Section P.1

• Know the rules for inequalities (p.2), particularly: (i) a > b =>  -a  <  -b and (ii) a > b > 0 or 0 > a > b => 1/a < 1/b.
• Know the different types of intervals and their graphs  (p.3).
• Know the definition of absolute value and its properties.
• Remember that |x - a| is the distance between x and a.
• Know what a centered interval is: |x - a| < b  <=>  -b < x -a  < b,

• Remember the distance formula (p.10).
• Know the point-slope and slope-intercept formulas for the equation of a straight line.
• Remember that two straight lines are perpendicular if their slopes p and q satisfy pq = -1.

• Remember the definition of a function and its natural domain and range.
• In finding domain and range of functions, remember that:   (i) the square root of a real number is defined only when that real number is nonnegative;   (ii) the value of the square root is also nonnegative; and (iii) we cannot divide a  number by 0.
• Be familiar with the graphs of elementary functions (p.21).
• Understand composition of functions: (f o g) (x) = f(g(x)) and (g o f) (x) = g(f(x)).
• Understand: if x= 0 is in the domain of an odd function, then f(0) = 0; both x and -x must lie the domain of a function that is odd or even; and not all functions are odd or even.

• Know that to shift the graph of f(x) to the right by a positive number a, you change f(x) to f(x - a); to shift the graph of f(x) to the left by a positive number a, change f(x) to f(x + a); to shift the graph of f(x) up by the positive number a, change f(x) to f(x) + a; and to shift the graph of f(x) down by a positive number a,  change f(x) to f(x) - a.
• Know the standard equation for a circle (1) (p.28) and how to complete the square to find the center and radius of a circle, Example 9 (p.29).
• Understand how to graph parabolas, Example 12 (p.31).

• Know how to convert angles in degrees into radians using the conversion factor pi/180 before evaluating trigonometric functions.
• Remember selected values of trigonometric functions, Table 2 (p.39).
• Remember the graphs of sin(x), cos(x), and tan(x) (p. 40).
• Know that the values of sin(x) and cos(x) lie in the closed interval [-1,1] for all real values of x; that they are periodic functions with period 2 pi; that sin(x) is an odd function; and that cos(x) is an even function.
• Remember trigonometric identity (2), (p.41). You do not need to remember the angle sum or double angle formulas (p.42) or the law of cosines (p.43). If you need them, we will supply them to you.

• Know the average rate of change and slope of the secant line (p.52-53).
• Know the informal definition of limit (p.55).
• Be sure to simplify an expression by doing the necessary algebra before finding the limit. For example, cancel any common factor from the numerator and denominator of a rational function before you calculate the limit.
• Review Examples 8 and 9 (p.56-57).

• Know the properties of limits and Example 1 (p.61-62).
• Know how to eliminate zero denominators algebraically and Examples 4 and 5 (p.64).
• Know the Sandwich theorem and Example 7 (p.64).

• Understand but do not memorize the formal definition of limit (p.70).
• For finding deltas for given epsilons: review the steps highlighted on p.72 and in Examples 5 and 6 (p.71-73).
• You do not need to know how to prove the theorems.

• Review the various definitions for left-hand, right-hand, and infinite limits. You do not need to memorize the definitions.
• Know that a rational function with no common factors has infinite limits as x approaches the zeros of the denominator. Review Examples 4-6 (p.81-82).

• Remember the definition of continuity (p.87) and the continuity test (p.89).
• Know the following: for a removable discontinuity at x = c, f(x) must have a limit L as x approaches c, but f(c) is either undefined or different from L; the discontinuity is removed by the continuous extension of f(x) to F(x) (p.92); and the new function F(x) coincides with f(x) except at x = c, where F(c) = L.
• Know that for a jump discontinuity, f(x) has different one-sided limits as x approaches c from the left and right, but for an infinite discontinuity, f(x) has an infinite limit.
• Review the Intermediate Value Theorem (p.93). Know that if a continuous function f(x) changes sign in an interval [a,b], then it must have a root in that interval.

• Remember the definition of the slope of a curve and its tangent line (p.99) and how to calculate them.
• Know that to find the equation of the tangent line, evaluate the tangent slope at the given point and then use the point-slope formula for a straight line.
• Understand that, for a moving particle, the instantaneous speed at t = a is the slope of the line tangent to the displacement curve  y = f(t) at t= a.

• Know the definition of the derivative (p.109) and how to calculate the derivative of a given function at a given point using the definition. Review Examples 1 and 2 (p.110-112) and the alternative formula for calculating derivatives (p.117).
• Know when a function does not have a derivative at a point (p.114-115).
• Remember that differentiable functions are continuous but continuous functions are not necessarily differentiable.
• Skip graphing f' from estimated values (p.112-113).

• Know the differentiation rules: remember them (especially the product and quotient rules) and learn how to use them, but you do not need to be able to prove them.
• Understand the evaluation of derivatives at a given point: f'(a) is the value of the derivative function f'(x) at x = a.
• Understand second and higher order derivatives and how to compute them (p.128).

• Know the average and instantaneous rate of change of a function (p.131).
• Understand motion along a line: how to calculate velocity, speed, and acceleration from a given displacement function. Know the difference between speed and velocity: positive velocity => forward motion, negative velocity => backward motion. Review Example 3 (p.134).
• You do not need to memorize the free-fall equations (p.135).
• Understand the application of rate of change in other areas, such as Economics, Chemistry, and Biology.