Download Linear equations

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Pre-Calculus Review
Interval notation -Closed interval
Open interval
[ a, b ]
( a, b )
Quadratic formula -Slope--
a<x<b
a<x<b
x=
 b  b 2  4ac
2a
( x1 , y1 ) ( x2 , y2 )
y  y1
m 2
x 2  x1
Linear equations -Point-slope form -Slope-intercept form -Vertical line -Horizontal line --
y - y1 = m(x - x1)
y = mx + b
x=a
(no slope)
y=b
(zero slope)
Parallel lines have the same slope
Perpendicular lines have slopes that are negative reciprocals
 x, x  0
x 
 x , x  0
Absolute value --
Distance between 2 points --
( x1  x 2 ) 2  ( y1  y 2 ) 2
Domain and Range -- domain - set of all x values (independent variable)
range - set of all y values (dependent variable)
a variable y depends on a variable x in such a way that each value of x
determines exactly one value of y (For each x there is only one y , y  f (x) )
One-to-One Functions-- a function in which for each y value there is exactly one x value
Function --
D
Function
R
D
R
One-to-One Function
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D
R
Not a Function
2
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Example 6: Sketch a graph of f(x) =
3
x  2  1. Label two points on the curve.
Example 7: Sketch the graph of the function f(x) = x3 – 12x2 + 35x..
x 2  3x  40
Example 8: Sketch the graph of the function f ( x)  2
?
x  2 x  35
Example 9: Solve x 3 + 3x 2 –x -3 < 0
Example 10: Solve 2 x 1  4
Example 11: A corral is to be built using the side of a barn as one of the sides of
the corral. The builder has 600 feet of fence. What are the dimensions of the corral
that will maximize the area of the constructed corral? Find both length and the
width.
Example 12: If f(x) = x 2 +kx +2 and f(4)=30 then find the value of k.
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Piecewise function -- the formula for the function changes depending on the value of x
1  x, x  1
Example 13: Graph f (x) 
x2 , x  1
 1, x  1
Example 14: Graph f (x) 
3 x  2, x  1
7  2 x, x  1
Symmetry --
f ( x)  x 2 )
f ( x)  x 3 )
x - axis (x, y) and (x, - y) on graph ; not a function (i.e. y 2  x )
y - axis f(-x) = f(x) ; even functions (i.e.
origin f(-x) = - f(x) ; odd functions(i.e.
Composite function --
 f  g x  f gx
Example 15: f ( x) 
2
and g ( x)  x 3
2
4 x
a. Find the Domain and Range for f (x) and g (x ) .
b. f ( g ( x)) 
c. g ( f ( x)) 
d. Does f (x) or g(x) have any symmetry?( even , odd)
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Example 16: f ( x)  ln(7  x) and
g ( x)  e x  7 .
a. Find the Domain and Range for f (x) and g (x ) .
b. f ( g ( x)) 
c. g ( f ( x)) 
d. Does f (x) or g(x) have any symmetry?( even , odd)
Inverse Functions-- A function f (x) has an inverse f 1 ( x) on a Domain D if f (x) one-to-one
over the Domain. To find the inverse of a function switch the positions of x
and y and then solve for y. When f ( x) and f 1 ( x) are graphed they will be
reflections of one another across the line y  x . Domain of f (x) becomes
Range of f 1 ( x) and Range of f (x) becomes Domain of f 1 ( x) . Also
f ( f 1 ( x))  f 1 ( f ( x))  x
Example 17: Find the inverse of f ( x)  2 x  3
Example 18: Find the inverse of f ( x) 
x3
x2
Example 19: Find the inverse of f(x) = 9 – x 2 for all x > 2 then find domain of
the inverse f -1 (x)
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Exponential Functions and Logarithms
a  1, exponential growth
Basic Exponential Function-(If k<0 then the values of ‘a’ that
cause growth or decay switch)
y  ka x , k  0
0  a  1, exponential decay
Example 20: Write a function in the form f ( x)  ab x where a>0 and b>0
if f (0)  4 and f (3)  32
Example 21: Graph y  2 x
Example 22: Graph y  2 x
Example 23: Graph y  2 x  2
Example 24: Graph y  ln( x)
Example 25: Graph y   ln( x  4)
Example 26: Graph y  ln(3  x)
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Exponential Functions Used to Compound Interest and Calculate Population-r
B  P0 (1  ) t n or B  P0 e r t
n
Example 27: The population in Knoxville is 500,000, and it is increasing at a rate of
3.75% per year.
(a) Estimate the population in 2010
(b) When will the population be 750,000
Example 28: How long does it take to double an investment with r  6.25%,
(a) Compounded quarterly.
(b) Compounded continuously.
Example 29: The half-life of a radioactive substance is 50 days starting at 10 grams.
(a) Express the amount of substance as a function of time
(b) When are there 3 grams of the substance remaining?
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Rules for Exponents--
(1) a x  a y  a x  y
ax
(2) y  a x  y
a
(3) (a x ) y  a xy
(4) a x  b x  (a  b) x
(5)
Rules for Logarithm--
ax  a 
 
bx  b 
x
(1) log a xy  log a x  log a y
x
(2) log a  log a x  log a y
y
(3) log a x y  y  log a x
ln x
(4) log a x 
ln a
y
(5) log a a  y
(6) y  log a x  x  a y
Trigonometry
y = sin x
y = cos x
D = ( - , )
R = [ -1, 1 ]
D = ( - ,  )
R = [ -1, 1 ]
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Period = 2
Period = 2
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y = tan x =
sin x
cos x
D = { x : x  /2  k }
y = cot x =
cos x
sin x
D = { x : x   k}
y = sec x =
1
cos x
y = csc x =
1
sin x
R = ( - ,  )
Period = 
R = ( -  ,  ) Period = 
D = { x : x  /2  k } R = (-  , -1 ] U [ 1,  ) Period = 2 
D = { x : x   k}
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R = (-  , -1 ] U [ 1,  ) Period = 2 
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y  cos 1 x
D = [-1,1]
R = [0,]
y  sin 1 x
D = [-1,1]
  
,
R= 
 2 2 
y  tan 1 x
D = (, )
   
, 
R= 
 2 2
y  sec 1 x
D = x 1
R = 0  y  , y 
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
2
13


 y ,y 0
2
2
y  csc 1 x
D = x 1
R=
y  sec 1 x
D = (, )
R = 0,  
A: All trig functions are positive in Quadrant I
STUDENTS
ALL
II I
III IV
TAKE
S:
T:
sin and csc are positive in Quadrant II
tan and cot are positive in Quadrant III
CALCULUS
C:
cos and sec are positive in Quadrant IV
 3
Example 30: Find the values of the six trigonometric at angle   cos 1   
 5
Example 31: Find the values of the six trigonometric at angle   
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
4
14
Example 32: Solve for tan x  1 where 0  x   .
2
Example 33: Find the exact value for the
sin
2
3
Example 34: Find the exact value for the following:
a. sin 3/4
b. cos -
c. tan 11/4
d. sec(-240)
e. cos 60
f. sin (-120)
Example 35: Give the exact value for the expression in radians:
a. Tan-1 3
b. Cos-1 (-1/2)
Example 36: On what interval is y  tan 1 ( x) a function?
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f ( x)  A sin  B  x  C    D
| A | = amplitude
2
= period
B
C = horizontal shift
D = vertical shift
Depending on the trig function the “2” in the above equation could be different. In place of
“2” will be the period of the trig function.
Example 37: Find the Period, Amplitude, and Graph one Period of the function
y  3 sin( 2 x   )  1
Example 38: A Ferris wheel is 35 meters in diameter and can be boarded at ground
level. The wheel completes one full revolution every 5 minutes. At t = 0 you are in the
three o’clock position and ascending. You then make two full complete revolutions and
return to the boarding platform.
a. Find a sin or cos equation to represent the situation of the form h = f(t)
b. Draw one period of h = f(t). Be sure to determine the appropriate interval for t, t  0.
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cos (x) = cos (- x) -- even function
sin (- x) = - sin (x) -- odd function
sin2x + cos2x = 1
1 + tan2x = sec2x
1 + cot2x = csc2x
sin 2x = 2 sin x cos x
cos 2x = cos2x - sin2x
cos2x = (1 + cos 2x)/2
sin2x = (1 - cos 2x)/2
sin (x + y) = sin x cos y + cos x sin y
sin (x - y) = sin x cos y - cos x sin y
cos(x + y) = cos x cos y - sin x sin y
cos(x - y) = cos x cos y + sin x sin y
Parametric Equations
If x and y are given as functions x = f(t) , y = g(t) over an interval of t-values, then the set
of points (f(t), g(t)) defined by these equations is a curve (or parametric curve) in the
coordinate plane. The equations are parametric equations for the curve, and we say that
they curve has been parameterized. The variable t is the parameter of the curve, and its
domain I = [a, b] = [tmin, tmax] is the parameter interval. The point (x1(a),y1(a)) is the
initial point of the curve, and (x1(b),y1(b)) is the terminal point of the curve. Parametric
equations are often used to describe the path of an object in the coordinate plane based on
time(t).
Identities --
Eliminating the Parameter-- It is often possible to eliminate the parameter(t) so that you can
find the Cartesian equation for the parameterized curve. The Cartesian equation can help you to
find the path that the parameterized curve lies along but the graph of the Cartesian equation is not
necessarily the graph of the parameterized curve. This is because the parameterized curve is
dependent on the parameter(t).
Example 39: Eliminate the parameter and graph
x(t )  2t  5 and
x(t )  2t  6 and
y (t )  4t  7, 1  t  4
y (t )  2t  6, 1  t  4
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Example 40: Particle A moves along the path determined by the parametric
equation x  t  2 y  (t  2) 2 . Particle B moves along the path determined by the
3
3
parametric equation x  t  4 y  t  2 . Find the exact time that the particles collide.
2
2
This occurs when the particles are at the same point at the same time
Circles and Ellipses-- Parametric Equations for Circles and Ellipses have two basic forms.
x(t )  a cos t  k
y (t )  b sin t  h
or
x(t )  a sin t  k
y (t )  b cos t  h
If a  b , it is a circle with center ( h, k ) and r  a  b
(In many examples h and k equal zero)
If a  b , it is an ellipse with center ( h, k )
When graphing circles and ellipses there are three basic possibilities.
(a) You get only a portion of a circle or ellipse.
(b) You get a complete circle or ellipse
(c) The graph you see is a complete circle or ellipse but in reality you are tracing over the
figure more than once.
Your starting point and the direction in which the graph is created can be altered based on the
parametric equations that you are using. Often, the best way to graph these parametric equations
to plot some points to find the initial point, terminating point, and some points in between. After
a while you will be able to graph circles and ellipses without needing to plot points.
Example 41: Graph x(t )  3 cos t  2
y (t )  3 sin t  1, 0  t  2
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y (t )  4 sin t , 0  t  
Example 42: Graph x(t )  2 cos t
Example 43: Graph x(t )  sin t
y (t )  cos t , 0  t  4
Example 44: Graph x(t )   cos( 2t )
y (t )  2 sin( 2t ), 0  t  
Example 45: Graph x(t )   sin t  2
y (t )  cos t  2, 0  t  
Example 46: Graph x(t )  3cos( t )
y (t )  5sin( t ) , 0  t  4 .
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Limits
Definition of a Limit: If f (x) tends to a single number L as x approaches a , then
lim f ( x)  L . In other words, as the values of x approach a , then the values of the function
xa
f(x), the values of y, in that “neighborhood” approach L .
Steps in finding a limit:
1. Evaluate the function at x  a .
2. Algebraically manipulate the function so it can be evaluated at x  a .
3. Make a table of values near x  a .
4. Draw a graph of the function.
Theorem:
1. If f(x) is any polynomial function and c is any real number then
lim = f (c )
x c
2. If f(x) and g(x) are polynomials and c is any real number then
lim
x c
f ( x ) f (c )
=
provided that g(c)0
g ( x ) g (c )
lim f ( x)L
lim g ( x)
Theorem: xc
and xc
 M (c can be any value to apply these properties, including   )
lim
1. The limit of a sum is the sum of the limits. xc
 f ( x)  g ( x)   L  M
lim
2. The limit of a difference is the difference of the limits. xc
lim
3. The limit of a product is the product of the limits. xc
 f ( x)  g ( x)   L  M
 f ( x)  g ( x)   L  M
4. The limit of a quotient is the quotient of the limits provided the limit of the denominator
 f ( x)   L
lim 

g ( x)  M

is not zero. xc
5. The limit of an nth root is the nth root of the limits.

lim f ( x )
xc
n  L
n
6. The limit of a constant times a function is the constant times the limit of the function.

lim k  f ( x)
xc
kL
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Example 47: Find lim x 2  2 x
x 3
Example 48: Find lim
x3
x2
Example 49: Find lim
x2  9
.
x3
Example 50: Find lim
x2  4
=
x2
Example 51: Find lim
2
=
x2
x 3
x 3
x2
x2
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Theorem:
If lim  lim  L , then lim exists and lim = lim  lim  L
x c
x c
x c
Example 52: Find lim
x 3
x c
x c
x c
5
 x  3 2
x 2  16
, x4
x4
Example 53: Find lim
x4
6
Example 54: Find lim
x2
Example 55: Find lim
x0
, x=4
x2
x2  4
x
x
1
1

Example 56: Find lim 4  x 4
x 0
x
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The Squeezing ( or Sandwich) Theorem -- Suppose that g(x) < f(x) < h(x) for all x = c in some
lim g( x)
h( x) = L. Then lim f ( x) L .
interval about c and that xc
= lim
xc
xc
Important Limits:
lim sin x
1
x 0
x
1
0
x 
x
lim
x
 1
 e
x   1 
x

lim
lim
x 0
End Behavior of rational functions of the form f ( x) 
tan x
0
x
Ax m  ...
as x   -Bx n  ...
 A
 A
f ( x)  lim   x mn 
1) If m  n , slant asymptote is y    x ( m n ) , therefore xlim
 
x   B
 
B
2) If m  n , horizontal asymptote is y 
A
A
f ( x) 
, therefore xlim
 
B
B
3) If m  n , horizontal asymptote is y  0 , therefore
3x 2  2 x  5
Example 57: Find lim
x  
2 x2  6
2 x 4  3x

Example 58: Find lim
x 
x2
Example 59: Find xlim

1

x2
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lim f ( x)  0
x 
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Continuity
Continuity at a point--The function f(x) is continuous at x = c iff all of the following are true:
f ( c) exists
1)
lim f ( x )
2)
exists
xc
lim f ( x) f (c)
3)
xc
Continuous Function—A function is considered to be “continuous” if it is continuous at every
point in its domain.
Example 60: Find the value for k for which the following piecewise function will be
continuous:
x2 ,
x2
f ( x) 
k  x, x  2
Intermediate Value Theorem: Let f (x) be continuous over a closed interval a, b . Then for
every value d such that if f (a)  d  f (b) there exists a value c where a  c  b such that
f (c )  d .
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