Transcript Chapter 6 Review Important Terms, Symbols, and Concepts

Chapter 13 Review
Important Terms, Symbols, Concepts
 13.1 Antiderivatives and Indefinite Integrals
 A function F is an antiderivative of a function f
if F ’(x) = f (x)
 If F and G are both antiderivatives of f, they differ by a
constant: F(x) = G(x) + k for some constant k.
 We use the symbol  f ( x ) dx , called an indefinite integral,
to represent the family of all antiderivatives of f, and we
write
 f ( x)dx  F ( x)  C
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 13.1 Antiderivatives and Indefinite Integrals
(continued)
 The symbol  is called an integral sign, f (x) is the
integrand, and C is the constant of integration.
 Indefinite integrals of basic functions are given in this
section.
 Properties of indefinite integrals are given in the section; in
particular, a constant factor can be moved across an
integral sign. However, a variable factor cannot be moved
across an integral sign.
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 13.2 Integration by Substitution
 The method of substitution (also called the change-ofvariable method) is a technique for finding indefinite
integrals. It is based on the following formula, which is
obtained by reversing the chain rule:
 f '[ g ( x)]g '( x)  f [ g ( x)]  C


This formula implies the general indefinite integral
formulas in this section.
Guidelines for using the substitution method are given by
the procedure in this section.
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 13.2 Integration by Substitution (continued)
 In using the method of substitution it is helpful to employ differentials
as a bookkeeping device:
 The differential dx of the independent variable x is an arbitrary
real number.
 The differential dy of the dependent variable y is defined by dy =
f ’(x) dx.
 13.3 Differential Equations; Growth and Decay
 An equation is a differential equation if it involves an unknown
function and one or more of its derivatives.
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 13.3 Differential Equations; Growth and
Decay (continued)
dy
 3x(1  xy 2 )
 The equation dx
is a first-order differential
equation because it involves the first derivative of the
unknown function y, but no second or higher-order
derivative.
 A slope field can be constructed for the differential
equation above by drawing a tangent line with slope
3x(1 + xy2) at each point (x, y) of a grid. The slope field
gives a graphical representation of the functions that are
solutions of the differential equation.
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 13.3 Differential Equations; Growth and
Decay (continued)
dQ
 rQ (in words: the rate
 The differential equation
dt
at which the unknown function Q increases is proportional
to Q) is called the exponential growth law. The constant
r is called the relative growth rate. The solutions to the
exponential growth law are the functions Q(t) = Q0ert
where Q0 denotes Q(0), the amount present at time t = 0.
These functions can be used to solve problems in
population growth, continuous compound interest,
radioactive decay, blood pressure, and light absorption.
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 13.3 Differential Equations; Growth and
Decay (continued)
 A table in this section gives the solutions to other first-order
differential equations used to model the limited or logistic
growth of epidemics, sales and corporations.
 13.4 The Definite Integral
 If the function f is positive on [a, b], then the area between
the graph of f and the x axis from x = a to x = b can be
approximated by partitioning [a, b] into n subintervals
[xk-1, xk] of equal length x = (b-a)/n, and summing the
areas of n rectangles.
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 13.4 The Definite Integral (continued)
 The process of summing the areas of n rectangles can be
accomplished by left sums, right sums, or, more generally,
by Riemann sums:
n
In a Riemann sum, each
 Left sum
Ln   f ( xk 1 )x
ck is required to belong
k 1
n
to the subinterval
Rn   f ( xk )x
 Right sum
[xk-1, xk]. Left sums and
k 1
right sums are special
n
cases of Riemann sums.
 Riemann sum S n   f (ck )x
k 1
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 13.4 The Definite Integral (continued)
 The error in an approximation is the absolute value of
the difference between the approximation and the actual
value. An error bound is a positive number such that the
error is guaranteed to be less than or equal to that number.
 Theorem 1 in this section gives error bounds for the
approximation of the area between the graph of a positive
function f and the x axis, from x = a to x = b, by left sums
or right sums, if f is either increasing or decreasing.
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 13.4 The Definite Integral (continued)
 If f (x) > 0 and is either increasing or decreasing on
[a, b], then its left and right sums approach the same
real number I as n  .
 If f is a continuous function on [a, b], then the
Riemann sums for f on [a, b] approach a real number
limit I as n  .
 Let f be a continuous function on [a, b]. The limit I of
Riemann sums for f on [a, b] is called the definite
integral of f from a to b, denoted
 f ( x)dx
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 13.4 The Definite Integral (continued)
 The integrand is f (x), the lower limit of integration is a,
and the upper limit of integration is b.
f ( x ) dx represents
 Geometrically, the definite integral
the cumulative sum of the signed areas between the graph
of f and the x axis from x = a to x = b.
 Properties of the definite integral are given in this section.

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 13.5 The Fundamental Theorem of Calculus

If f is a continuous function on [a, b] and F is any
antiderivative of f, then
f ( x ) dx = F(b) - F(a)

The fundamental theorem gives an easy and exact method for
evaluating definite integrals, provided we can find an
antiderivative F(x) of f (x). In practice, we first find an
antiderivative F(x) (when possible) using techniques for
computing indefinite integrals, then we calculate F(b) - F(a).
If it is impossible to find an antiderivative we must resort to
left or right sums or other approximation methods.

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 13.5 The Fundamental Theorem of Calculus
Graphing calculators have built-in numerical approximation
routines, more powerful than left or right sum methods, for
calculating the definite integral.
 If f is a continuous function on [a, b], then the average
value of f over [a, b] is defined to be
b
1
f ( x)dx

ba a
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