§3.3 Curve Sketching
limits involving infinity: Limits at infinity and infinite limits
Our goal is to see how limits involving infinity may be
interpreted as graphical features.!!
§3.3Vertical Asymptotes
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§3.3 Vertical Asymptotes
The graph approaches the horizontal line y=1 as x increase
or decrease without bound.
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f ' (1)
--------
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++++++
-1
Asymptote
--------
1
Max
Type of concavity
Sign of
--------
--------
-1
No inflection
++++++
2
inflection
7. The vertical asymptote (dashed line) breaks the graph
into two parts. join the features in each separate part by a
smooth curve to obtain the completed graph.
§3.4 Optimization
Absolute Maxima and Minima of a Function
Let f be a
Function defined on an interval I that contains the number c.
Then
f(c) is the absolute maximum of f on I if f(c)  f(x) for all x in I
f(c) is the absolute minimum of f on I if f(c)  f(x) for all x in I
Absolute extrema often
coincide with relative
extrema but not always!
§3.4 The Extreme Value Property
The Extreme Value Property
A function f(x) that is
continuous on the closed Interval a  x  b attains its
absolute extrema on the interval either at an endpoint of the
interval (a or b) or at a critical number c such that a<c<b.
§3.4 The Extreme Value Property
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§3.4 Absolute Extrema
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§3.4 Two General Principle of Marginal
Analysis
§3.4 Two General Principle of Marginal
Analysis
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§3.4 Two General Principle of
Marginal Analysis
Explanation in Economics The marginal cost (MC) is approximately the same as
the cost of producing one additional unit. If the additional unit costs less to produce
than the average cost (AC) of the existing units (If MC<AC), then this lessexpensive unit will cause the average cost per unit to decrease. On the other hand,
if the additional unit costs more than the average cost of the existing units (if
MC>AC), then this more-expensive unit will cause the average cost per unit to
increase. However (if MC=AC), then the average cost will neither increase nor
decrease, which means (AC)’=0
§3.4 Price Elasticity of Demand
§3.4 Price Elasticity of Demand
§3.4 Price Elasticity of Demand
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§3.4 Price Elasticity of Demand
§3.4 Price Elasticity of Demand
Summary
Applications of the derivative
A. Increasing and decreasing functions, critical point,
relative maxima and minima: First Derivative
B. Concavity, inflection point: Using the second
derivative to test for concavity and inflection points
Second derivative test for relative extrema
C. Limits involving infinity: Vertical Asymptotes and
Horizontal Asymptotes (rational functions)
D. Optimization: Absolute maximum and absolute
minimum, marginal analysis criterion for maximum
profit and minimal average cost, elasticity of demand