MAC2233-1

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FUNCTION
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DEFINITIONS
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ALGEBRAIC FUNCTIONS
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VERTICAL LINE TEST
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LINEAR FUNCTIONS
f(x) mx b (Slope-intercept form)
y intercept
Independent variable
Slope
Dependent variable
ax by c 0 (General form)
(y2 - y1) m(x2 - x1) (Point slope form)
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QUADRATIC FUNCTION
f(x) ax2 bx c
y intercept
Independent variable
Dependent variable
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SKETCHING QUADRATIC FUNCTIONS
f(x)
f(x)
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GENERAL SHAPE OF A POLYNOMIAL
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VERTICAL ASYMPTOTES
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HORIZONTAL ASYMPTOTES
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EXPONENTIAL FUNCTION
y f(x) ax
Independent Variable x ? D
Base 0 lt a lt 1 Decreasing Function 1
lt a lt ? Increasing Function
Dependent Variable f(x) gt 0
Inverse Function Logarithmic Function
x log b y
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EXPONENTIAL MODEL
P(t) P0 a t
Independent Variable t gt 0
Base 0 lt a lt 1 Decay Factor 1 lt a lt ?
Growth Factor
Initial Condition at t 0 P0 gt 0
Dependent Variable P(t) gt 0
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LOGARITHMIC FUNCTIONS
COMMON LOGARITHMS
NATURAL LOGARITHMS
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PROPERTIES OF LOGARITHMS
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RATES OF CHANGE
Instantaneous Rate of Change Tangent Line
P2
Average Rate of Change Secant Line
P1
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DIFFERENCE FUNCTIONS
Secant Line
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BASIC ECONOMIC MODEL
Ct Cv Cf
Cost Function
Ct Total Cost Cv Variable cost nc n
number produced c unit cost of production Cf
Fixed Cost
Revenue Function
R np
n number sold p selling price
Profit Function
P R - Ct
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BASIC ECONOMIC MODEL

Break Even Points
R
CT
Fixed Cost
P
n
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INTEREST FORMULAE
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LOGISTICS MODEL
LLimit of growth aconstant derived from
initial conditions kconstant derived from
boundary conditions ttime
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DEFINITION OF A LIMIT
Left Hand Limit
Right Hand Limit
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PROPERTIES OF LIMITS
1. For f(x) c (a constant),
2. For f(x) x,
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LIMIT USING DIFFERENCE QUOTIENT
(xh. f(xh))
(x, f(x))
h
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THE DERIVATIVE
The instantaneous rate of change of f(x) at x.
The slope of the tangent line to the graph of
f(x) at x
The limit of the difference equation as h (or ?x)
approaches 0.
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FACTS ABOUT THE DERIVATIVE

• A derivative of a function, f(x), can be found
  for any value of x where the derivative exists.

• Since the derivative is a limit, then the limit
  must exist.

• The derivative itself is a function, f(x)

• When the derivative function, f(x), is
  evaluated at some value of xc, the resulting
  value is the slope of the line tangent to the
  original function at xc.

• When the derivative function, f(x), is
  evaluated at some value of xc, the resulting
  value is the instantaneous rate of change of the
  function, f(x), at xc.

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DIFFERENTIATION
The process of determining the derivative of a
function is called differentiation.
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RULES OF DIFFERENTIATION
1. Constant Function Rule
2. Power Function Rule
3. Constant Multiple Rule
4. Sum and Difference Rule
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RULES OF DIFFERENTIATION (cont.)
5. Product Rule
6. Quotient Rule
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FUNCTION DIFFERENTIABLE AT A POINT
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DIFFERENTIALS
Since
Then
And
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AVERAGE AND MARGINAL ANALYSIS
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BASIC ECONOMIC MODEL

Break Even Points
R
CT
Fixed Cost
P
n
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CHAIN RULE
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PROPERTIES OF LOGARITHMS
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DERIVATIVE OF LOGARITHMIC FUNCTION
Natural Logarithmic Function
General Logarithmic Function
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DERIVATIVE OF EXPONENTIAL FUNCTION
Natural Exponential Function
General Exponential Function
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IMPLICIT DIFFERENTIATION
Let f (x) y n
Since
Then
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GRAPH DEFINITIONS
Absolute Maximum
P
Relative Maximum
P
P
I
P
c
c
c
c
c
c
Relative Minimum
P
P
Absolute Minimum
c critical value P critical point I
inflection point
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Graphic Analysis 1
f(x)
f(x) 3x2 6x
cicritical value picritical point pici,
f(ci) Iinflection point
f(x) 6x 6
f(x) x3 3x2 - 4
x
c1 c2 P1
I p2
f(x) f(x) f(x) Concavity
Increasing ? Decreasing ? Increasing
gt0 0 lt 0 0 gt 0
lt0 0 gt0
Down ? Up
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Graphic Analysis 2
f(x)
cicritical value picritical point pici,
f(ci) Iinflection point
f(x) .38x3 - 9.19x2 42.85x 1105
P1 I
p2 c1
c2
f(x) 2.28x 18.38
f(x) 1.14x2 18.38x 42.85
x
f(x) f(x) f(x) Concavity
Inc. ? Decreasing
? Increasing
gt0 0 lt 0
0 gt 0
lt0 0
gt0
Down ?
Up
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Graphic Analysis 3
f(x)
f(x) 60x2 -20x3
cicritical value picritical point pici,
f(ci) Iinflection point
p2
I p1 c1
c2
f(x) 5x4 x5
x
f(x) 20x3 -5x4
f(x) f(x) f(x) Concavity
Decreasing ? Increasing ?
Decreasing
lt0 0 gt 0 0
lt 0
gt0 0 gt0 0 lt0
Up ? Up Down
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GRAPHIC ANALYSIS AND CURVE SKETCHING

• Establish Dx.
• Find x and y intercepts, if possible
• Find end points.
• Find f(x)
• Find critical values f(x) 0.
• Find where f(x) is increasing f(x) gt 0 and
• decreasing f(x) lt 0.
• Locate extrema.
• Find f(x)
• Locate inflection points.
• Find where f(x) is concave up f(x) gt 0 and
• concave down f(x) lt 0.

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Graphic Analysis 4
f(x)
cicritical value picritical point pici,
f(ci) Iinflection point
f(x) x 1 2x - 3
x
f(x) -1 . (2x 3)2
f(x) 4 . (2x 3)3
f(x) f(x) f(x) Concavity
Decreasing ? Decreasing
lt0 ?
lt0
lt0 ?
gt0
Down ?
Up
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SKETCHING A GRAPH
P1 o

c1
f(x)
cicritical value picritical point pici,
f(ci) Iinflection point
I
f(x) .82x3 33.09x2 333.75x 435.28

o o
f(x) 4.92x 66.18
x
f(x) 2.46x2 66.18x 333.75
f(x) f(x) f(x) Concavity
Increasing ? Decreasing
gt0 0 lt 0
lt0 0 gt0
Down ? Up
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SUMMARY OF FIRST DERIVATIVE

• f(x) is used to find
• Slope of tangent line at any point.
• Instantaneous rate of change.
• Critical values.
• Intervals of increasing f(x)
• Extrema

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SUMMARY OF SECOND DERIVATIVE

• F(x) is used to find
• Increasing and decreasing behavior of the rate of
  change.
• Concavity of f(x).
• Direction of extrema, maximum or minimum.
• Inflection points f(x) 0.

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SUMMARY OF GRAPHIC ANALYSIS AND CURVE SKETCHING
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ANTIDERIVATIVE AND INDEFINATE INTEGRALS
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INDEFINATE INTEGRAL FORMULAE
Power Rule For Antiderivatives
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PROPERTIES OF INDEFINATE INTEGRALS
k dx k dx kx C
Constant Rule
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AREA UNDER A CURVE
Right Sum
Left Sum
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TRAPAZOIDAL RULE
1/2 0 5/4 1 9/8
1 1/2 2 5/4 13//8
3/2 1 13/4 2 21//8
2 3/2 5 13/4 31//8
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DEFINITION OF THE INDEFINATE INTEGRAL
(over the interval a to b)
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PROPERTIES OF DEFINATE INTEGRALS
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FUNDAMENTAL THEOREM OF DEFINATE INTEGRALS
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DISTANCE MODEL FOR A FALLING BODY
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SUBSTUTUTION POWER RULE
n ? -1
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TIPS ON u-SUBSTITUTION

• u is usually a radicand, a denominator, or an
  expression in parenthesis.
• Compute dx u du
• After the u-substitution into the integral, no
  factors can contain an x or other variable.
• Integrate
• Rewrite the antiderivative in terms of x, or the
  original independent variable.

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INDEFINATE INTEGRAL FORMULAE
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INTEGRATION BY PARTS
or
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AREA BETWEEN CURVES
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AREA BETWEEN TWO INTERSECTING CURVES
Compute the intersections of f(x) and g(x) to
find a and b
g(x)
b
x
a
f(x)
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SUPPLY, DEMAND AND EQUILIBRIUM
Consumer Surplus
Producer Surplus
p - price
S(q) Supply Function
CS
Equilibrium Point
p1
PS
D(q) Demand Function
q - quantity
q1
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AVERAGE VALUE FUNCTION
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CONTINUOUS FLOW OF MONEY