Transcript Calculus, 9 edition th Varberg, Purcell & Rigdon

Calculus, 9th 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.
2x  7x  4
2
2x2  7x  4  0
(2 x  1)( x  4)  0
(2 x  1  0) AND( x  4  0), x  1 / 2  x  4
OR (2 x  1  0)AND( x  4  0), x  1 / 2  x  4
(,4)  (1 / 2, )
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
Ax  Bx  Cy  D  0
2
y  a ( x  h)  k
2
Ay  By  Cx  D  0
2
x  a( y  k )  h
2
Cubic Functions
• Reflects through the origin
y  ax  bx  cx  d
3
2
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)
sin ( x)  cos ( x)  1
2
2