Calculus, 9 th edition Varberg, Purcell & Rigdon

Chapter 0 Preliminaries

.1 Real Numbers, Estimation, & Logic

In calculus, the principle numbers are real numbers. Be able to calculate with rational numbers (expressed as either repeating or terminating decimals) or irrational numbers (decimals that do NOT terminate or repeat) Be able to ESTIMATE answers before pushing a button on a calculator! Use good mental mathematics. Much done in math must be proven, and different methods of proof can be employed.

0.2 Inequalities and Absolute Value

Solving inequalities Solve by comparing the inequality to zero, factor if possible, and solve.

Solving Absolute Value Consider absolute value as distance, if the distance is greater than a constant, you must get further away in both directions. If the distance is less than a constant, the solution values must be within a certain range of values.

0.3 The Rectangular Coordinate System

Cartesian Coordinate System Graphs are done in the x-y system. You can find distance between any 2 points using Pythagorean theorem and midpoint of 2 any 2 points simply as the average. In both instances, a graph is often helpful in understanding the situation, prior to calculating.

Linear Equations General form: Ax + By + C = 0 Slope-intercept form: y = mx + b Point-slope form y – y1 = m(x – x1)

0.4 Graphs of Equations

Quadratic functions Graphs to a parabola Vertex at (h,k) Graph has reflection symmetry

Cubic Functions Reflects through the origin

0.5 Functions & Their Graphs

Functions Domain (x-values): real numbers which can be placed for x Range (y-values): real numbers which are created from the values for x Even functions: Reflect through the y-axis, f(x) = f(-x) Odd functions: Reflect through the origin, f(x) = -f(-x)

0.6 Operations on Functions

Functions can be added, subtracted, multiplied or divided Only consideration? Operations cannot result in a zero denominator Composition of functions: When g is composed on f, the range of f becomes the domain for g.

0.7 Trigonometric Functions

For all pts, (x,y) on the unit circle: sin t = y, cos t = x, tan t = y/x t = real number (length of arc on unit circle) that corresponds to pt (x,y) y = sin x y = cos x

Other trig functions sec x = 1/cos x csc x = 1/sin x cot x = 1/tan x Pythagorean identity (main one, others may be developed from this one)