Transcript Chapter 14 Review

Chapter 14 Review
Important Terms, Symbols, Concepts
 14.1 Area Between Curves
 If f and g are continuous and f(x) > g(x) over the
interval [a, b], then the area bounded by y = f (x) and
y = g(x) for a < x < b is given by
b
A   [ f ( x)  g ( x)]dx
a
Barnett/Ziegler/Byleen Business Calculus 11e
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Chapter 14 Review
 14.1 Area Between Curves (continued)
 A graphical representation of the distribution of
income among a population can be obtained by
plotting data points (x,y) where x represents the
cumulative percentage of families at or below a given
income level, and y represents the cumulative
percentage of total family income received.
Regresssion analysis can be used to find a function
called a Lorenz curve that best fits the data.
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Chapter 14 Review
 14.1 Area Between Curves (continued)
 A single number, the Gini Index, measures income
concentration:
1
Gini Index =
2  [ x  f ( x)]dx
0

A Gini index of 0 indicates absolute equality - all families
share equally in the income. A Gini index of 1 indicates
absolute inequality - one family has all of the income and
the rest have none.
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Chapter 14 Review
 14.2 Applications in Business and Economics
 Probability Density Functions. If any real number x in an
interval is a possible outcome of an experiment, then x is
said to be a continuous random variable. The probability
distribution of a continuous random variable is described
by a probability density function f that satisfies
 f (x) > 0 for all real x.
 The area under the graph of f (x) over the interval
(-, ) is exactly 1.
d
 The probability that c < x < d is
 f ( x ) dx
c
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Chapter 14 Review
 14.2 Applications (continued)
 Continuous Income Stream. If the rate at which income is
received - its rate of flow - is a continuous function f (t) of
time, then the income is said to be a continuous income
stream. The total income produced by a continuous income
b
stream from t = a to t = b is
 f (t ) dt
a

The future value of a continuous income stream that is
invested at rate r, compounded continuously for 0 < t < T, is
T
FV   f (t )e r (T t ) dt
0
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Chapter 14 Review
 14.2 Applications (continued)
 Consumers’ and Producers’ Surplus. If ( x , p) is a
point on the graph of a price-demand equation p = D(x),
then the consumers’ surplus at a price level of p is
x


CS   D( x)  p dx
0

Similarly, for a point ( x , p) on the graph of a pricesupply equation p = S(x), the producers’ surplus at a
price level of p is
x


PS   p  S ( x) dx
0
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Chapter 14 Review
 14.3 Integration by Parts
 Some indefinite integrals, but not all, can be found by
means of the integration by parts formula:
 udv  uv   vdu

Select u and dv with the help of the guidelines in the
section.
 14.4 Integration Using Tables
 A table of integrals is a list of integration formulas that
can be used to find indefinite or definite integrals of
frequently encountered functions. Such a list appears in
Table II of Appendix C.
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