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Calculus, 9 th edition

Calculus, 9 th edition. Varberg, Purcell & Rigdon. Chapter 0. Preliminaries. .1. Real Numbers, Estimation, & Logic. In calculus, the principle numbers are real numbers.

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Calculus, 9 th edition

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  1. Calculus, 9th edition Varberg, Purcell & Rigdon

  2. Chapter 0 • Preliminaries

  3. .1 • Real Numbers, Estimation, & Logic

  4. 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.

  5. 0.2 • Inequalities and Absolute Value

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

  7. 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.

  8. 0.3 • The Rectangular Coordinate System

  9. 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.

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

  11. 0.4 • Graphs of Equations

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

  13. Cubic Functions • Reflects through the origin

  14. 0.5 • Functions & Their Graphs

  15. 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)

  16. 0.6 • Operations on Functions

  17. 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.

  18. 0.7 • Trigonometric Functions

  19. 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

  20. 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)

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