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x a can make f ( x ) arbitrarily large (and positive) by taking x sufficiently close to a (on either side of a) without letting x = a . There is a similar definition for lim f x a ( x ) = -¥ except we make f ( x ) arbitrarily large and negative. Graph is a line with point (0; b) and slope m. Graph is a circle with radius r and center (h; k). a2 b2 Graph is an ellipse with center (h; k) with vertices a units right/left from the center and vertices b units up/down from the center. The graph is a parabola that opens up if a > or down if a < 0 and has a vertex at (h; k). center with slope . In this section, we describe a general technique for solving first-order equations. We begin with linear equations and work our way through the semilinear, quasilinear, and fully non-linear cases. We start by looking at the case when u is a function of only two variables as that is the easiest to picture geometrically. To write algebraic expressions and equations, assign a variable to represent the unknown number. In the table below, the letter “x” is used to represent the unknown. In translation problems, the words sum, total, difference, product and quotient imply at least two parts – use parentheses when a sum or difference is multiplied. The reader should check that for the perturbation of the right-handsidebusedearlier,therelativeerrorsδx/x and δx/x satisfy the inequality δx x ≤ cond(A) δb b and comes close to equality. These can be derived from the definition of ln x as the inverse of the function ex, the definition of e, and the rules of exponents we reviewed at the start of lecture. For example, suppose X represents the roll of a fair 6-sided die, and Y represents whether the roll is even (0 if even, 1 if odd). Clearly, the value of Y tells us something about the value of X and vice versa.