MAE 301 Review for Midterm I

1. Be able to explain what an "advanced perspective" is, using a specific example like the division algorithm, or the addition formulas for sin and cos, or the solution of linear equations.

2.1.1 Know the definition of a rational number (p 2-4) and be able to prove that the sum of two rationals is always a rational (p 2-5). Understand the geometrical argument placing root 2 on the number line (fig 3 p 2-12) and know the proof that root 2 is irrational (p 2-8). Be able to generalize to the square root of any prime, and to the positive n-th root of any prime. (prob 6 p 2-10).

2.1.2 Understand how the decimal representation of a number x gives a series of nested intervals with rational endpoints, each of which contains x (example 1 p 2-15). Understand why a repeating decimal represents a rational number, and be able to compute the number (in form p/q, p and q integers) given the decimal (prob 1 p 2-19).

2.1.4 Know the proof that e is irrational.

2.2.1 Know the arithmetic of complex numbers. Be able to explain it in terms of the (a,b) representation as well as the a + ib representation (see also 4.1.1). Understand complex conjugate and absolute value (p 2-44). Be able to go back and forth between the reactangular ((a,b)) representation and the polar ([r,theta]) representation of a complex number (p 2-47). Be able to use the rectangular form to explain addition, subtraction and absolute value geometrically (p 2-50), and the polar form to explain multiplication and division geometrically (p 2-53).

3.1.1 Understand the two definitions of "function" (the dynamic definition that says the function is a rule or process "associating..." (p 3-3) and the static definition that says a function is a certain subset of a cartesian product (p 3-5)) and be able to show that they are equivalent.

3.2.1 Understand the differences between the families of functions: polynomial/rational, exponential, logarithmic, trigonometric and be able to tell from the graph of a function what type it is. (p 3-24) Understand why the graph of a function must pass the "vertical line test" (p 3-26). For trigonometric functions be able to draw a "trigonometric circle" (p 3-28) to identify the sine, cosine and tangent of an angle.

3.2.2 Understand composition of functions and inverse functions (p 3-35, 3-37). Understand the "horozontal line test" and be able to determine (by drawing the graph, by calculus, or other method) whether a given function is invertible or not. Understand why it is useful to restrict the domain of a non-invertible function in order to produce a function which is invertible (f(x)=x2 and the examples on p 3-38).

3.3.2 Understand that a polynomial of degree n is determined by n+1 coefficients, and therefore that its graph in the (x,y)-plane can be made to go through any n+1 points if no two lie on a vertical line. Be able to fit a parabola to three given points (p 3-71). Understand Lagrange interpolation (project 4 p 3-84) -watch for typos in the last line on the page: the "r"s should be "q"s. Be able to apply it to get a cubic with graph passing through 4 given points.

4.2.1 Be able to calculate the inverse of a 2 by 2 matrix with nonzero determinant (p 4-20). Know how to use matrix operations to solve linear systems of 2 equations in 2 unknowns (p 4-19, 4-20).

4.3.2 Be able to solve simple equations involving logarithms exponentials and trigonometric functions by using the inverse functions (p 4-31). (Be sure you know how to solve a quadratic by hand, and know how to use a calculator to get approximate solutions to higher-order polynomial equations and to difficult non-polynomial equations.)

October 16 2001