous systems with constant coefficients) | 11.1 |

2013-04-03 | Constrained max/min problems (Lagrange multipliers) | 10.6 |

2013-03-22 | More applications - max/min problems </del>with constraints and Lagrange's method of multipliers</del> | 10.6 |

2013-03-20 | Properties of the gradient, applications (max/min problems) | 10.5-10.6 |

2013-03-15 | More on vector-valued functions. The chain rule, implicit differentiation and directional derivatives | 10.4-10.5 |

2013-03-13 | Differentiability and linearization. Vector-valued functions and differentiation. | 10.4 |

2013-03-08 | Partial derivatives, tangent planes, | 10.3-10.4 |

2013-03-06 | Level curves, limits and continuity, | 10.1-10.2, |

2013-03-06 | Level curves, limits and continuity, partial derivatives | 10.1-10.3 |

2013-03-01 | Analytic geometry, functions of more than one variable | 9.4,10.1 |

2013-02-27 | Analytic geometry, | 9.4, |

2013-02-22 | More on eigenvalues and eigenvectors (iterated maps), Leslie matrices | 9.3.3, 9.2.5 |

2013-02-20 | Eigenvalues and eigenvectors, | 9.3, |

2013-02-15 | More on determinants, linear maps, eigenvalues and eigenvectors | note, 9.3 |

2013-02-13 | More on inverse matrices, determinants of matrices larger than \( 2 \times 2 \) | note, 9.2 |

2013-02-08 | Matrix algebra - multiplication and inverse matrices | 9.2 |

2013-02-06 | Linear systems of equations, matrices | 9.1-9.2 |

2013-02-01 | Linear first order differential equations, linear systems of equations | note, 9.1 |

2013-01-30 | Stability | 8.2.1 |

2013-01-25 | Differential equations, stability | 8.1, 8.2.1 |

2013-01-23 | Partial fraction decomposition, differential equations | 7.3.2, 8.1 |

2013-01-18 | Techniques of integration: Integration by parts, partial fraction decomposition | 7.2, 7.3.2 |

2013-01-16 | Techniques of integration: substitution (change of variables) and integration by parts | 7.1-7.2 |