Test 3 Review — Math 308-503 (Spring 2011)

General Information

Calculators. You may use scientific calculators to do numerical calculations — logs, exponentials, and so on. You may not use any calculator that has the capability of doing algebra or calculus, or of storing course material.

Other devices. You may not use cell phones, computers, or any other device capable of storing, sending, or receiving information.

Structure and Coverage There will be 4 to 6 questions, some with multiple parts. The problems will be similar to homework problems, examples done in class and worked out in the text. The material covered is from the following sections in the text: 5.1-5.4, 5.5 & 5.6 (except for periodic functions, which we skipped), 5.7, 5.8, A.1, and A.2.

Topics covered

Laplace Transforms

• You will be given the table of Laplace transforms on p. 325 of the text. Be able to derive any of the formulas in the table, except for entries 4, 15, 17 and 18. You will be allowed to use this table for other Laplace transform problems.
• Be able to find Laplace transforms, either by using the table or by direct calculation, for the functions given in 5.1-5.8 — again, except the periodic ones in 5.5 and 5.6. (The functions include discontinuous functions, impulse function and convolutions.)
• Be able to find the convolution f∗g (Section 5.8).

Inverse Laplace Transforms

• Know how to expand a rational function in partial fractions. You may either be asked to set up the form of the expansion, without evaluating coefficients, or, in simple cases, to find the coefficients as well (Section 5.3).
• Be able to use partial fraction expansions and the table to find inverse Laplace transforms (Sections 5.3-5.8).
• Be able to use the convolution theorem (entry 15 in the table and Section 5.8) to find inverse Laplace transforms of products.
• Be able to use the Laplace transform to find solutions to initial value problems (IVP) (Sections 5.4-5.8).

Matrices

• The Algebra of Matrices (A.1)
  • Know the algebra of matrices — how to add, multiply, and take the transpose of matrices.
  • Know these special matrices: 0, the zero matrix; I_n×n, the n×n identity matrix; diagonal matrices; upper and lower triangular matrices.
  • Know the definition of the inverse of a matrix.

• Linear Systems and Row Reduction (A.2)
  • Be able to put a system into matrix form, Ax = b, and augmented forms [A| b].
  • Know the row reduction algorithm described in my Notes on Row Reduction, including the notation for row operations. Be able to use it to row reduce a matrix, solve systems, and find the rank of a matrix.
  • Be able to determine the number of solutions a system has by comparing rank(A) and rank([A| b]). See p. 560 of A.2.
  • Be able to define the terms linear combination, span, linear independence and linear dependence.