# M 274 - Fall 2017 - Exam 2

The following information is given. Virtually everything on this sheet is needed for the exam. Make sure you understand how to use each result on this sheet.

The exam is written to take 45 minutes. Use your time wisely.

The exam covers Sections 7.2-7.9. Expectations from each section are listed below.

- 7.2 - Know the definition of the Laplace transform and be able to compute a transform using the definition. Know the Laplace transform is linear. Know the definition of exponential order, piecewise continuity, and the theorem on existence of the transform.
- 7.3 - The important properties are included on the given information. Be sure that you can use them.
- 7.4 - Know the basic forms for partial fraction decomposition. Know how to complete the square. Know how to use the given table.
- 7.5 - Know how to solve an initial value problem using Laplace transforms. There will not be initial value problems with initial data not at zero (example 3 in your text). There will not be initial value problems with non-constant coefficients (example 4 in your text). However, both of these examples may show up on the final exam when we have more time.
- 7.6 - Know how to express a piecewise defined function using unit step functions and then determine its Laplace transform, and find the inverse of such. Be sure to use the translation properties on the given table.
- 7.7 - Theorem 9 is given on the table. Make sure you understand what the notation means and be able to use it.
- 7.8 - Know the definition of the convolution and be able to compute a convolution. Be able to use the convolution property on the given table.
- 7.9 - Understand the definition of the Dirac delta function and be able to use the property on the given table.

Particulars.

- The exam is 75 points.
- Many of the problems have been crafted so that the algebraic manipulation is minimized.
- You will be asked to determine the Laplace transform of various continuous and piecewise continuous functions.
- You will be asked to determine the inverse Laplace transform of various functions.
- You will be asked to solve, or partially solve, a number of initial value problems.
- "You may leave your solution in terms of a convolution," means we expect something along the line of f(t)*g(t).