As noted in Theorem 1, the sign of the second derivative on an
interval indicates the concavity of the graph on that interval.
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As illustrated in Figure 4, a point of inflection is the point on a
curve where the concavity changes from concave up to concave
down or concave down to concave up. The second derivative
equals zero at a point of inflection.
This is formalized in Theorem 2.
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Theorem 3 gives us a second means of justifying that a minimum or
maximum occurs at a critical point.
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Optimization – the process of finding the “best”
solution to a problem, e.g.,
• the largest area that a given length of fence can
enclose
• the path that minimizes travel time
• the dimensions that maximize the volume of a
box made from a rectangle of cardboard
The process of optimization usually involves three
steps:
1. Choose variables.
2. Determine the function.
3. Optimize the function.
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In some cases, we may know the derivative and wish to find the
function itself. A function F (x) whose derivative is f (x) is called
an antiderivative of f (x).
Theorem 1 gives us a method of finding the general antiderivative
of a function, namely:
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The process of antidifferentiation is denoted by the Leibniz symbol ∫
which specifies finding the indefinite integral of a function.
Just as we have rules for differentiation, we also have rules for
integration, of which the first is:
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In Figure 2, we see that the slope of
the graph of y = ln |x| is x–1 for all x.
This leads to Theorem 3, which
covers the exception to Theorem 2.
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The linearity rules for differentiation have their counterparts for
integration as noted in Theorem 4.
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Recalling derivatives of the basic trigonometric functions, we can
recognize the basic trigonometric integrals.
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Recalling the Chain Rule, we have the following trigonometric
integrals.
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Remembering that the exponential function is its own derivative, we
see how the indefinite integral of ex is ex + C.
Applying the Chain rule in reverse to the exponential function,
we obtain:
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Since we used rectangles of common width, the approximate area
under the curve becomes:
Using the second rule of
linearity of summations,
we obtain
Some texts refer to this method as the Rectangular Approximation
Method or RAM.
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The summations we use for the left-hand and midpoint approximations
are respectively:
The Rectangular Approximation Methods may further be abbreviated
as RRAM, LRAM, and MRAM to indicate using the y-values on the
right, left, and midpoint respectively.
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The largest subinterval in a Riemann sum is known as the norm of
the partition.
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A function is integrable over [a, b] if all of the Riemann sums (not
just the endpoint and midpoint approximations) approach one and
the same limit L as the norm of the partition tends to zero, which we
may write as:.
N
L  lim R  f , P , C   lim
P 0
P 0
 f c  x
i
i
i 1
Assuming |R(f, P, C) – L| gets arbitrarily small as the norm||P|| tends
to zero, the limit is called the definite integral of f (x) over [a, b].
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R eferring to

b
a
f  x  dx
The definite integral is usually referred to as the integral of f over
[a, b].
The function f (x) inside the integral symbol is called the integrand.
The numbers a and b are called the limits of integration.
The independent variable in the function is used as the variable
of integration.
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As illustrated in Figure 3, when f (x) is not positive for all x on [a, b],
the definite integral yields the signed area between the graph and the
x-axis. Signed area is defined as:
If the graph in Figure 3 represented the velocity of a particle, then the
integral from a to b would tell us the net displacement or movement
of the particle from its starting point.
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Figure 8 illustrates the definite
integral of f (x) = C over [a, b]
for some C > 0. The integral
may be evaluated by using
Theorem 2.
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Similar to limits, definite integrals have linearity properties as noted
in Theorem 3.
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If we reverse the limits of integration, we change the sign of the
signed area yielded by the definite integral as noted in the following
definition:
If the upper and lower limits of integration equal one another, the
width of the interval is zero and
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This theorem is illustrated in Figure 10.
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Figure 11 illustrates Theorem 5, the Comparison Theorem.
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If a function has a lower bound m and an upper bound M on [a, b],
then the Comparison Theorem may be written algebraically as:
This is illustrated in Figure 12.
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The Fundamental Theorem of Calculus ties together the operations of
differentiation and integration in a relationship. Specifically, the
integral of f (x) evaluated from a to b is equal to the antiderivative of
f (x) evaluated at x = b minus the antiderivative of f (x) evaluated at
x = a.
W hen evaluating a definite integral, w e frequently have
an interm ediate step as follow s:
b
a f
 x  dx 
w here F  x 
b
a
F x
b
a
 F b   F a 
indicates w e are evaluating F  x  betw een a and b .
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Recalling that the antiderivative of f (x) = x–1 is F(x) = ln x, the FTC
tells us:
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Basic Antiderivative Formulae

x dx 
n
x
n 1
n 1
C

dx
 ln x  C
x
 e dx  e  C
x
x
 sin xdx   cos x  C
 cos xdx  sin x  C
 sec xdx  tan x  C
 csc xdx   cot x  C
 sec x tan xdx  sec x  C
 csc x cot xdx   csc x  C
2
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2
If we take the derivative of the area function, we obtain the equation
of the function under whose graph we are finding the area, or the
derivative of the area function is the original function as stated in
Theorem 1.
S im ilarly, applying the C hain R ule:
d
dx
gx
a
f  t  dt  f  g  x   g   x 
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If s′(t) is both positive and negative on [t1, t2], the integral will total
the signed areas and give us the net change in s. This is the basis of
Theorem 1.
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Theorem 2 formalizes what we observed in Figure 2.
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R em em bering that the derivative of ln x is
1
and that ln 1  0,
x
and that ln is not defined for x  0, w e can state:
Figure 1 shows a graphical interpretation of this statement.
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The following integral formulas apply to the inverse trigonometric
function derivatives we learned in Section 3.9.
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R em em bering that to find the derivative of y  b , w e use
x
y  bx  e
ln b
x
=e
x ln b
, thence y    ln b  e
x ln b
w e evaluate the follow ing integral:
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  ln b  e
ln b
x
  ln b  b ,
x
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A frequently asked question is: “When will this process produce
twice the original amount?” This question can be answered by
solving 2 = 1 ekt for t.
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In the case of exponential decay, the half-life of the substance is
calculated by using:
Archeologists use the half-life of carbon-14 to date organic materials.
The process was developed following World War II.
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In Chapter 5, we learned that the definite integral gives us the
signed area between the graph of the function and the x-axis. If we
want to find the area between two graphs and f (x) > g (x) on [a, b],
then we can use the definite integral to find this area.

b
f
a
 x  dx  a g  x  dx  a  f  x   g  x   dx
b
b
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Sometimes it is more convenient to integrate with respect to y.
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More formally,
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For some urban planning functions, it might be appropriate to consider
the population density P as a function of distance r from the urban
center P (r).
To calculate the population living between r1 and r2 miles of the
urban area, we would use the integral:
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Using the fact that the volume of a solid is equal to the integral of
the cross sectional area over the interval, we obtain:
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When calculating the volume of the solid obtained by rotating the
region between curves f (x) and g(x), where f (x) > g(x) > 0 on [a, b],
about the x-axis, use the following formula:
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If the region is rotated about an axis other than the x- or y-axis, use
the washer method, calculating the inner and outer radii from the
stated axis. In Figure 7, the outer radius is R = g(x) – (–3) = g(x) + 3
and the inner radius is r = f (x) – (–3) = f (x) + 3
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Notice how the sketch in Figure 8 helps us determine the inner and
outer radii and what to subtract from what to obtain them.
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As shown in Figure 9, the same process works for axes parallel to
the y-axis.
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If we divide the interval [a, b] into N even intervals, the area is found
using the Trapezoidal Rule
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• Differential Equations
• Slope Fields