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Unformatted text preview: 1 PHY 3221 Spring 2011 Mathematics Self-Assessment Steven Detweiler The lack of mathematical sophistication is a leading cause of difficulty for students in Classical Mechanics. An official pre-requisite of phy3321 is phy2048, phy2049 and the math requirements include MAC 2311, 2312 and 2313 (Vector Calculus). These math courses together cover derivatives and integrals of trig and log functions, series and sequences, analytic geometry, vectors and partial derivatives and multiple integrals. We will casually be using math from all of these subjects. None of these should be completely unfamiliar to you. The following discussions and questions are grouped by subject and in approximate order of difficulty—easiest first. These are representative of the level of mathematics which I expect. You should feel very comfortable with mathematics at this level, at least through section C on calculus. Section D, on differential equations, is probably more difficult for you but important. The answers to the questions are not always given. If you do not know that your answer is correct then you are not comfortable with mathematics at this level. Sections E, F, G, H and I are more advanced than is necessary as a prerequisite for this course. But, these ideas will often be discussed in class. If you understand the Taylor series in section G, then you are likely to find section H, on calculators, interesting and amusing. Don’t be surprised if my discussions seem confusing at first: To understand math and physics often requires multiple, multiple readings while working out algebraic details with paper and pencil in hand. Section I involves an ordinary differential equation that has an interesting application to radioactivity. A. Algebra Solve for x : f ( x ) = ax 2 + bx + c = 0 . For what value of x is f ( x ) a maximum or a minimum? Make a sketch of the function y ( x ) where y = mx + b and where m and b are constants. What are the meanings of the constants m and b in terms of your sketch? Factor: ( a 2 + 4 ab + 4 b 2 ) and ( a 2- 9 b 2 ) B. Vector Algebra Let ~ A = 1ˆ ı + 2ˆ + 3ˆ and ~ B = 4ˆ ı + 5ˆ + 6ˆ . What is | ~ A | ? What is ~ A · ~ B ? 2 What is ~ A × ~ B ? What is the cosine of the angle between ~ A and ~ B ? If ~ A · ~ C = 10 and the angle between ~ A and ~ C is 30 ◦ , then what is the magnitude of ~ C ? C. Calculus If x , v and a are constants and x ( t ) = x + v t + 1 2 at 2 then what is dx/dt ? What is d 2 x/dt 2 ? If a < 0, does the function x ( t ) curve up or down? If x is negative when t = 0 and x is positive when t is very large: then for precisely which values of t is x positive? Evaluate the derivative d dt A cos( ωt + φ ) where A , ω and φ are constant....
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This note was uploaded on 12/15/2011 for the course PHY 3221 taught by Professor Chan during the Spring '08 term at University of Florida.
- Spring '08