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- The definite integral
In the following exercises, approximate the average value using Riemann sums
L
100 and
R
100 . How does your answer compare with the exact given answer?
[T]
over the interval
the exact solution is
the exact average is between these values.
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In the following exercises, compute the average value using the left Riemann sums
L
N for
How does the accuracy compare with the given exact value?
[T]
over the interval
the exact solution is
The exact answer
so
L
100 is not accurate.
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[T]
over the interval
the exact solution is
The exact answer
so
L
100 is not accurate to first decimal.
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Suppose that
and
Show that
Use
Then,
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Show that the average value of
over
is equal to 1/2 Without further calculation, determine whether the average value of
over
is also equal to 1/2.
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Show that the average value of
over
is equal to
Without further calculation, determine whether the average value of
over
is also equal to
so divide by the length 2
π of the interval.
has period
π , so yes, it is true.
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Explain why the graphs of a quadratic function (parabola)
and a linear function
can intersect in at most two points. Suppose that
and
and that
Explain why
whenever
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Suppose that parabola
opens downward
and has a vertex of
For which interval
is
as large as possible?
The integral is maximized when one uses the largest interval on which
p is nonnegative. Thus,
and
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Suppose
can be subdivided into subintervals
such that either
over
or
over
Set
- Explain why
- Then, explain why
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Suppose
f and
g are continuous functions such that
for every subinterval
of
Explain why
for all values of
x .
If
for some
then since
is continuous, there is an interval containing
t
0 such that
over the interval
and then
over this interval.
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Suppose the average value of
f over
is 1 and the average value of
f over
is 1 where
Show that the average value of
f over
is also 1.
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Suppose that
can be partitioned. taking
such that the average value of
f over each subinterval
is equal to 1 for each
Explain why the average value of
f over
is also equal to 1.
The integral of
f over an interval is the same as the integral of the average of
f over that interval. Thus,
Dividing through by
gives the desired identity.
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Suppose that for each
i such that
one has
Show that
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[T] Compute the left and right Riemann sums
L
10 and
R
10 and their average
for
over
Given that
to how many decimal places is
accurate?
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[T] Compute the left and right Riemann sums,
L
10 and
R
10 , and their average
for
over
Given that
to how many decimal places is
accurate?
so the estimate is accurate to two decimal places.
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Estimate
using the left and right endpoint sums, each with a single rectangle. How does the average of these left and right endpoint sums compare with the actual value
The average is
which is equal to the integral in this case.
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Estimate
by comparison with the area of a single rectangle with height equal to the value of
t at the midpoint
How does this midpoint estimate compare with the actual value
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From the graph of
shown:
- Explain why
- Explain why, in general,
for any value of
a .
a. The graph is antisymmetric with respect to
over
so the average value is zero. b. For any value of
a , the graph between
is a shift of the graph over
so the net areas above and below the axis do not change and the average remains zero.
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If
f is 1-periodic and
is it necessarily true that
for all
A ?
Yes, the integral over any interval of length 1 is the same.
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Questions & Answers
evolutionary history and relationship of an organism or group of organisms
AI-Robot
the study of living organisms and their interactions with one another and their environments
AI-Robot
HOW CAN MAN ORGAN FUNCTION
the diagram of the digestive system
allimentary cannel
Ogenrwot
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
Genetics is the study of heredity
Misack
how does twins formed?
Misack
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
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Source:
OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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