1.3 The fundamental theorem of calculus  (Page 2/11)

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$f\left(c\right)=\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)dx,$

and the proof is complete.

Finding the average value of a function

Find the average value of the function $f\left(x\right)=8-2x$ over the interval $\left[0,4\right]$ and find c such that $f\left(c\right)$ equals the average value of the function over $\left[0,4\right].$

The formula states the mean value of $f\left(x\right)$ is given by

$\frac{1}{4-0}{\int }_{0}^{4}\left(8-2x\right)dx.$

We can see in [link] that the function represents a straight line and forms a right triangle bounded by the x - and y -axes. The area of the triangle is $A=\frac{1}{2}\left(\text{base}\right)\left(\text{height}\right).$ We have

$A=\frac{1}{2}\left(4\right)\left(8\right)=16.$

The average value is found by multiplying the area by $1\text{/}\left(4-0\right).$ Thus, the average value of the function is

$\frac{1}{4}\left(16\right)=4.$

Set the average value equal to $f\left(c\right)$ and solve for c .

$\begin{array}{ccc}8-2c\hfill & =\hfill & 4\hfill \\ \hfill c& =\hfill & 2\hfill \end{array}$

At $c=2,f\left(2\right)=4.$

Find the average value of the function $f\left(x\right)=\frac{x}{2}$ over the interval $\left[0,6\right]$ and find c such that $f\left(c\right)$ equals the average value of the function over $\left[0,6\right].$

$\text{Average value}=1.5;c=3$

Finding the point where a function takes on its average value

Given ${\int }_{0}^{3}{x}^{2}dx=9,$ find c such that $f\left(c\right)$ equals the average value of $f\left(x\right)={x}^{2}$ over $\left[0,3\right].$

We are looking for the value of c such that

$f\left(c\right)=\frac{1}{3-0}{\int }_{0}^{3}{x}^{2}dx=\frac{1}{3}\left(9\right)=3.$

Replacing $f\left(c\right)$ with c 2 , we have

$\begin{array}{ccc}{c}^{2}\hfill & =\hfill & 3\hfill \\ c\hfill & =\hfill & \text{±}\sqrt{3}.\hfill \end{array}$

Since $\text{−}\sqrt{3}$ is outside the interval, take only the positive value. Thus, $c=\sqrt{3}$ ( [link] ).

Given ${\int }_{0}^{3}\left(2{x}^{2}-1\right)dx=15,$ find c such that $f\left(c\right)$ equals the average value of $f\left(x\right)=2{x}^{2}-1$ over $\left[0,3\right].$

$c=\sqrt{3}$

Fundamental theorem of calculus part 1: integrals and antiderivatives

As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. The theorem is comprised of two parts, the first of which, the Fundamental Theorem of Calculus, Part 1 , is stated here. Part 1 establishes the relationship between differentiation and integration.

Fundamental theorem of calculus, part 1

If $f\left(x\right)$ is continuous over an interval $\left[a,b\right],$ and the function $F\left(x\right)$ is defined by

$F\left(x\right)={\int }_{a}^{x}f\left(t\right)dt,$

then ${F}^{\prime }\text{(}x\right)=f\left(x\right)$ over $\left[a,b\right].$

Before we delve into the proof, a couple of subtleties are worth mentioning here. First, a comment on the notation. Note that we have defined a function, $F\left(x\right),$ as the definite integral of another function, $f\left(t\right),$ from the point a to the point x . At first glance, this is confusing, because we have said several times that a definite integral is a number, and here it looks like it’s a function. The key here is to notice that for any particular value of x , the definite integral is a number. So the function $F\left(x\right)$ returns a number (the value of the definite integral) for each value of x .

Second, it is worth commenting on some of the key implications of this theorem. There is a reason it is called the Fundamental Theorem of Calculus. Not only does it establish a relationship between integration and differentiation, but also it guarantees that any integrable function has an antiderivative. Specifically, it guarantees that any continuous function has an antiderivative.

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absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
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