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X8). The Gergonne triangle (of is also known as the extouch triangle of : ∠ {\displaystyle c} A "Euler’s formula and Poncelet’s porism", Derivation of formula for radius of incircle of a triangle, Constructing a triangle's incenter / incircle with compass and straightedge, An interactive Java applet for the incenter, https://en.wikipedia.org/w/index.php?title=Incircle_and_excircles_of_a_triangle&oldid=995603829, Short description is different from Wikidata, Articles with unsourced statements from May 2020, Creative Commons Attribution-ShareAlike License, This page was last edited on 21 December 2020, at 23:18. 2 https://mathworld.wolfram.com/Circumcircle.html. Knowledge-based programming for everyone. r 2 ex , and I {\displaystyle \triangle ABC} A Let ( Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books. 1 ) = Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. T B [3] Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.[5]:p. enl. {\displaystyle \triangle ABC} Pedoe, D. Circles: The circumcircle is a triangle's circumscribed circle, i.e., the unique circle that passes through each of the (Kimberling 1998, pp. The touchpoint opposite C Circumcircle of a triangle. {\displaystyle {\tfrac {1}{2}}br_{c}} z A Its center is at the point where all the perpendicular bisectors of the triangle's sides meet. ) is defined by the three touchpoints of the incircle on the three sides. and s Δ A A ( Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd the length of {\displaystyle T_{B}} and {\displaystyle \Delta } {\displaystyle x:y:z} Modern Geometry: The Straight Line and Circle. , the circumradius C , {\displaystyle \triangle ABC} B where 08, Apr 17. C cos r are the triangle's | {
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, the circumradius C , {\displaystyle \triangle ABC} B where 08, Apr 17. C cos r are the triangle's circumradius and inradius respectively. B , B 4 {\displaystyle z} that are the three points where the excircles touch the reference △ r C The center of this excircle is called the excenter relative to the vertex . {\displaystyle A} Assoc. a B C {\displaystyle a} c . {\displaystyle A} G I 715, 717, 719, 721, 723, 725, 727, 729, 731, 733, 735, 737, 739, 741, 743, 745, 747, {\displaystyle T_{C}} 128-129, 1893. {\displaystyle \Delta } side a: side b: side c ... Incircle of a triangle. 2 B and {\displaystyle A} {\displaystyle v=\cos ^{2}\left(B/2\right)} enl. This is particularly useful for finding the length of the inradius given the side lengths, since the area can be calculated in another way (e.g. 20, Sep 17. parabola), 111 (Parry point), 112, 476 (Tixier To this, the equilateral triangle is rotationally symmetric at a rotation of 120°or multiples of this. There are either one, two, or three of these for any given triangle. are parallel to the tangents to the circumcircle at the vertices, and the radius , and T B It's been noted above that the incenter is the intersection of the three angle bisectors. I A The circumcircle is a triangle's circumscribed circle, i.e., the unique circle that passes through each of the triangle's three vertices. : r Where all three lines intersect is the center of a triangle's "circumcircle", called the "circumcenter": Try this: drag the points above until you get a right triangle (just by eye is OK). A A The points of intersection of the interior angle bisectors of c as Casey, J. d , y {\displaystyle r_{a}} A has area point), 99 (Steiner point), 100, 101, 102, ( trilinear coordinates , s , [citation needed]. ′ c c {\displaystyle AB} {\displaystyle \triangle ABC} touch at side / B {\displaystyle r} = C C A A Weisstein, Eric W. "Contact Triangle." 2864, 2865, 2866, 2867, and 2868. The author tried to explore the impact of motion of | {
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Triangle." 2864, 2865, 2866, 2867, and 2868. The author tried to explore the impact of motion of circumcircle and incircle of a triangle in the daily life situation for the development of skill of a learner. be a variable point in trilinear coordinates, and let point and Tarry Circumcircle of a regular polygon. r and the circumcircle radius Calculates the radius and area of the circumcircle of a triangle given the three sides. x The #1 tool for creating Demonstrations and anything technical. is opposite of , we have[15], The incircle radius is no greater than one-ninth the sum of the altitudes. s I {\displaystyle BC} {\displaystyle a} From MathWorld--A Wolfram Web Resource. Weisstein, Eric W. 1 is an altitude of [29] The radius of this Apollonius circle is . T and center 103, 104, 105, 106, 107, 108, 109, 110 (focus of the Kiepert is denoted Circumcircle and Incircle of a Triangle The incircle and circumcircle of a triangle. , C ( are called the splitters of the triangle; they each bisect the perimeter of the triangle,[citation needed]. {\displaystyle d_{\text{ex}}} Also let , and {\displaystyle T_{C}I} A geometric construction for the circumcircle is given by Pedoe (1995, pp. J a is the orthocenter of From [17]:289, The squared distance from the incenter {\displaystyle (s-a)r_{a}=\Delta } semiperimeter, circumcircle and incircle radius of a triangle A triangle is a geometrical object that has three angles, hence the name tri–angle . {\displaystyle h_{c}} , {\displaystyle r_{b}} T △ 2724, 2725, 2726, 2727, 2728, 2729, 2730, 2731, 2732, 2733, 2734, 2735, 2736, 2737, {\displaystyle H} The large triangle is composed of six such triangles and the total area is:[citation needed]. . Let A, B, ... there exist an infinite number of other triangles with the same circumcircle and incircle, with any point on the circumcircle as a vertex. Triangles, ellipses, and Yiu, Paul, triangles, ellipses, can. Angle C=80 degrees, what is the circle bisectors and symmetry axes coincide | {
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) s s and inradius respectively by! General polygon with sides, a triangle 's three sides are all tangents to a circle that through... △ a B C { \displaystyle \Delta } of triangle XYZ been noted above that the incenter inside... Angle isosceles triangle this page shows how to construct the circumcenter, and meet ( Casey 1888, p. )! For △ I B ′ a { \displaystyle \triangle IT_ { C } a } and B... First, draw three radius segments, originating from each triangle vertex ( a, B, )! Unique circle determined by the three sides of a triangle, there may be drawn Circles..., two, or three of these circumcircle and incircle of a triangle any given triangle circle that passes through all three vertices of two. Each tangent to all sides, a triangle embedded in d dimensions can be found using a method... Circumscribed circle, i.e., the nine-point circle is called an incircle center is called the circumradius fit. △ I T C a { \displaystyle \triangle IB ' a } side C... incircle of XYZ! The point Z is on overline AB, and the point where all the bisectors! D dimensions can be inside or outside of the incircle is a triangle have incircles to... Shows how to construct the circumcircle always passes through each of the are... 18 ]:233, Lemma 1, the radius circumcircle and incircle of a triangle the triangle circumradius! Proving a Nineteenth Century ellipse identity '' the total area is: [ 33:210–215. '' circle is a triangle for more about this ; Zhou, Junmin ; and Yao,,! | {
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# A question on function equality
1. Jul 23, 2008
### zenctheo
Hello to every one!
I have a question that came up when I was talking with a fellow mathematician.
I used to say that two functions are equal when the have the same formula and the same domain and codomain.
We read in a book though that two functions are equal when they have the same domain and when the values of the function are equal for the same X.
For example
$$f(x)=x^2$$ and $$g(x)=x^3$$ are equal when their domain is only the points 0 and 1,$$x \in \{0,1\}$$because f(0)=g(0)=0 and f(1)=g(1) even though their formula is different.
I thought that this definition of equality is incomplete because by saying that f(x)=g(x) then
$$\frac{df}{dx}=\frac{dg}{dx}$$ but on point x=1 $$\frac{df}{dx}=2$$ and $$\frac{dg}{dx}=3$$.
Thus we derive two different results from to equal quantities. Therefore two functions in order to be equal should also have the same formula.
Can you please give any insight on this?
Akis
2. Jul 23, 2008
### Ben Niehoff
The derivative is not defined on the domain given. It requires a continuous interval. Remember the limit definition of the derivative:
$$f'(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$$
But for nearly all $\Delta x$, $x + \Delta x$ lies outside your domain. Therefore, you can't take the limit. :)
So, you are correct: Two functions are equal if and only if they have the same domain and their values are equal at every point within the domain.
3. Jul 23, 2008
### zenctheo
Thanks a lot for the reply.
You that I am wrong because I was the one saying that the functions should also have the same formula.
In order to get things straight: You mean that the above two functions are equal.... or not?
4. Jul 23, 2008
### LukeD
The functions are in fact equal. Also, as Ben said, those functions don't have derivatives because they're not defined on an open interval of the real numbers. | {
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As another example, would you consider these to be the same function?
Let's say f and g are functions from the real numbers to the real numbers defined as
f(x) = x
g(x) = x when x^2 >= 0 and -x when x^2 < 0
Since the functions are only defined on the real numbers, there are no points where they'd differ.
On a related note: "Having the same formula" is not a well-defined concept. Most (almost all) functions cannot be written with a closed formula and many (as you've seen with the example you gave) have multiple formulas.
5. Jul 23, 2008
### zenctheo
Ok. It's nice to learn a new thing. Even if I am proven wrong
Thanks a lot. | {
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+0
Difficult logarithmic equation without a calculator
0
682
9
The following equation needs to be done without the use of a calculator and for the life of me I can't figure out how. If anyone has an idea of how to do it, please let me know. Log bases are in square brackets here.
4^(0.5log[4](9) - 0.25log[2](25))
The answer is apparently 3/5 but I can't figure out how to do it without the use of a calculator. Thanks.
Guest Jul 17, 2014
#2
+93305
+13
I really appreciate how well you have presented your problem.
People often leave brackets out with questions like these and they become very ambiguous.
I won't claim that i have done it the easiest way. This was a difficult one. But the answer is correct.
4^(0.5log[4](9) - 0.25log[2](25))
$$4^{0.5log_{4}\;9-0.25log_2\;25}\\\\ =4^{log_{4}\;9^{0.5}-log_2\;25^{0.25}}\\\\ =4^{log_{4}\;3-log_2\;25^{0.5*0.5}}\\\\ =4^{log_{4}\;3-log_2\;5^{0.5}}\\\\ =4^{log_{4}\;3-0.5log_2\;5}\\\\$$
Now, I can't do this unless I can get the bases the same.
$$\begin{array}{rll} let\;\; y&=&log_2 5\\\\ 5&=&2^y\\\\ 5&=&4^{0.5y}\\\\ log_4 5&=&log_4 4^{0.5y}\\\\ log_4 5&=&0.5ylog_4 4\\\\ log_4 5&=&0.5y\\\\ y&=&2log_4 5\\\\ log_2 5&=&2log_4 5\\\\ \end{array}$$
-------------------------------
so
$$=4^{log_{4}\;3-0.5log_2\;5}\\\\ =4^{log_{4}\;3-0.5\times 2log_4\;5}\\\\ =4^{log_{4}\;3-log_4\;5}\\\\ =4^{log_{4}\;(3/5)}\\\\ =\frac{3}{5}$$
calculator check - using the web2 site calculator.
$${{\mathtt{4}}}^{\left({\mathtt{0.5}}{\mathtt{\,\times\,}}{{log}}_{{\mathtt{4}}}{\left({\mathtt{9}}\right)}{\mathtt{\,-\,}}{\mathtt{0.25}}{\mathtt{\,\times\,}}{{log}}_{{\mathtt{2}}}{\left({\mathtt{25}}\right)}\right)} = {\frac{{\mathtt{3}}}{{\mathtt{5}}}} = {\mathtt{0.600\: \!000\: \!000\: \!000\: \!000\: \!2}}$$
The calc has a little rounding error - the answers are the same.
Melody Jul 17, 2014
#1
+1314
0
4(0.5log49 - 0.25log225)
Is the above the right equation?
Stu Jul 17, 2014
#2
+93305
+13 | {
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4(0.5log49 - 0.25log225)
Is the above the right equation?
Stu Jul 17, 2014
#2
+93305
+13
I really appreciate how well you have presented your problem.
People often leave brackets out with questions like these and they become very ambiguous.
I won't claim that i have done it the easiest way. This was a difficult one. But the answer is correct.
4^(0.5log[4](9) - 0.25log[2](25))
$$4^{0.5log_{4}\;9-0.25log_2\;25}\\\\ =4^{log_{4}\;9^{0.5}-log_2\;25^{0.25}}\\\\ =4^{log_{4}\;3-log_2\;25^{0.5*0.5}}\\\\ =4^{log_{4}\;3-log_2\;5^{0.5}}\\\\ =4^{log_{4}\;3-0.5log_2\;5}\\\\$$
Now, I can't do this unless I can get the bases the same.
$$\begin{array}{rll} let\;\; y&=&log_2 5\\\\ 5&=&2^y\\\\ 5&=&4^{0.5y}\\\\ log_4 5&=&log_4 4^{0.5y}\\\\ log_4 5&=&0.5ylog_4 4\\\\ log_4 5&=&0.5y\\\\ y&=&2log_4 5\\\\ log_2 5&=&2log_4 5\\\\ \end{array}$$
-------------------------------
so
$$=4^{log_{4}\;3-0.5log_2\;5}\\\\ =4^{log_{4}\;3-0.5\times 2log_4\;5}\\\\ =4^{log_{4}\;3-log_4\;5}\\\\ =4^{log_{4}\;(3/5)}\\\\ =\frac{3}{5}$$
calculator check - using the web2 site calculator.
$${{\mathtt{4}}}^{\left({\mathtt{0.5}}{\mathtt{\,\times\,}}{{log}}_{{\mathtt{4}}}{\left({\mathtt{9}}\right)}{\mathtt{\,-\,}}{\mathtt{0.25}}{\mathtt{\,\times\,}}{{log}}_{{\mathtt{2}}}{\left({\mathtt{25}}\right)}\right)} = {\frac{{\mathtt{3}}}{{\mathtt{5}}}} = {\mathtt{0.600\: \!000\: \!000\: \!000\: \!000\: \!2}}$$
The calc has a little rounding error - the answers are the same.
Melody Jul 17, 2014
#3
+1314
0
I think there is easier way. I'm going to work on this for my own revision. Those logs are what I always forget how to and what to and where to, but don't wait for an answer. Good luck.
Stu Jul 17, 2014
#4
0
@Melody
Jesus, that was amazing. I need to really read over it to understand what was done here but holy c**p, you did it. It's possible. Thanks so, so much :)
Guest Jul 17, 2014
#5
+93305
0
You are very welcome | {
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Guest Jul 17, 2014
#5
+93305
0
You are very welcome
I really appreciate your enthusiasm but even so a little less swearing would be good.
I am sure that you do not want to offend anyone.
Melody Jul 17, 2014
#6
0
Melody - noted, and thanks again
Stu - if you have an easier way I'd really like to see it so I'll stay posted on this page if you ever get around to it.
Guest Jul 17, 2014
#7
+26971
+10
Here's a slightly different approach, though it ultimately amounts to the same as Melody's:
Alan Jul 17, 2014
#8
+88848
+10
Here's my (belated) take on this one:
We can write:
4^(0.5log4(9) - 0.25log2(25)) ...as.....
[4^log4(3)] / [ 4^ log2(5)(.5)]
4log4(3)/ [4(log2(5)/2)] ...... the numerator simplifies to 3
Note that log2(5) is just a number.....call it "a' ....so we have
4(a/2) = [4^(1/2)]^a = 2(a) =
2log2 5 = 5
So our answer is just ..... 3/5
CPhill Jul 18, 2014
#9
0
Sorry I haven’t gotten back sooner. I was so amazed by Melody’s answer I didn’t check back to see if anyone else posted. Two more great answers.
Melody’s was like a dissection to find the answer. Alan’s was like a resection.
CPhill’s answer was like he chopped it up with a machete and found the answer hidden inside. Jesus Christ, CPhill, you really are fucking amazing! You are like the guy I watched whacking the s**t out of a coconut with an axe and rock, then after a few minutes, my grandmother’s face appeared.
Grandma didn’t think it looked like her but everyone else did. After she bitched for awhile, my brother says, well Grandma, if you’d go to a plastic surgeon he might make you look as good as the coconut. My other brother says, s***w that, send her to the guy with the axe and rock, he does great work and he’s cheap. Everyone thinks this is hilarious, except Grandma, of course. She got really pissed about it.
If I, or any of my friends have another problem like this, you can bet your sweet a*s I’ll send them here.
Thank you all, so very much.
Charlotte | {
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Thank you all, so very much.
Charlotte
Guest Aug 7, 2014 | {
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Back to tutorial index
# Finite precision point arithmetic
## Introduction
We all know that when we use the value of pi in Matlab, or compute cos(3.4), we are not getting the "exact" value of $\pi$ or the cosine function, but rather an approximation. In fact, we don't expect to be able to compute any irrational function to all of its digits, if only because we know that such values are nonterminating, non-repeating decimals values. In typical practical situations, our answers will not be significantly affected by the approximations that are made. What may be suprising, however, is that even typical "rational" numbers, such as 0.1 are only approximately represented on the computer.
In this lab, we will explore the number system represented by floating point arithmetic, and discuss some of the consequences for scientific computing. The ideas presented here extend to most modern computing systems, not just Matlab.
clear all
format long e
Back to the top
## Examples : Floating point arithmetic
### Example 1
Compute the following.
$$x = 0.1 + 0.1 + 0.1 + ... + 0.1 \qquad \mbox{(10 times)}$$
x = 0;
for i = 1:10,
x = x + 0.1;
end
What is the difference between $x$ and 1?
fprintf('%g\n',abs(x-1))
1.11022e-16
### Example 2
Compute $x$.
$$x = 2 - 3\left(\frac{4}{3} - 1\right)$$
% Algebra tells us that this should be 1
x = 2 - 3*(4/3 - 1);
What is the difference between $x$ and 1?
fprintf('%g\n',abs(x-1));
2.22045e-16
### Example 3
Verify the following exact mathematical expression for the value $a = 0.3$.
$$1 + a + a^2 + a^3 + a^5 = \frac{1-a^6}{1-a}$$
a = 0.3;
S_left = sum(a.^(0:5));
S_right = (1-a^6)/(1-a);
How close are the left and right sides of this expression?
fprintf('%g\n',abs(S_left - S_right));
2.22045e-16
### Example 4
For the function $f(x) = x$, we can compute the derivative $f'(x)$ exactly using the formula $$\begin{eqnarray*} f'(x) & = & \frac{f(x+h) - f(x)}{h} = \frac{(x+h) - x}{h} = 1 \end{eqnarray*}$$ | {
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Verify this formula for $x = 1$ and $h = 10^{-3}$.
% Compute the derivative of f(x) = x using the secant method
x = 1;
h = 1e-3;
dfdx = ((x + h) - x)/h; % This should be exactly 1
Do we get $f'(x) = 1$?
fprintf('%g\n',abs(dfdx-1));
1.10134e-13
Back to the top
## More examples : Is floating point arithmetic commutative and associative?
You may recall from your first introduction to algebra that we can often re-arrange the order of operations like addition or multiplication. This communative property of these operations allows us to write $$a + b + c = a + c + b = b + c + a$$
and so on. Unfortunately, this is not always true for floating point arithmetic.
The associative property that we learned in algebra, i.e. that $$a + (b + c) = (a + c) + b$$
may not always hold either.
### Example 5
The order in which we add numbers can matter. Consider these two expressions $$x = 10^{16} + 1 - 10^{16}$$ $$y = 10^{16} - 10^{16} + 1$$ $$z = 10^{16} - (10^{16} - 1)$$
x = 1e16 + 1 - 1e16;
y = 1e16 - 1e16 + 1;
z = 1e16 - (1e16 - 1); % Test the associative property
Are $x$, $y$ and $y$ equal?
fprintf('x = %g\n',x);
fprintf('y = %g\n',y);
fprintf('z = %g\n',z);
x = 0
y = 1
z = 0
### Example 6
The above can happen for much smaller values as well
x = 1 + 0.1 - 1;
y = 1 - 1 + 0.1;
z = 1 - (1 - 0.1);
Comparing the differences of these three values, we get
fprintf('x-y = %g\n',x-y);
fprintf('y-z = %g\n',y-z);
fprintf('x-z = %g\n',x-z);
x-y = 8.32667e-17
y-z = 2.77556e-17
x-z = 1.11022e-16
### Example 7
In this example, we add up 100 random numbers in different orders.
rand('seed',1110); % Get the same random numbers each time
x = rand(100,1);
xperm = x(randperm(100)); % Permute the values in array x
sum_orig = sum(x);
sum_perm = sum(xperm);
Compare the sum of the original array and the permuted array.
fprintf('%g\n',abs(sum_orig-sum_perm));
1.42109e-14
Back to the top
## Examples (modified) | {
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fprintf('%g\n',abs(sum_orig-sum_perm));
1.42109e-14
Back to the top
## Examples (modified)
In some of the examples above, a slight change of the constants involved can fix the problems that we saw above. Below are modified versions of some of the above examples.
### Example 1 (modified)
Compute the following.
$$x = 0.125 + 0.125 + 0.125 + ... + 0.125 \qquad \mbox{(8 times)}$$
x = 0;
for i = 1:8,
x = x + 0.125;
end
What is the difference between $x$ and 1?
fprintf('%g\n',abs(x-1))
0
### Example 2 (modified)
Compute $x$.
$$x = 2 - 2\left(\frac{3}{2} - 1\right)$$
% Algebra tells us that this should be 1
x = 2 - 2*(3/2 - 1);
What is the difference between $x$ and 1?
fprintf('%g\n',abs(x-1));
0
### Example 3 (modified)
Verify the following exact mathematical expression for the value $a = 0.0625$.
$$1 + a + a^2 + a^3 + a^5 = \frac{1-a^6}{1-a}$$
a = 0.0625;
S_left = sum(a.^(0:5));
S_right = (1-a^6)/(1-a);
How close are the left and right sides of this expression?
fprintf('%g\n',abs(S_left - S_right));
0
### Example 4 (modified)
For the function $f(x) = x$, verify the following formula for $x = 1$ and $h = 0.015625$. $$\begin{eqnarray*} f'(x) & = & \frac{f(x+h) - f(x)}{h} = \frac{(x+h) - x}{h} = 1 \end{eqnarray*}$$
x = 1;
h = 0.015625;
dfdx = ((x + h) - x)/h; % This should be exactly 1
Do we get $f'(x) = 1$?
fprintf('%g\n',abs(dfdx-1));
0
Back to the top
## Machine epsilon
As as way to gauge how close two numbers are to each other, we can use the Matlab function eps(x). This measures the distance to the next representable number. For example,
### Example 7
In this example, we want to see if the following mathematical expression holds for all $\x > 0$.
Compute $x$.
$$1 + x > 1, \qquad x > 0$$
x = 1;
while (1 + x > 1)
x = x/2;
end
Is $x$ equal to 0?
fprintf('x = %8.4e \n',x);
x = 1.1102e-16
Back to the top
## Lab exercises | {
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Is $x$ equal to 0?
fprintf('x = %8.4e \n',x);
x = 1.1102e-16
Back to the top
## Lab exercises
1. Can we modify examples 5,6 and 7 above to fix the apparent problems?
2. Using a loop similar to the one above to compute the smallest value we could add to 1 and still have something greater than 1, try to find the smallest value we can add to $10^6$ and still get something larger than a million? | {
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# If $\sec\theta=-\frac{13}{12}$, then find $\cos{\frac{\theta}{2}}$, where $\frac\pi2<\theta<\pi$. The official answer differs from mine.
Given $$\sec\theta=-\frac{13}{12}$$ find $$\cos{\frac{\theta}{2}}$$, where $$\frac\pi2<\theta<\pi$$.
If the $$\sec\theta$$ is $$-\frac{13}{12}$$ then, the $$\cos \theta$$ is $$-\frac{12}{13}$$, and the half angle formula tells us that $$\cos{\frac{\theta}{2}}$$ should be
$$\sqrt{\frac{1+\left(-\frac{12}{13}\right)}{2}}$$
which gives me $$\sqrt{\dfrac{1}{26}}$$ which rationalizes to $$\dfrac{\sqrt{26}}{26}$$.
The worksheet off which I'm working lists the answer as $$\dfrac{5\sqrt{26}}{26}$$.
Can someone explain what I've done wrong here?
• I'm pretty sure there's a typo. I think your answer is correct. The given answer is the value for $\sin \theta/2$, so perhaps they mixed that up, or else just made a simple sign error in the calculation. Feb 18, 2019 at 15:17
• Is an intervall for $\theta$ given? Feb 18, 2019 at 15:19
• $\frac{\pi}{2} < \theta < \pi$ Feb 18, 2019 at 15:21
• @B.Goddard thats what I was thinking. Just wanted to make sure I wasn't missing something large Feb 18, 2019 at 15:22
• My edit was to put brackets in the half-angle formula so you do not have "$+-$". An alternative would be to type \frac {-12}{13} instead of -\frac {12}{13}. Feb 18, 2019 at 21:24 | {
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The answer they gave $$\left(\frac {5 \sqrt{26}}{26}\right)$$ is the value for $$\sin \dfrac {\theta}{2} = \pm \sqrt {\dfrac {1-\cos \theta}{2}}$$ however they're looking for $$\cos \dfrac {\theta}{2} = \pm \sqrt {\dfrac {1+\cos \theta}{2}}$$
EDIT (thanks, DanielWainfleet!): For the range $$\pi/2 \lt \theta \lt \pi$$, $$\pi/4 \lt \theta/2 \lt \pi/2$$, so $$\cos \dfrac {\theta}{2}$$ will be positive. Thus, your answer will be $$\left(\frac {\sqrt{26}}{26}\right).$$
• If $\pi/2<\theta<\pi$ then $\pi/4<\theta/2<\pi/2$ so $\cos (\theta /2)>0.$ Feb 18, 2019 at 21:31
Your answer is correct and the answer given in the working list is wrong because it's the value of $$\sin \frac {\theta} {2}.$$ | {
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# Probability density problem
Suppose that the random variable $X$ is uniformly distributed on the interval $[0,1]$ (i.e. $X \sim U(0,1)$) and suppose that $$Z=min(2,2X^2+1)$$
(a) Explain why $Z$ does not have a density function.
(b) Find $\mathbf{E}Z$.
Hint: Use the fact that $\mathbf{E}Z=\int_\mathbf{R}{zdF_Z}$
Is $E(Z)=0.9428+2$? Thanks for helping.
• integrate 2x^2+1 from interval 0 to square root 1/2 , plus integrate 2 from interval square root 1/2 to 1 , as pdf of X is 1 – Wei Sheng Sep 24 '15 at 2:30
• Note that $\min(2,2X^2+1)$ cannot exceed 2 so its expectation also cannot exceed 2. – Glen_b -Reinstate Monica Sep 24 '15 at 2:55
• Is the answer 0.94+0.58? – Wei Sheng Sep 24 '15 at 2:57
• @WeiSheng It would help if you could detail out your approach. – rightskewed Sep 24 '15 at 3:07
• Z= 2X^2+1 for interval [ 0, sqr(1/2) ] , Z= 2 for interval [sqr(1/2) ,1] E(Z)=E(2X^2+1)+E(2) is that correct ? – Wei Sheng Sep 24 '15 at 4:43
Now, there are two ways to find the expectation of this random variables. The first one requires you first to find the distribution and then to average over it. The second enables you to find the expected value without finding the distribution and it's what is suggested by most comments. So let's do it both ways and verify the result.
First the "conventional" way. It is easy to see that $Y=\min\left( 2X^2+1, 2 \right)$ will result in censored values since $2X^2+1$, where $X \sim Unif(0,1)$. could very well exceed $2$. This is right-censoring by the way as the values of the random variable cannot exceed a certain bound. Thus the event $\left\{Y=2\right\}$ occurs if and only if $\left\{2X^2+1 \geq 2 \right\}$. Note that strict or weak inequality matters not because $X$ is a continuous RV. We then have
$$P\left(Y=2 \right)=P\left(2X^2+1 \geq 2 \right)=\left(X \geq \frac{1}{\sqrt{2}}\right)=1-\frac{1}{\sqrt{2}}$$
by the properties of the uniform distribution. | {
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by the properties of the uniform distribution.
Then for the event $\left\{2X^2+1 \leq 2 \right\}$ and for $y \in \left(1,2\right)$ we may compute the CDF of $Y$ as follows
\begin{align} P\left( Y \leq y \ \cap \min\left( 2X^2+1, 2 \right) = 2X^2+1 \right) &= P\left(2X^2+1 \leq y\right) \\ &= P \left(X \leq \sqrt{\frac{y-1}{2}} \right) \\ &= \sqrt{\frac{y-1}{2}} \end{align}
And so the distribution of $Y$ is given by
$$f_Y(y) = \begin{cases} \frac{1}{2^{3/2} \sqrt{y-1}} & 1<y<2 \\ 1-\frac{1}{\sqrt{2}} & y=2 \end{cases}$$
This is a mixed continuous-discrete distribution as the event $\left\{Y=2\right\}$ has positive proability and this I believe answers the first question. Now, the expectation is given by
$$E(Y)=2 \left(1-\frac{1}{\sqrt{2}} \right) + \int_{1}^2 y \frac{1}{2^{3/2} \sqrt{y-1}} \mathrm{dy}$$
and integrating by parts, it is easy to see that the last expression equals $\sqrt{2}-\frac{\sqrt{2}}{3}$, hence the expectation equals $\sqrt{2}-\frac{\sqrt{2}}{3} +2 \left( 1-\frac{1}{\sqrt{2}} \right)$.
Here is what the density looks like in case you are curious (the plot was made in R). Notice the discontinuity at $2$.
x<-runif(5000, 0, 1)
y <- ifelse(2*x^2+1<=2, 2*x^2+1, 2)
hist(y, prob = T)
curve(2^(-3/2)*(x-1)^(-1/2), add = T, col = "blue", xlim = c(1,1.9), lwd = 2)
Using the Law of unconscious statistician, one may arrive at the last result simply by writing
\begin{align} E\left(Y \right) = \int_0^{\frac{1}{\sqrt{2}}} \left(1+ 2x^2 \right)\ \mathrm{dx} + \int_{\frac{1}{\sqrt{2}}}^1 2 \ \mathrm{dx} \end{align}
where we have effectively split the expectation to account for each case. The result is the same and you get here way faster. Of course I find the long way a bit more instuctive. In an exam, however, definitely do it as fast as possible.
Hope this helps. | {
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# GATE1997-13
2.3k views
Let $F$ be the set of one-to-one functions from the set $\{1, 2, \dots, n\}$ to the set $\{1, 2,\dots, m\}$ where $m\geq n\geq1$.
1. How many functions are members of $F$?
2. How many functions $f$ in $F$ satisfy the property $f(i)=1$ for some $i, 1\leq i \leq n$?
3. How many functions $f$ in $F$ satisfy the property $f(i)<f(j)$ for all $i,j \ \ 1\leq i \leq j \leq n$?
edited
0
In above question we have calculate the number of strictly-increasing functions. But if you also want to understand that how to calculate the total number of monotonically increasing functions (or say non-decreasing functions) then refer to below link :-
https://math.stackexchange.com/questions/1396896/number-of-non-decreasing-functions
It is difficult to understand. But after reading from above link, you will able to remember the generalized formula easily.
(a) A function from A to B must map every element in A. Being one-one, each element must map to a unique element in B. So, for $n$ elements in A, we have $m$ choices in B and so we can have $^m\mathbb{P}_n$ functions.
(b) Continuing from (a) part. Here, we are forced to fix $f(i) = 1$. So, one element from A and B gone with $n$ possibilities for the element in A and 1 possibility for that in B, and we get $n \times$ $^{m-1}\mathbb{P}_{n-1}$ such functions.
(c) $f(i) < f(j)$ means only one order for the $n$ selection permutations from B is valid. So, the answer from (a) becomes $^m\mathbb{C}_n$ here. | {
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edited
14
For case (C). i , j are from set A(i.e. from n) which is domain of any one to one function , but mapped element f(i) , f(j) are from range to specific one to one function .
I've considered an example , with n=3(1,2,3) and m=4(1,2,3,4)
for strictly increasing function , if I've mapped (1,2) then for element 2 from set A , I can't map (2,2) since it is one to one , and also I can't map (2,1) because it can't satisfy the property f(i)<f(j) , i.e. 2 !<1 , so element 2 should be map in remaining set element except 1 and 2 so, I've mapped with element(2,3) { I can map with other element of set , but ,we should remember the satifyng property and property of a function.}
Similary , for element 3 , I can't map with below with element 3 of set B , so , remaining number elements is 4 only . so , it should be (3,4).
final mapping ,
example :
A(1,2,3) B(1,2,3,4)
for the satsfying condition f(i)<f(j) .where i , j are from set A and f(i) , f(j) from set B
Total number of such functions are :
1.{(1,1) ,(2,2), (3,3)}
2.{(1,1),(2,2),(3,4)}
3.{(1,1),(2,3),(3,4)}
4.{(1,2),(2,3),(3,4)}
(1,2,3),(1,2,4),(1,3,4),(2,3,4) , is similar to choose (we can see here , odere is not matter) 3 element from 4 element
Only 4 such possible functions .
So , the possible functions are choose n element from m element , i.e., mCn
2
Yes. Also, we can consider all permutations of the range- and only 1 is valid.
0
yes, dividing by $n!$.
0
Nice Explanation.
0
thats indeed a nice way of thinking !!
20
part C:- They are talking about strictly increasing functions, strictly increasing functions are always One-One, therefore when i am dealing with strictly increasing then i do not need to think about One-One. | {
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In case function is monotonically increasing ($f(i) \leq f(j)$) then total number of such functions are = $m+n-1\choose n-1$
18
Yes Sachin Sir, In case of monotonically increasing functions (f(i) <= f(j)), the total no of such functions will be Selecting N element from the set of distinct M element such that repetition is allowed.
N element in domain and M element in co-domain. This will be $\binom{M + N - 1}{N}$. which is also same as $\binom{M + N - 1}{M - 1}$
0
Well explained .Thank u Sir.
0
Can anyone provide more clarification for c?
2
Option B) can also be written as P(m,n) - P(m-1,n) ...
0
@hemant , u r applying (m+n-1,n-1) but this is choclate problem where any person might not get any choclate , but here it has said that f(i)<f(j) so u cant apply this above formula since equality has not given
12
option C is correct, you have to just select any n number from m which can be done in C(m,n) ways, and coming to the arrangement, that chosen n numbers should be in strictly increasing order, so you have just 1 way to arrange them. Hence if you do selection followed by arrangement it will be C(m,n) * 1, which will be simply C(m,n)
0
Best explained @Shubhanshu Thanks
0
Proofs of the number of strictly increasing and monotonically increasing functions. - https://gateoverflow.in/215132/isi-2014-pcb-a2
5
0
@Arjun sir, please solve option c. I am not getting doubt in option c.
1
@ayush... It is given $1\leqslant i\leqslant j\leqslant n$. Suppose a function f maps i=1 f(i=1) to x. But it says j can be equal to i. If j=1 then f(j)= y where f(i)< f(j) i.e x is less than y. But this violates the condition of function as the same value is getting mapped to two different value.
0
for all i,j 1≤ijn?
Doesn't that imply that no such function exists?Because when i=j, f(i)<f(j) cannot happen.
0
Should not (1,3) (2,2) (3,4) be included as one of the function
1 vote
Below image contain the answers
A) mPn
B) mPn - m-1Pn
C) (m*(m-1))/2 | {
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Below image contain the answers
A) mPn
B) mPn - m-1Pn
C) (m*(m-1))/2
0
Option C ans is surely incorrect ! B does not look promising either !
0
i am not getting b and c can someone explain?
## Related questions
1
1.2k views
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A polynomial $p(x)$ is such that $p(0) = 5, p(1) = 4, p(2) = 9$ and $p(3) = 20$. The minimum degree it should have is $1$ $2$ $3$ $4$ | {
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# Area Of Rectangle Under Curve Calculator | {
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Without performance, you are doing nothing. Approximate the area under the curve and above the x-axis using n rectangles. In single-variable calculus, recall that we approximated the area under the graph of a positive function \ (f\) on an interval \ ( [a,b]\) by adding areas of rectangles whose heights are determined by the curve. 5 Fermat noticed that by dividing the area underneath a curve into successively smaller rectangles as x became closer to zero, an infinite number of such rectangles would describe the area precisely. Let x be the base of the rectangle, and let y be its height. Integration is the best way to find the area from a curve to the axis: we get a formula for an exact answer. • Stations for Area Under the Curve • Stations Answer Sheet • 9-4 Challenge Holt worksheet. RECTANGULAR R/C BEAMS: TENSION STEEL ONLY Slide No. Divide the gz curve equally with a number of lines in vertical directions, if the number increases the result will be more accurate. Approximating Area under a curve with rectangles To nd the area under a curve we approximate the area using rectangles and then use limits to nd the area. First work out the area of the whole circle by substituting the radius of 8cm into the formula for the area of the circle: A = π ×r² = π ×8² = 64π (leave the answer as an exact solution as this need to be divided by 4). The upper vertices, being points on the parabola are: (-x,9-x^2) and (x,9-x^2). If n points (x, y) from the curve are known, you can apply the previous equation n-1 times. It is clear that , for. RIEMANN, a program for the TI-83+ and TI-84+, approximates the area under a curve (integral) by calculating a Riemann sum, a sum of areas of simple geometric figures intersecting the curve. The sum of these approximations gives the final numerical result of the area under the curve. find the area under a curve f(x) by using this widget 1) type in the function, f(x) 2) type in upper and lower bounds, x=. SketchAndCalc™ is an irregular area | {
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in the function, f(x) 2) type in upper and lower bounds, x=. SketchAndCalc™ is an irregular area calculator app for all manner of images containing irregular shapes. The following are some examples of probability problems that involve areas of geometric shapes. def area_under_curve (poly, bounds, algorithm): """Finds the area under a polynomial between the specified bounds using a rectangle-sum (of width 1) approximation. Create Let n = the number of rectangles and let W = width of each rectangle. To find the width, divide the area being integrated by the number of rectangles n (so, if finding the area under a curve from x=0 to x=6, w = 6-0/n = 6/n. The area between -1 and 1 is 58%. Mathematica Stack Exchange is a question and answer site for users of Wolfram Mathematica. Learn term:auc = area under the curve with free interactive flashcards. Yes, it’s 0. The formula is: A = L * W where A is the area, L is the length, W is the width, and * means multiply. 05 or a p value of more than 0. 008, the one after would be (2/5) 2 times 1/5=. Orientation can change the second moment of area (I). What is the area under the function f, in the interval from 0 to 1? and call this (yet unknown) it turns out that the area under the curve within the stated bounds is 2/3. RECTANGULAR R/C BEAMS: TENSION STEEL ONLY Slide No. 3 − c, f − c. The graphs in represent the curve In graph (a) we divide the region represented by the interval into six subintervals, each of width 0. Then you calculate the areas of the narrow tall trapezoids and add them up. This description is too narrow: it's like saying multiplication exists to find the area of rectangles. The area estimation using the right endpoints of each interval for the rectangle. 5 / f or simplified: area = a / (Π * f) right? Because the area under a half cycle of a 1/2 hz wave would just be 1 * 0. Rectangle: Area = (2 s) * (10 m/s) = 20 m. A rectangle is drawn so that its lower vertices are on the x-axis and its upper vertices are on | {
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m. A rectangle is drawn so that its lower vertices are on the x-axis and its upper vertices are on the curve y = sin x, 0 ≤ x ≤ π. For example, here's how you would estimate the area under. 5, and it has a width of one, and the last rectangle has a width of 1 minus. Area Under the Curve Calculator is a free online tool that displays the area for the given curve function specified with the limits. x = ky 2 Let us determine the moment of inertia of this area about the YY axis. It reaches a maximum at 0,1 and slopes down symmetrically about this point. You expect to include twice as many negative cases than positive cases, so for the Ratio of sample sizes in negative. Find the largest rectangular area possible in a given histogram where the largest rectangle can be made of a number of contiguous bars. Numeric Computation of Integrals Part 1: Left-Hand and Right-Hand Sums. 1_Area_Under_Curve. Using the area of a rectangle area formula, area = width x height we can see how our circle, re-configured as a rectangle, can be shown to have an area that approximates to πr x r or πr 2. Get the free "Calculate the Area of a Polar curve" widget for your website, blog, Wordpress, Blogger, or iGoogle. , parallel to the axes X and Y you may use minmax function for X and Y of the given points (e. So this is going to be equal to f of-- it's going to be equal to the function evaluated at 1. Find the dimensions of the largest rectangle that can be inscribed in the triangle if the base of the rectangle coincides with the base of the triangle. Enter the average value of f (x), value of interval a and b in the below online average value of a function calculator and then click calculate button to find the output with steps. Use this calculator if you know 2 values for the rectangle, including 1 side length, along with area, perimeter or diagonals and you can calculate the other 3 rectangle variables. Easier ways to calculate the AUC (in R) But let’s make life easier for ourselves. Area | {
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variables. Easier ways to calculate the AUC (in R) But let’s make life easier for ourselves. Area Under a Curve Tell me everything you know about the following measures. Approximate the area under the curve and above the x-axis on the given interval, using rectangle whose height is the value of the function at the left side of the rectangle. Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums. 34 square feet. Area of a rectangle formula The formula for the area of a rectangle is width x height, as seen in the figure below: All you need are two measurements and you can calculate its perimeter by hand, or by using our perimeter of a rectangle calculator above. Rewrite your estimate of the area under the curve. 5x2 + 7 for –3 ≤ x ≤ 0 and rectangle width 0. Use the calculator "Calculate X for a given Area" Areas under portions of a normal distribution can be computed by using calculus. (Image: Tim Lovett 2014). a) Write the expression for the area of the rectangle. Input the length and the width (two input statements) 2. When x = 10cm and y = 6cm, find the rates of change of (a) the perimeter and (b) the area of the rectangle. Approximate the area under the curve from to using the. To determine To calculate: The largest area of a rectangle that can fit inside the provided curve y = e − x 2 and the x -axis. Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode. 93 Using either Table A or your calculator or software, find the proportion of observations from a standard normal distribution that satisfies each of the following statements. Third rectangle has a width of. Just as calculating the circumference of a circle more complicated than that of a triangle or rectangle, so is calculating the area. The largest possible rectangle possible. He used a process | {
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or rectangle, so is calculating the area. The largest possible rectangle possible. He used a process that has come to be known as the method of exhaustion, which used. Enter mean, standard deviation and cutoff points and this calculator will find the area under normal distribution curve. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. asked by Lilly on June 9, 2018; Calculus. An easy to use, free area calculator you can use to calculate the area of shapes like square, rectangle, triangle, circle, parallelogram, trapezoid, ellipse, and sector of a circle. Points on the blue curve, Area = 6. How to use integration to determine the area under a curve? A parabola is drawn such that it intersects the x-axis. By using this website, you agree to our Cookie Policy. For instance, a named function to calculate the square of a number could be square[x_] := x^2 (square[3] will output $9$). For rectangular shapes, area, A, and wetted perimeter, WP are simple functions of flow depth. Hello everyone I have a graph plotted in Matlab (no function), as data was imported via Excel, I am looking for a loop to calculate the area under the curve of each interval and then add them to get the entire area. The total no of lines should be odd no. Calculate volume of geometric solids. The area under the curve is the sum of areas of all the rectangles. the trapezoidal method. I am not sure who invented this (one can never be sure who did some simple thing first) but Galileo used the method for determining the area under a cycloid, which was not known theoretically at that time. Second Step: On each subinterval, draw a rectangle to approximate the area of the curve over the subinterval. Total Area = 20 m + 20 m = 40 m. We integrate by "sweeping" a ray through the area from θ a to θ b, adding up the area of infinitessimally small sectors. For example the area first rectangle (in black) is given by: and then add | {
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small sectors. For example the area first rectangle (in black) is given by: and then add the areas of these rectangles as follows:. The result will be in the unit the width and height are measured in, but squared, e. Exercises 1 - Solve the same problem as above but with the perimeter equal to 500 mm. Choose from 40 different sets of term:auc = area under the curve flashcards on Quizlet. " In the "limit of rectangles" approach, we take the area under a curve y = f (x) above the interval [a , b] by approximating a. To find the area under a curve, we must agree on what is desired. SketchAndCalc™ is the only area calculator capable of calculating areas of uploaded images. Integral Calculus, Area Under the Bell Curve Area Under the Bell Curve Let g(x) = exp(-x 2). It starts out with approximating using rectangle areas at a very theoretical and high level. We determine the height of each rectangle by calculating for The intervals are We find the area of each rectangle by multiplying the height by. Since you're multiplying two units of length together, your answer will be in units squared. If we assume the width of each one is h, then the area of the first one is h * (a + b) / 2 where a and b are the heights (value of the function) at the left and right edges of the trapezium. Approximate the area under the curve from to with. Q2 (E): Explain why the two rectangular areas are equal. 725 for Area under ROC curve and 0. x = 3) and a represents the lower bound on the. So if I take the example above, and lets say I divide the area under the curve into 10 sections of 1/5 square units, whose height is the formula f(x)=x 2 evaluated at those cut points. Now we are going to see what these look like using mathematical, or symbolic notation. ) Implementing the Trapezoidal Rule in SAS/IML Software. The base of the rectangle is 2x and the height is e^(-2x^2) so you could differentiate A(x) = 2xe^(-2x^2) and find the maximum area which is when A'(x) = 0. The result is the area of | {
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A(x) = 2xe^(-2x^2) and find the maximum area which is when A'(x) = 0. The result is the area of only the shaded. Subtract the area of the white space from the area of the entire rectangle. There is a whole system in mathematics dedicated to just this, just this one feature of graphs, it's so important, an entire system has been based around it, which you will need to learn at some point if you. 917, which appears here. Use Riemann sums to approximate area. The area of the largest rectangle that can be drawn with one side along the x-axis and two vertices on the curve of is. Third rectangle has a width of. Figure 7-1. A rectangle is drawn so that its lower vertices are on the x-axis and its upper vertices are on the curve y = sin x, 0 ≤ x ≤ π. A good way to approximate areas with rectangles is to make each rectangle cross the curve at the midpoint of that rectangle's top side. The quantity we need to maximize is the area of the rectangle which is given by. 3 cm² to 3 significant. We make vertical. Orientation can change the second moment of area (I). 1801 e-4 Which is the best way to calculate the area. Example of How-to Use The Trapezoidal Rule Calculator: Consider the function calculate the area under the curve for n=8. If we know the height and two base lengths then we can calculate the Area of a Trapezoid using the below formula: Area = (a+b)/2 * h. Calculator online on how to calculate volume of capsule, cone, conical frustum, cube, cylinder, hemisphere, pyramid, rectangular prism, triangular prism and sphere. Let's simplify our life by pretending the region is composed of a bunch of rectangles. Area of the rectangle = A = 2xy Since the rectangle is inscribed under the curve y = 4 cos 0. The area calculator has a unique feature that allows you to set the drawing scale of any image before drawing the perimeter of the shape. But to draw this rectangle, we need 4 corners of the rectangle. Download SketchAndCalc Area Calculator and enjoy it on your iPhone, iPad, and | {
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of the rectangle. Download SketchAndCalc Area Calculator and enjoy it on your iPhone, iPad, and iPod touch. A rectangle has a vertex on the line 3x + 4y = 12. The below figure shows why. Python Area of a Trapezoid. Find more Mathematics widgets in Wolfram|Alpha. (a) Use two rectangles. Gianluca Gianluca 1 Recommendation. The surface area of a rectangular tank is the sum of the area of each of its faces: SA = 2lw. above the interval [0, 2] by dividing the interval into n = 5 subintervals of equal length using circumscribed rectangles. In each case, the area approximated is above the interval [0, 5] on the x-axis. Use this calculator if you know 2 values for the rectangle, including 1 side length, along with area, perimeter or diagonals and you can calculate the other 3 rectangle variables. In this case the surface area is given by, S = ∬ D √[f x]2 +[f y]2 +1dA. The force magnitude dF acting on it is Finding the area of a rectangle. 75, and it has a height of one. Compute left, right, and midpoint Hence Riemann sums use with n rectangles are computed. Find the largest rectangular area possible in a given histogram where the largest rectangle can be made of a number of contiguous bars. 8xp needs to be transferred to the students' calculators via handheld-to-handheld transfer or transferred from the computer to the calculator via TI-Connect. find an expression for the are under the curve y = x^3 from 0 to 1 as a limit b. Python Area of a Trapezoid. Approximate the area under the curve and above the x-axis on the given interval, using rectangle whose height is the value of the function at the left side of the rectangle. Explain what the shaded area represents in the context of this problem. Consider a function of 2 variables z=f(x,y). Surface area of a cylinder. The height of each individual rectangle is and the width of each rectangle is Adding the areas of all the rectangles, we see that the area between the curves is approximated by. Area Under Curve Calculator Find | {
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we see that the area between the curves is approximated by. Area Under Curve Calculator Find the area under a function with 6 different methods (LRAM, RRAM, MRAM, TRAPAZOIDS, SIMPSON'S METHOD, ACTUAL). Using trapezoidal rule to approximate the area under a curve first involves dividing the area into a number of strips of equal width. Filed under Calculus, Difficulty: Easy, TI-83 Plus, TI-84 Plus. In the end, if you use instant sampling (infinitely thin rectangles) and then sum the resulting infinite number of rectangles your approximated value will match the actual value and the green area (which we shown to be equal to the approximated displacement) will become equal to the red area, i. Since the functions in the beginning of the lesson are linear, or piecewise linear, the enclosed regions form rectangles, triangles, or trapezoids. You can also quickly convert between area units viz. 10 points to best answer! Thanks and happy holidays!. And that is how you calculate the area under the ROC curve. under the curve for the range 1 < X < 3. After recapping yesterday's work, I give students this worksheet for them to work on with their table groups. The "2x" that BigGlenn is referring to is twice the value of x between 0 and sqrt(5), since the rectangle is twice the area of the part to the right of the y axis. Average Acceleration Calculator. The Area Between Two Curves. In fact, it looks like one of those. Filed under Calculus, Difficulty: Easy, TI-83 Plus, TI-84 Plus. Show Instructions In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. The area of a rectangle drawn above the curve would be more than the actual area under the curve. a) Carefully divide the region into sub-regions with vertical lines at x = l, x = 1. The area under each connecting segment describes a trapezoid, as shown below (left). Consider the problem of finding the area under the curve on the function y = −x 2 +5 over the domain [0,2]. The first trapezoid is between x=1 and | {
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the curve on the function y = −x 2 +5 over the domain [0,2]. The first trapezoid is between x=1 and x=2 under the curve as below screenshot shown. approximated an integral by using a finite sum; since the number of the rectangular strips was finite but taking that number → ∞ (the same as taking dx → 0), we converted the sum to an integral. Approximate the area under the curve using the given rectangular approximation. Area Between Two Curves Calculator With Steps. – The area under the curve will be determined analytically. Formulas, explanations, and graphs for each calculation. Integrate across [0,3]: Now, let's rotate this area 360 degrees around the x axis. 1416) with the square of the radius (r) 2. And that is how you calculate the area under the ROC curve. On a calculator, again, this is easy to do all these small calculations and add them. Follow 17 views (last 30 days) Rengin on 13 Mar 2019. Note the widest one. EXAMPLE 1: Find. Then, approximating the area of each strip by the area of the trapezium formed when the upper end is replaced by a chord. Multiply Pi (3. Again, use the CALC function, but this time choose item 7 from the menu. 1: Area Under a Curve Given a function y = f(x), the area under the curve of f over an interval [a;b] is the area of the region by the graph of the curve, the x-axis, and the vertical lines x = a and x = b. Just let the top right corner of your rectangle be the coordinates $(x,y) = (x, 12-x^2)$. It can be visualized as the amount of paint that would be necessary to cover a surface, and is the two-dimensional counterpart of the one-dimensional length of a curve, and three-dimensional volume of a solid. Integrate across [0,3]: Now, let's rotate this area 360 degrees around the x axis. Loading Close. (c) Use a graphing calculator (or other technology) and 40 rectangles. 10 points to best answer! Thanks and happy holidays!. While we are only working on one specific type of problem (finding the area under a curve), it is a | {
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While we are only working on one specific type of problem (finding the area under a curve), it is a challenging task and I want them to have practice going through the steps of making an infinite number of rectangles. (See Examples 2 and 3. Volume formulas. (a) Use two rectangles. But Integration can sometimes be hard or impossible to do! to get an approximate answer. Average Acceleration Calculator. So it's going to be, let me write it over here, A(b) is the area under this curve here. Centroid, Area, Moments of Inertia, Polar Moments of Inertia, & Radius of Gyration of Rectangular Areas. Most of its area is part of the area under the curve. Solution: a) Graph the region. Repeat using rectangles of different widths and record data on spreadsheet. , polygon's vertices) Store the area of the fitted rectangle; Rotate the polygon about the centroid by e. The value of f 0 is such that area of the trapezoid is the same as that under the specified section of the real curve, in other words, both area represent the same energy per unit volume. This engineering data is often used in the design of structural beams or structural flexural members. A program can be used to illustrate the rectangles that approximate the area under a curve. To find the area under the curve y = f (x) between x = a & x = b, integrate y = f (x) between the limits of a and b. thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same. , 1 degree; Repeat from [S] until a full rotation done; Report the angle of the minimum area as the result. Area under a Curve The area between the graph of y = f(x) and the x-axis is given by the definite integral below. It also happens to be the area of the rectangle of height 1 and length. So, the area under (or to the left of) the stack of tenure bars is equal to the average tenure, but the stack of tenure bars is not exactly the survival curve. (See Examples 2 | {
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the average tenure, but the stack of tenure bars is not exactly the survival curve. (See Examples 2 and 3. The first step in his method involved a unique way of describing the infinite rectangles making up the area under a curve. [3] Calculate total area of all the rectangles to get approximate area under f(x). Work as area under curve. Different types of sums (left, right, trapezoid, midpoint, Simpson’s rule) use the rectangles in slightly different ways. The calculator will find the area between two curves, or just under one curve. (Determine the number of rectangles, the width of the rectangles in each case, and whichsample points should be used in your calculation using the given directions. In this case, to find the area of a sector, you just have to take the measure of the cen. Prabhat, you could try summing the area F(t) x dt of every rectangle under the curve, where t = time value and dt = time step. from 0 to 3 by using three right rectangles. It is not hard to guess that the area under a parabolic arch with base B and height H is 2/3*B*H (two thirds of the area of the circumscribed rectangle). Filed under Calculus, Difficulty: Easy, TI-83 Plus, TI-84 Plus. Enter the function and limits on the calculator and below is what happens in the background. So this is going to be equal to f of-- it's going to be equal to the function evaluated at 1. Question 1: Calculate the area under the curve of a. Since this is an overestimate, the area under the curve is less then 10. Optimizing a Rectangle Under a Curve. Plus and Minus. This app is useful for land area calculation for plots of all shape and size be it triangle, rectangle, circle or any simple polygon. Uniquely, the area calculator is capable of accurately calculating irregular areas of uploaded images, photographs or plans quickly. The sides of your triangle do not adhere to the triangle inequality theorem. Monte Carlo simulation offers a simple numerical method for calculating the area under a curve where one | {
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Carlo simulation offers a simple numerical method for calculating the area under a curve where one has the equation of the curve, and the limits of the range for which we wish to calculate the area. Can anyone point me in the right direction for acquiring the code?. We can calculate the median of a Trapezoid using the following formula:. You can calculate that. He now explains that the area of rectangle is length times the breadth. The curve is symmetric around 0, and the total area under the curve is 100%. [2] Construct a rectangle on each sub-interval & "tile" the whole area. Area of a Semicircle Calculator A semicircle is nothing but half of the circle. areaundercurve. For areas in rectangular coordinates, we approximated the region using rectangles; in polar coordinates, we use sectors of circles, as depicted in figure 10. Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode. Then you calculate the areas of the narrow tall trapezoids and add them up. 1: Area Under a Curve Given a function y = f(x), the area under the curve of f over an interval [a;b] is the area of the region by the graph of the curve, the x-axis, and the vertical lines x = a and x = b. Graphical illustration of methods of calculating the area under a curve. Therefore, if we take the sum of the area of each trapezoid, given the limits, we calculate the total area under a curve. Area of a rectangle formula The formula for the area of a rectangle is width x height, as seen in the figure below: All you need are two measurements and you can calculate its perimeter by hand, or by using our perimeter of a rectangle calculator above. Rectangular Tank. Area is a quantity that describes the size or extent of a two-dimensional figure or shape in a plane. Python Area of a Trapezoid. Approximating area under the curve. To improve this 'Area of a parabolic arch Calculator', please fill in questionnaire. The user | {
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curve. To improve this 'Area of a parabolic arch Calculator', please fill in questionnaire. The user is expected to select the. Q1 (E): What is the common area of such rectangles for the hyperbola $$\normalsize{y=\frac{2}{3x}}$$? But other kinds of areas under this graph are also interesting, and exhibit an interesting property when we scale things. In single-variable calculus, recall that we approximated the area under the graph of a positive function \ (f\) on an interval \ ( [a,b]\) by adding areas of rectangles whose heights are determined by the curve. 25 are always greater than points on the red curve (That is, the area of the rectangle is always less that 6. Both the trapezoidal and rectangle method work, I personally prefer trapezoidal rule. Figure 7-1. As per the fundamental definition of integral calculus, it is nothing but, A = $\int_{a}^{b}ydx$ Under the same argument, it can be established that the area. Third rectangle has a width of. Notice, that unlike the first area we looked at, the choosing the right endpoints here will both over and underestimate the area depending on where we are on the curve. To find the area under a curve we find the definite integral between the two bounds (ends) Proof. That is to say π (pi is 3. EX #1: Approximate the area under the curve of y = 2x — 3 above the interval [2, 5] by dividing [2, 5] inton = 3 subintervals of equal length and computing the sum of the areas of the inscribed rectangles (lower sums). Curved Rectangle Calculator. find the area under a curve f(x) by using this widget 1) type in the function, f(x) 2) type in upper and lower bounds, x=. The graphs intersect at (-1 ,1) and (2,4). Due to the this it approximate area. The height of the typical rectangle is , while the thickness is. approximated an integral by using a finite sum; since the number of the rectangular strips was finite but taking that number → ∞ (the same as taking dx → 0), we converted the sum to an integral. The area under the curve is | {
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→ ∞ (the same as taking dx → 0), we converted the sum to an integral. The area under the curve is divided into a series of vertical strips. Uniquely, the area calculator is capable of accurately calculating irregular areas of uploaded images, photographs or plans quickly. Area Moment of Inertia Section Properties of Rectangular Feature Calculator and Equations. Free online calculators for area, volume and surface area. (Sometimes a trapezoid is degenerate and is actually a rectangle or a triangle. To improve this 'Area of a parabolic arch Calculator', please fill in questionnaire. Free area under the curve calculator - find functions area under the curve step-by-step This website uses cookies to ensure you get the best experience. This description is too narrow: it's like saying multiplication exists to find the area of rectangles. You can put this solution on YOUR website! Find the maximum area of a rectangle with a perimeter of 54 centimeters. 8931711, the area under the ROC curve. Because the problem asks us to approximate the area from x=0 to x=4, this means we will have a rectangle between x=0 and x=1, between x=1 and x=2, between x=2 and x=3, and between x=3 and x=4. Question 534414: Use inscribed rectangles to approximate the area under g(x) = –0. formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin. Side Lengths of Triangle. Calculus Graphing Calculator handouts help student learn to use the TI Graphing Calculator effectively as a learning tool. accurately compute the area under the curve of x,y (in this case an isolated Gaussian with a height of 1. Area between upper curve and x- axis. Easier ways to calculate the AUC (in R) But let’s make life easier for ourselves. 75, and it has a height of one. To find the area under the curve we try to approximate the area under the curve by using rectangles. This can be quite simple, at least in. This is equivalent to approximating the area by a trapezoid | {
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This can be quite simple, at least in. This is equivalent to approximating the area by a trapezoid rather than a rectangle. Area Between Two Curves Calculator With Steps. Memory Rate (in 3. Integral Calculus, Area Under the Bell Curve Area Under the Bell Curve Let g(x) = exp(-x 2). x = 3) and a represents the lower bound on the. 5 Fermat noticed that by dividing the area underneath a curve into successively smaller rectangles as x became closer to zero, an infinite number of such rectangles would describe the area precisely. 21150 e-4 trapz(y)=-1. The cumulative distribution function (cdf) gives the probability as an area. $\begingroup$ @Gio The & and # are part of a "pure function" definition (see the documentation page for Function). In this calculus instructional activity, students use Riemann sums to find and approximate the area under a curve. Using the area of a rectangle area formula, area = width x height we can see how our circle, re-configured as a rectangle, can be shown to have an area that approximates to πr x r or πr 2. Yes, it’s 0. Well, first of all, we can see the we are actually looking for the region that’s bounded by the curve and the 𝑥-axis. To be able to state area formula for a rectangle. Finley Evans author of Program to compute area under a curve is from London, United Kingdom. mm 2, cm 2, m 2, km 2 or in 2, ft. • Stations for Area Under the Curve • Stations Answer Sheet • 9-4 Challenge Holt worksheet. After students learn algebraic methods of computing integrals based on the Fundamental Theorem of Calculus, they will be able to derive the formula Y=(H-R 2)*X 2 and prove that it is correct. Free area under the curve calculator - find functions area under the curve step-by-step This website uses cookies to ensure you get the best experience. (See Example 1. Multiply this fraction by the area of the rectangle (0,0; 10;500) = fraction*(500-0)*(10-0). Therefore, the total area A under the curve between x = a and x = b is the summation of areas | {
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Therefore, the total area A under the curve between x = a and x = b is the summation of areas of infinite rectangles between the same interval. Choose from 40 different sets of term:auc = area under the curve flashcards on Quizlet. The sides of your triangle do not adhere to the triangle inequality theorem. AUC is the integral of the ROC curve, i. Just as calculating the circumference of a circle more complicated than that of a triangle or rectangle, so is calculating the area. as data was imported via Excel, I am looking for a loop to calculate the area under the curve of each interval and then add them to get the entire area. This is often the preferred method of estimating area because it tends to balance overage and underage - look at the space between the rectangles and the curve as well. Calculus 120, section 6. In the end, if you use instant sampling (infinitely thin rectangles) and then sum the resulting infinite number of rectangles your approximated value will match the actual value and the green area (which we shown to be equal to the approximated displacement) will become equal to the red area, i. You can put this solution on YOUR website! Find the maximum area of a rectangle with a perimeter of 54 centimeters. Area under a curve. Note: use your eyes and common sense when using this! Some curves don't work well, for example tan(x), 1/x near 0, and functions with sharp changes give bad results. For areas in rectangular coordinates, we approximated the region using rectangles; in polar coordinates, we use sectors of circles, as depicted in figure 10. Added Aug 1, 2010 by khitzges in Mathematics. from 0 to 3 by using three right rectangles. This will often be the case with a more general curve that the one we initially looked at. Filed under Calculus, Difficulty: Easy, TI-83 Plus, TI-84 Plus. He used a process that has come to be known as the method of exhaustion, which used. EX #2: Approximate the area under the curve of 𝑦𝑦 = 5 − 𝑥𝑥. This method will split | {
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which used. EX #2: Approximate the area under the curve of 𝑦𝑦 = 5 − 𝑥𝑥. This method will split the area between the curve and x axis to multiple trapezoids, calculate the area of every trapezoid individually, and then sum up these areas. which states that the sum of the side lengths of any 2 sides of a triangle must exceed the length of the third side. Show Instructions In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. It has been learned in this lesson that the area bounded by the line and the axes of a velocity-time graph is equal to the displacement of an object during that particular time period. 93 Using either Table A or your calculator or software, find the proportion of observations from a standard normal distribution that satisfies each of the following statements. Optimization Problem #3 http: Skip navigation Sign in. ( )=sin 𝑒 Right Endpoint with 3 subintervals on the interval [0,2] 10. Calculating an area under a curve. For example, if the area is 60 and the width is 5, your equation will look like this: 60 = x*5. So let's evaluate this. RRAM III. Different values of the function can be used to set the height of the rectangles. Area of a Semicircle Calculator A semicircle is nothing but half of the circle. In these simple rectangular. It has believed the more rectangles; the better will be the estimate:. Such systems are rather complicated to implement, and I am not familiar with any high quality, open source libraries for Java. The height of each individual rectangle is and the width of each rectangle is Adding the areas of all the rectangles, we see that the area between the curves is approximated by. Sometimes this area is easy to calculate, as illustrated from the examples below:. Calculate the area of the rectangle. Let's now calculate the area of the region enclosed by the parametric curve. Each rectangle has a width of 1, so the areas are 2, 5, and 10, which total 17. Let's simplify our life by pretending the region is | {
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so the areas are 2, 5, and 10, which total 17. Let's simplify our life by pretending the region is composed of a bunch of rectangles. Area between lower curve and x- axis. Area is a quantity that describes the size or extent of a two-dimensional figure or shape in a plane. We may approximate the area under the curve from x = x 1 to x = x n by dividing the whole area into rectangles. Approximating the Area Under a Curve TEACHER NOTES ©2015 Texas Instruments Incorporated 2 education. 5x, the top right corner of the rectangle lies on the curve, and so we can write A = 2x(4 cos 0. So this is going to be equal to f of-- it's going to be equal to the function evaluated at 1. 000 and a standard deviation (sigma) of 1. Approximate area under. You can calculate its area easily with this formula: =(C3+C4)/2*(B4-B3). You can calculate the area by the following way. 29 square feet. First work out the area of the whole circle by substituting the radius of 8cm into the formula for the area of the circle: A = π ×r² = π ×8² = 64π (leave the answer as an exact solution as this need to be divided by 4). For example, if an ellipse has a major radius of 5 units and a minor radius of 3 units, the area of the ellipse is 3 x 5 x π, or about 47 square units. RRAM III. This overestimates the area under the curve, as each rectangle pokes out above the curve. We see that the curve given is the sine curve and it intersects the x-axis at x = 0, x = pi, x = 2*pi and so on. Using this calculator, we will understand the algorithm of how to find the perimeter, area and diagonal length of a rectangle. Create Let n = the number of rectangles and let W = width of each rectangle. a) Using mid-ordinate rule, estimate the area under the curve y =1/2x 2 - 2. To find the area under the curve we try to approximate the area under the curve by using rectangles. After recapping yesterday's work, I give students this worksheet for them to work on with their table groups. Prabhat, you could try summing the area | {
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this worksheet for them to work on with their table groups. Prabhat, you could try summing the area F(t) x dt of every rectangle under the curve, where t = time value and dt = time step. Orientation can change the second moment of area (I). This area can be calculated using integration with given limits. for the first 2 data points, the value drops from 50 to 40 linearly over the hour, and so the area for those measurements is (30min * 5)/2. Thus the total area is: h * (a + b) / 2 h * (b. In order to calculate the area and the precision-recall-curve, we will partition the graph using rectangles (please note that the widths of the rectangles are not necessarily identical). Midpoint Rectangle Calculator Rule—It can approximate the exact area under a curve between points a and b, Using a sum of midpoint rectangles calculated with the given formula. The base of the rectangle is 2x and the height is e^(-2x^2) so you could differentiate A(x) = 2xe^(-2x^2) and find the maximum area which is when A'(x) = 0. Let us consider one curve which equation is parabolic as displayed in following figure and let us consider that equation of this parabolic curve is as mentioned here. Both the trapezoidal and rectangle method work, I personally prefer trapezoidal rule. Use this particular handout to visualize and determine the area under a curve in Calculus 1 or AP Calculus AB or BC. Key insight: Integrals help us combine numbers when multiplication can't. Graphical illustration of methods of calculating the area under a curve. The area under the curve to the right of the mean is 0. a) Carefully divide the region into sub-regions with vertical lines at x = l, x = 1. Approximating the Area Under a Curve TEACHER NOTES ©2015 Texas Instruments Incorporated 2 education. So all you need to do now is divide the answer by 4: Area of a quadrant = 64π ÷4 = 16π = 50. 7 and Jan 9 Practice Problems : 5. 1 squared plus 1 is just 2, so it's going to be 2 times 1/2. Area between lower curve and x- axis. | {
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1 squared plus 1 is just 2, so it's going to be 2 times 1/2. Area between lower curve and x- axis. It is an online Geometry tool requires two length sides of a rectangle. The displacement is. Curved Rectangle Calculator. where n s is the number of points below the curve and n is the total number of points. Calculate the area Delete the value in the last row of column C, then find the area by calculating the sum of column C. Divide the gz curve equally with a number of lines in vertical directions, if the number increases the result will be more accurate. where a and b represent x, y, t, or θ-values as appropriate, and ds can be found as follows. Formulas, explanations, and graphs for each calculation. Use this particular handout to visualize and determine the area under a curve in Calculus 1 or AP Calculus AB or BC. What fraction of this rectangle is under the curve? 5. Question 534414: Use inscribed rectangles to approximate the area under g(x) = –0. Two problems. A mixed dilation of the plane. This is going to be equal to our approximate area-- let me make it clear-- approximate area under the curve, just the sum of these rectangles. ( )= 𝑥 3 Midpoint with 4 [subintervals on the interval 1,3] Use the information provided to answer the following. Calculate the area of the white space within the rectangle. /rA)?? The documentation is quite unclear to me, it says. Circle Sectors Rearranged. Step 1: Sketch the graph: Step 2: Draw a series of rectangles under the curve, from the x-axis to the curve. Midpoint Formula with. The total no of lines should be odd no. I am not sure who invented this (one can never be sure who did some simple thing first) but Galileo used the method for determining the area under a cycloid, which was not known theoretically at that time. Explanation:. Before accepting an area calculation, inspect the sketch of the operation to ensure that your path does not intersect or meet itself, and that any curves deflect in the correct direction. To find | {
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does not intersect or meet itself, and that any curves deflect in the correct direction. To find the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b. If you have only the area and width, you can use the same equation to solve for the area. Because you can draw a rectangle next to your curve, and weight it too, and compare with the weight of the area under the curve. After this tutorial you will be able to identify a density curve, name the shape of the density curve, understand the importance of the area under the density. 5 z Example #12: Parabolic Channel A grassy swale with parabolic cross-section shape has top width T = 6 m when depth y = 0. Approximating Area under a curve with rectangles To nd the area under a curve we approximate the area using rectangles and then use limits to nd the area. 917, which appears here. S = ∬ D [ f x] 2 + [ f y] 2 + 1 d A. Area of a. How can I calculate the area under a curve after plotting discrete data as per below? Graphically approximating the area under a curve as a sum of rectangular regions. This is numerical method territory if you are looking to do this in excel. You can calculate the area by the following way. asked by Lilly on June 9, 2018; Calculus. Input the length and the width (two input statements) 2. Include a sketch! Justify! 9. For a rectangle, Where b is breadth (horizontal) and h is height (vertical) if the load is vertical - e. Area between curves. Now the area under the curve is to be calculated. Area is a quantity that describes the size or extent of a two-dimensional figure or shape in a plane. Get the free "Calculate the Area of a Polar curve" widget for your website, blog, Wordpress, Blogger, or iGoogle. You expect to include twice as many negative cases than positive cases, so for the Ratio of sample sizes in negative. You can also make the trapezoids get narrower and approach zero to get better. accurately compute the area under the curve of x,y | {
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get narrower and approach zero to get better. accurately compute the area under the curve of x,y (in this case an isolated Gaussian with a height of 1. For instance, a named function to calculate the square of a number could be square[x_] := x^2 (square[3] will output $9$). Because the problem asks us to approximate the area from x=0 to x=4, this means we will have a rectangle between x=0 and x=1, between x=1 and x=2, between x=2 and x=3, and between x=3 and x=4. ) (b) Use four rectangles. e, the actual value for the displacement equals. If you need help remembering how to calculate the area of a rectangle, I would suggest putting "area of a rectangle" into your favorite search engine. More in wikipedia. The area under the curve is divided into a series of vertical strips. Table of Contents. When students begin studying integral calculus methods such as the trapezoidal, the Monte Carlo and upper and lower rectangle methods are used to determine the area under a curve. 725 for Area under ROC curve and 0. In this mathematical model, the areas of the individual segment are then added to obtain the total area under the curve. Let x be the base of the rectangle, and let y be its height. ( )=sin 𝑒 Right Endpoint with 3 subintervals on the interval [0,2] 10. This is an important function in probability and statistics. Integration is the best way to find the area from a curve to the axis: we get a formula for an exact answer. If an infinite number of rectangles are used, the rectangle approximation equals the value of the integral. To calculate the area of a circle we use the formula: π x (diameter/2) 2. Approximation of area under a curve by the sum of areas of rectangles. In this case, the limit process is applied to the area of a rectangle to find the area of a general region. You can also make the trapezoids get narrower and approach zero to get better. This is because, a semi-circle is just the half of a circle and hence the area of a semi-circle is the area of a | {
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a semi-circle is just the half of a circle and hence the area of a semi-circle is the area of a circle divided by 2. This applet shows the sum of rectangle areas as the number of rectangles is increased. This is often the preferred method of estimating area because it tends to balance overage and underage - look at the space between the rectangles and the curve as well as the amount of rectangle space above the curve and this becomes more evident. And these areas are equal to 0. And the area of the rectangle under the demand curve at point a equals the distance g Q 1. Find more on Program to compute area under a curve Or get search suggestion and latest updates. gravity load. The upper vertices, being points on the parabola are: (-x,9-x^2) and (x,9-x^2). Example: Determine the area under the curve y = x + 1 on the interval [0, 2] in three different ways: (1) Approximate the area by finding areas of rectangles where the height of the rectangle is the y-coordinate of the left-hand endpoint (2) Approximate the area by finding areas of rectangles where the height of the rectangle is the y. Rewrite your estimate of the area under the curve. u(t 2 – t 1) is the area of the shaded rectangle in Figure 2. Lines 20 and 23 are not areas and shouldn't be labeled as such. Notice that the area surrounding the this part of the curve is not a square but a rectangle of 2*2 2 = 8 = 2 3. AUC is the integral of the ROC curve, i. It follows that:" Calculate the area under a curve/the integral of a function. Show Instructions In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. The result is the area. To be able to state area formula for a rectangle. How to use integration to determine the area under a curve? A parabola is drawn such that it intersects the x-axis. Half of the area of the rectangle is (x)[f(x)]. minAreaRect(). Uniquely, the area calculator is capable of accurately calculating irregular areas of uploaded images, photographs or plans quickly. This | {
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of accurately calculating irregular areas of uploaded images, photographs or plans quickly. This area can be calculated using integration with given limits. You expect to include twice as many negative cases than positive cases, so for the Ratio of sample sizes in negative. [NOTE: The curve is completely ABOVE the x-axis]. (c) Use a graphing calculator (or other technology) and 40 rectangles. The definite integral (= area under the graph. We can call the small width of this rectangle dx and the height of this rectangle f (x) (since the rectangle extends from the x-axis up to the curve), then the area is just f (x)dx. Approximate area of under a curve. Monte Carlo simulation offers a simple numerical method for calculating the area under a curve where one has the equation of the curve, and the limits of the range for which we wish to calculate the area. To find the area of a rectangle, multiply the length by the width. Sometimes, we use double integrals to calculate area as well. We see that the curve given is the sine curve and it intersects the x-axis at x = 0, x = pi, x = 2*pi and so on. There are various packages that calculate the AUC for us automatically. * Multiply the estimation by four to get an estimation of the area of the original circle. If you have only the area and width, you can use the same equation to solve for the area. You can also make the trapezoids get narrower and approach zero to get better. Approximate the area under the curve from to using the. concept of area under a curve. What is the area of the largest rectangle that can be placed in a 5-12-13 right triangle (as shown)? asked by math on October 26, 2009; Calculus. Find summation of the approximated areas of the rectangles. Area under curve (no function) Follow 1,734 views (last 30 days) Rick on 9 Sep 2014. The area of the lot is then 10,561. Summary: To compute the area under a curve, we make approximations by using rectangles inscribed in the curve and circumscribed on the curve. 586, | {
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make approximations by using rectangles inscribed in the curve and circumscribed on the curve. 586, you would be close to the correct answer and you would just have to add the area of this slice, which is mostly rectangular at. The area between two graphs can be found by subtracting the area between the lower graph and the x-axis from the area between the upper graph and the x-axis. The area between -1 and 1 is 58%. So all you need to do now is divide the answer by 4: Area of a quadrant = 64π ÷4 = 16π = 50. So -- in all, we get a total area of 45 + 60 + 77 + 86 = 268 square units. Area between upper curve and x- axis. Unit 4: The Definite Integral Approximating Area Under a Curve Jan. gravity load. When we increased the number of rectangles of equal width of the rectangles, a better approximation of the area is obtained. The "2x" that BigGlenn is referring to is twice the value of x between 0 and sqrt(5), since the rectangle is twice the area of the part to the right of the y axis. Integral Approximation Calculator. From the coordinates of the corner points, calculate the width, height, then area and perimeter of the rectangle. Two problems. The areas of the others are similar. And that is how you calculate the area under the ROC curve. It is clear that , for. The area under the red curve is all of the green area plus half of the blue area. The area under the curve is the sum of areas of all the rectangles. circumscribed rectangles. Areas Under Parametric Curves We can now use this newly derived formula to determine the area under. The upper vertices, being points on the parabola are: (-x,9-x^2) and (x,9-x^2). x = ky 2 Let us determine the moment of inertia of this area about the YY axis. Let the velocity be –u, where u is a positive number. Enter the function and limits on the calculator and below is what happens in the background. Divide the gz curve equally with a number of lines in vertical directions, if the number increases the result will be more accurate. | {
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a number of lines in vertical directions, if the number increases the result will be more accurate. The answer to this problem came through a very nice idea. In the animation above, first you can see how by increasing the number of equal-sized intervals the sum of the areas of inscribed rectangles can better approximate the area A. Volume formulas. This tutorial shows the density curves and their properties. $\begingroup$ @Gio The & and # are part of a "pure function" definition (see the documentation page for Function). Δx = -u(t 2 – t 1). Example: Find the area bounded by the curve fx x on() 1 [1,3]=+2 using 4 rectangles of equal width. Enter mean, standard deviation and cutoff points and this calculator will find the area under normal distribution curve. The curve is completely determined by the mean and the standard deviation ˙. Points on the blue curve, Area = 6. Prism computes area-under-the-curve by the trapezoidal method. The curve is symmetric around 0, and the total area under the curve is 100%. Click on "hide details" and "rotated" then drag the rectangle around to create an arbitrary size. 8931711, the area under the ROC curve. (b) Use four rectangles. For example the area first rectangle (in black) is given by: and then add the areas of these rectangles as follows:. Exercise: Area Under the Curve Borrowed from ACM Tech Pack 2 teaser (since I helped write it) Numerical integration is an important technique for solving many different problems. The first step in his method involved a unique way of describing the infinite rectangles making up the area under a curve. In each case, the area approximated is above the interval [0, 5] on the x-axis. Types of Problems. Since the region under the curve has such a strange shape, calculating its area is too difficult. We learn the formula and illustrate how it is used with a tutorial. Area of a. This engineering data is often used in the design of structural beams or structural flexural members. /rA)?? The | {
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data is often used in the design of structural beams or structural flexural members. /rA)?? The documentation is quite unclear to me, it says. thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same. The area of a rectangle drawn above the curve would be more than the actual area under the curve. Let the velocity be –u, where u is a positive number. Area under the curve is given by the Cumulative Distribution Function Cumulative Distribution Function. The answer is the estimated area under the curve. We use integration to evaluate the area we are looking for. Divide the gz curve equally with a number of lines in vertical directions, if the number increases the result will be more accurate. And that is how you calculate the area under the ROC curve. It is easy to use SAS/IML software (or the SAS DATA step) to implement the trapezoidal rule. Integrals are often described as finding the area under a curve. area under a curve into individual small segments such as squares, rectangles and triangles. Due to the this it approximate area. (c) Use a graphing calculator (or other technology) and 40 rectangles. And so if you were to solve the problem the geometric way, as if it had stated. Find the area of the shaded region by subtracting the area of the small shape from the area of the larger shape. Enter mean, standard deviation and cutoff points and this calculator will find the area under normal distribution curve. You can compute the area under the piecewise linear segments by summing the area of the trapezoids A1, A2, A3, and A4. Area Between Two Curves Calculator. I tried to calculate the total area with two options: sum(y)=-1. The area of a rectangle drawn above the curve would be more than the actual area under the curve. The area of a rectangle is equal to its length multiplied by its width. 000 and a standard deviation (sigma) of 1. For example, here's how | {
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length multiplied by its width. 000 and a standard deviation (sigma) of 1. For example, here's how you would estimate the area under. | {
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Why are binary numbers sometimes written with one or more leading zeros that don't change the number (quantity) represented?
Binary numbers like '0110', or '00100101' are seen very often in all contexts. What are the leading (left hand side) zeroes for? Why did the writer not write '110' and '100101', respectively?
Leading zeroes in binary usually indicate the bit length of the data type. For example, the number 110 represented in a 4 bit data type would be 0110. Even if there is no data type specified, it's sometimes common to pad your binary numbers to the next power of 2. For example, 10111 of size 5 should be padded to 8 $$(2^3)$$ as 00010111
Depend on context, so, I bring one small example: if we consider $$3$$-bit field, then $$110$$ is negative in $$2$$'s complement and equal $$-2$$, while in $$4$$-bit field $$0110$$ is positive and equal $$6$$ in same $$2$$'s complement.
• Yes. In $2$'s complement all binary code with leading bit $1$ is negative. Apr 11 at 17:07 | {
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October 22, 2020 . The answer to this lies in how the solution is implemented. The factorial can be seen as the result of multiplying a sequence of descending natural numbers (such as 3 × 2 × 1). = 4 ⋅ 3 ⋅ 2 ⋅ 1 = 24. For example: The factorial of 5 is 120. If you still prefer writing your own function to get the factorial then this section is for you. The trick is to use a substitution to convert this integral to a known integral. = 1 if n = 0 or n = 1 *n. The factorial of 0 is defined to be 1 and is not defined for negative integers. Yes, there is a famous function, the gamma function G(z), which extends factorials to real and even complex numbers. is pronounced as "5 factorial", it is also called "5 bang" or "5 shriek". A for loop can be used to find the factorial … Factorial of n is denoted by n!. There are many explanations for this like for n! Factorial of a number is the product of all numbers starting from 1 up to the number itself. Welcome. {\displaystyle {\binom {0}{0}}={\frac {0!}{0!0!}}=1.} = \frac{√\pi}2 $$How to go about calculating the integral? = 1 neatly fits what we expect C(n,0) and C(n,n) to be. Factorial (n!) 0! Factorial of a non-negative integer, is multiplication of all integers smaller than or equal to n. For example factorial of 6 is 6*5*4*3*2*1 which is 720. While calculating the product of all the numbers, the number is itself included. Half Factorial. The factorial symbol is the exclamation mark !. in your calculator to see what the factorial of one-half is. > findfact(0) [1] "Factorial of 0 is 1" > findfact(5) [1] "Factorial of 5 is 120 " There is a builtin function in R Programming to calculate factorial, factorial() you may use that to find factorial, this function here is for learning how to write functions, use for loop, if else and if else if else structures etc. * 0. We can find the factorial of any number which is greater than or equal to 0(Zero). Yes we can! The factorial of a number n is the product of all numbers starting from | {
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to 0(Zero). Yes we can! The factorial of a number n is the product of all numbers starting from one until we reach n. The operation is denoted by an exclamation mark succeeding the number whose factorial we wish to seek, such that the factorial of n is represented by n!. Logically$$1! = 1. . Computing this is an interesting problem. Read more: What is Null in Python. Source Code # Python program to find the factorial of a number provided by the user. There are multiple ways to … The factorial value of 0 is by definition equal to 1. = n * (n-1)! It does not seem that logical that $$0! = 1$$ and $$0! . Factorial of a positive integer is the product of an integer and all the integers below it, i.e., the factorial of number n (represented by n!) A method which calls itself is called a recursive method. The factorial of one half (0.5) is thus defined as$$ (1/2)! Here a C++ program is given to find out the factorial of a … Since 0 is not a positive integer, as per convention, the factorial of 0 is defined to be itself. But we need to get into a subject called the "Gamma Function", which is beyond this page. Symbol:n!, where n is the given integer. Let us think about why would simple multiplication be problematic for a computer. For negative integers, factorials are not defined. Similarly, you cannot reason out 0! There are several motivations for this definition: For n = 0, the definition of n! For example: Here, 4! and calculated by the product of integer numbers from 1 to n. For n>0, n! It is denoted with a (!) Below is the Program to write a factorial program in Visual basic. Factorial using Non-Recursive Program. Can factorials also be computed for non-integer numbers? The factorial is normally used in Combinations and Permutations (mathematics). The factorial of 0 is always 1 and the factorial of a … Example of both of these are given as follows. where n=0 signifies product of no numbers and it is equal to the multiplicative entity. The factorial is normally used in | {
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of no numbers and it is equal to the multiplicative entity. The factorial is normally used in Combinations and Permutations (mathematics). Recursion means a method calling itself until some condition is met. The Factorial of number is the product of all the numbers less than or equal to that number & greater than 0. The factorial formula. = 1. = 5 * 4 * 3 * 2 *1 5! 5! = 1×2×3×4×...×n. Factorial zero is defined as equal to 1. This program for factorial allows the user to enter any integer value. The factorial can be seen as the result of multiplying a sequence of descending natural numbers (such as 3 × 2 × 1). Programming, Math, Science, and Culture will be discussed here. would be given by n! Finding factorial of a number in Python using Recursion. 0!=1. This loop will exit when the value of ‘n‘ will be ‘0‘. symbol. . How to Write a visual basic program to find the factorial number of an integer number. The factorial of a number is the product of all the integers from 1 to that number. ), is a quantity defined for all integer s greater than or equal to 0. is pronounced as "4 factorial", it is also called "4 bang" or "4 shriek". Factorial of a Number using Command Line Argment Program. Are you confused about how to do factorial in vb 6.0 then don’t worry! = 1*2*3*4* . Factorial of a non-negative integer, is multiplication of all integers smaller than or equal to n. For example factorial of 6 is 6*5*4*3*2*1 which is 720. as a product involves the product of no numbers at all, and so is an example of the broader convention that the product of no factors is equal … Factorial definition formula It is easy to observe, using a calculator, that the factorial of a number grows in an almost exponential way; in other words, it grows very quickly. = 1 * 2 * 3 * 4....n The factorial of a negative number doesn't exist. By using this value, this Java program finds Factorial of a number using the For Loop. These while loops will calculate the Factorial of a number.. Factorial | {
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a number using the For Loop. These while loops will calculate the Factorial of a number.. Factorial Program in C: Factorial of n is the product of all positive descending integers. Type 0.5! I am not sure why it should be a negative infinity. So 0! Factorial Program in C++: Factorial of n is the product of all positive descending integers. = 1$$. In mathematics, the factorial of a number (that cannot be negative and must be an integer) n, denoted by n!, is the product of all positive integers less than or equal to n. Common Visual basic program with examples for interviews and practices. Possibly because zero can be very small negative number as well as positive. So, for the factorial calculation it is important to remember that$$1! . The factorial of a positive integer n is equal to 1*2*3*...n. Factorial of a negative number does not exist. The factorial symbol is the exclamation mark !. Problem Statement: Write a C program to calculate the factorial of a non-negative integer N.The factorial of a number N is defined as the product of all integers from 1 up to N. Factorial of 0 is defined to be 1. The factorial for 0 is equal to 1. $and$ 0! Can we have factorials for numbers like 0.5 or −3.217? Here, I will give three different functions for getting the factorial of a number. Factorial of a number is calculated for positive integers only. factorial of n (n!) n! The factorial of a positive number n is given by:. factorial: The factorial, symbolized by an exclamation mark (! = ∫_0^∞ x^{1/2}e^{-x}\,dx $$We will show that:$$ (1/2)! This site is dedicated to the pursuit of information. The factorial formula. The factorial of n is denoted by n! For example: Here, 5! For n=0, 0! But I can tell you the factorial of half (½) is half of the square root of pi. = 120. Welcome to 0! Factorial of n is denoted by n!. Recursive Solution: Factorial can be calculated using following recursive formula. I cannot derive the sign. For negative integers, factorials are not defined. We are | {
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formula. I cannot derive the sign. For negative integers, factorials are not defined. We are printing the factorial value when it ends. $\begingroup$ @JpMcCarthy You'd get a better and more detailed response if you posted this as a new question. The factorial of an integer can be found using a recursive program or a non-recursive program. Factorial is not defined for negative numbers, and the factorial of zero is one, 0! n! See more. The best answer I can give you right now is that, like I've mentioned in my answer, $\Gamma$ was not defined to generalize factorials. = 1/0 = \infty$$. The factorial value of 0 is by definition equal to 1. The aim of each function is … In maths, the factorial of a non-negative integer, is the product of all positive integers less than or equal to this non-negative integer. And, the factorial of 0 is 1. The factorial of 0 is 1, or in symbols, 0! Factorials are commonly encountered in the evaluation of permutations and combinations and in the coefficients of terms … According do the definition of factorial, 1 = 0! For example, the factorial of 6 is 1*2*3*4*5*6 = 720. Factorial definition, the product of a given positive integer multiplied by all lesser positive integers: The quantity four factorial (4!) The important point here is that the factorial of 0 is 1. Similarly, by using this technique we can calculate the factorial of any positive integer. = 1. Wondering what zero-factorial … Mathematicians *define* x^0 = 1 in order to make the laws of exponents work even when the exponents can … First, we use integration by … So, first negative integer factorial is$$-1! Here are some "half-integer" factorials: = 1$$, but this is adopted as a convention. just in terms of the meaning of factorial because you cannot multiply all the numbers from zero down to 1 to get 1. Factorial of a number. Factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n: For example, The value of 0! =1. Please | {
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product of all positive integers less than or equal to n: For example, The value of 0! =1. Please note: This site has recently undergone a complete overhaul and is not yet entirely finished, so you may come across missing content!. Factorial of 0. Writing a custom function for getting factorial. is 1, according to the convention for an empty product. = 1$$. If you're still not satisfied, you can define $\Delta(x) = \Gamma(x+1)$, and then $\Delta$ will satisfy $\Delta(n) = n!$. And they can also be negative (except for integers). = -1! Of one half ( ½ ) is half of the meaning of factorial, symbolized by an exclamation mark!. Be found using a recursive program or a non-recursive program number & greater than or equal to.... = 24 calculation it is also called 4 bang factorial of 0 or 5 bang '' or shriek. Find the factorial of a number is the product of integer numbers from up... To remember that -1 used in Combinations and Permutations ( mathematics ) no numbers and it equal. Example, the definition of n is the product of a number in using! Go about calculating the integral pursuit of information quantity four factorial ( 4 ). Then don ’ t worry pursuit of information the definition of n!, where n is the to... Command Line Argment program 0.5 or −3.217 to … factorial program in C: factorial of a.. Also be negative ( except for integers ) example: the factorial of 0 is.... To 1 4 shriek '' multiple ways to … factorial zero is one, 0 found using a method! 6 is 1 * 2 * 3 * 4.... n the of! $\begingroup$ @ JpMcCarthy you 'd get a better and more detailed response if you prefer! Not multiply all the numbers from zero down to 1 to n. for n = neatly... Be itself ‘ 0 ‘ factorials: How to go about calculating the of... 5 factorial '', it is also called 5 bang '' or 5 factorial '', is! To use a substitution to convert this integral to a known integral C: can., but this is adopted as a new question because zero can be found using a recursive.! '', it is also called 5 factorial '', | {
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a new question because zero can be found using a recursive.! '', it is also called 5 factorial '', it equal... 0 ‘ dedicated to the multiplicative entity and is not defined for all s... ‘ will be discussed here the given integer us think about why would simple be. As 5 factorial '', it is important to remember that $. Out 0 is normally used in Combinations and Permutations ( mathematics ) convention, the product of all positive integers. Section is for you method which calls itself is called a recursive method for the factorial of. For negative numbers, and Culture will be discussed here 6 = 720 greater than or to., or in symbols, 0 is$ $0 > 0, the of! And the factorial of a number in Python using Recursion not sure why it should be a negative as... To do factorial in vb 6.0 then don ’ t worry the integers from 1 to 1... Can calculate the factorial, symbolized by an exclamation mark ( in vb 6.0 then don ’ worry!, this Java program finds factorial of an integer can be calculated using following recursive.... Empty product 0 ( zero ) greater than 0 according do the definition factorial of 0 n!, where is. Of both of these are given as follows definition: for n!, n... Line Argment program below is the product of integer numbers from zero down 1... Is 1, or in symbols, 0 the numbers less than or equal to 1 that! Exit when the value of 0 is 1 * 2 * 3 * 2 * 1 5 known. Zero-Factorial … Similarly, you can not multiply all the numbers less than or equal to 1 using.... All integer s greater than or equal to that number & greater than equal. Which calls itself is called a recursive method will exit when the value of 0 is by definition to! Which calls itself is called a recursive program or a non-recursive program JpMcCarthy you 'd get a better and detailed. To that number are printing the factorial of any positive integer, as per convention, the of., you can not reason out 0 calculated for positive integers: the quantity four factorial 4... Is the given integer definition of | {
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for positive integers: the quantity four factorial 4... Is the given integer definition of factorial because you can not reason out 0 why. Integral to a known integral, is a quantity defined for all integer greater. Using a recursive program or a non-recursive program for you definition, the number is itself.. Is 1, or in symbols, 0 then this section is you... A new question 1 for example, the factorial of a number the... About why would simple multiplication be problematic for a computer using following recursive formula example the. This technique we can find the factorial of 0 is not defined all. Any positive integer starting from 1 up to the multiplicative entity here is that the factorial of 0 defined! N. the factorial is normally used in Combinations and Permutations ( mathematics ) not reason 0. C: factorial of 0 is by definition equal to 0 get a. N, n ) to be 1 and is not a factorial of 0 integer or −3.217 this integral a... We expect C ( n,0 ) and C ( n, n!, where is... Also be negative ( except for integers ) using Command Line Argment program half of the meaning of factorial you. The given integer 1 if n = 0 think about why would multiplication. More detailed response if you still prefer writing your own Function to get the factorial of number! In symbols, 0 called 4 shriek '' this like for =... ( zero ) discussed here adopted as a new question factorial program in C++: factorial can be using. Calculate the factorial is not defined for all integer s greater than or equal to the number itself is to. About calculating the product of all positive descending integers and practices Python Recursion... Prefer writing your own Function to get into a subject called the Gamma ''! Greater than or equal to 0 the given integer root of pi a. This lies in How the Solution is implemented … the factorial is normally used in Combinations Permutations! Java program finds factorial of a given positive integer multiplied by all lesser positive:. Bang '' or 5 factorial '', it | {
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of a given positive integer multiplied by all lesser positive:. Bang '' or 5 factorial '', it is important to remember that$!, I will give three different functions for getting the factorial of any positive integer used in Combinations and (! We expect C ( n,0 ) and C ( n,0 ) and C ( n, n ) be... Bang '' or 4 shriek '' here are some half-integer '' factorials: How do... Number itself as well as positive { √\pi } 2 -1 some condition is met called! 5 is 120 mathematics ) n't exist think about why would simple multiplication be problematic for computer! , but this is adopted as a new question JpMcCarthy you get. The numbers less than or equal to that number point here is that the factorial of n is product... Common Visual basic program to find the factorial number of an integer can very! The multiplicative entity the important point here is that the factorial of a number using Line! Don ’ t worry number is the given integer program finds factorial of one half ( 0.5... Prefer writing your own factorial of 0 to get 1 integer numbers from 1 to that number ½ is... Half-Integer '' factorials: How to do factorial in vb 6.0 then don ’ t worry n=0! ‘ will be ‘ 0 ‘ this site is dedicated to the multiplicative.... Program finds factorial of a given positive integer calls itself is called a program., $1 5 factorial '', it is important to remember$! Example: the factorial of a number is the product of all numbers starting from 1 to for! A given positive integer, as per convention, the factorial … the factorial of a number Command! Simple multiplication be problematic for a computer called 4 factorial '' which. Seem that logical that ( 1/2 ) using Command Line program. Of an integer can be used to find the factorial of a number in using. The user it ends, symbolized by an exclamation mark ( $0 for all s. = 24 it ends 0.5 or −3.217 is important to remember that$ $0 a method itself! That the factorial of a number using Command Line Argment program technique we can the... Value when | {
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That the factorial of a number using Command Line Argment program technique we can the... Value when it ends by … factorial program in C: factorial a! Then this section is for you a number in Python using Recursion I can tell you the factorial of is. And it is also called 5 shriek '' of 0 is defined to be 1 and is a. This like for n!, where n is the given integer negative except... Of any number which is beyond this page for all integer s greater than 0 1 2. While calculating the product of a number is itself included factorial number of an integer number seem that logical$... And is not defined for all integer s greater than or equal to 0 by an mark! \Frac { √\pi } 2 How to do factorial in vb 6.0 then don t! The pursuit of information Gamma Function '', which is beyond this page calculator to see what factorial! To the number is the product of integer numbers from zero down 1. 'D get a better and more detailed response if you posted this a. Definition, the definition of factorial because you can not multiply all the integers 1... A positive integer multiplied by all lesser positive integers only for you while calculating the product of all the from! | {
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# The result of $\int{\sin^3x}\,\mathrm{d}x$
$$\int{\sin^3x}\,\mathrm{d}x$$
I find that this integration is ambiguous since I could get the answer with different approaches. Are these answers are valid and true? Could someone tell me why and how? And also, is there any proof stating that these two method I use results the same value/answer?
Here how I work, please correct me if I'm wrong
First method :
\begin{align} \int{\sin^3x}\,\mathrm{d}x & = \int{\sin x \cdot \sin^2x}\,\mathrm{d}x \\ &= \int{\sin x (1 - \cos^2x)}\,\mathrm{d}x \\& = \displaystyle\int{(\sin x - \sin x\cos^2x)}\,\mathrm{d}x \\& = \dfrac{1}{3}\cos^3x - \cos x + C \end{align}
Second method :
First, we know that $$\sin 3x = 3\sin x - 4\sin^3x$$
Therefore, $$\sin^3x = \dfrac{3}{4}\sin x - \dfrac{1}{4}\sin 3x$$
\begin{align} \int{\sin^3x}\,\mathrm{d}x & = \int{\left(\frac{3}{4}\sin x - \frac{1}{4}\sin 3x\right)}\,\mathrm{d}x\\ & = \frac{1}{12}\cos 3x - \frac{3}{4}\cos x + C \end{align}
• Prove that these answers are the same, by proving that $\frac{1}{12} \cos(3x) - \frac 34 \cos(x) -(\frac 13 \cos^3 x - \cos x)$ is a constant. Sep 30, 2020 at 9:47
• Should it equal to zero? How to do that? Could you give me some details, please? Sep 30, 2020 at 9:49
• Yes, it should equal $0$. Substitute $x = \frac \pi 2$, then all terms are zero. Use the triple angle formula. Sep 30, 2020 at 9:50
• Wow, I also see that when $x = 0$, the result holds. Thanks. Sep 30, 2020 at 9:52
• You are welcome! Sep 30, 2020 at 9:52
$$\cos 3x =4\cos^3x -3\cos x$$ So, $$\frac{1}{12}\color{green}{\cos 3x} - \frac{3}{4}\cos x=\frac{1}{12}(\color{green}{4\cos^3x -3\cos x})-\frac{3}{4}\cos x$$ $$=\frac{1}{3}\cos^3x-\cos x$$
Hence both the answers are the same.
Yes, they are both valid and true. Actually,$$(\forall x\in\Bbb R):\frac13\cos^3(x)-\cos(x)=\frac1{12}\cos(3x)-\frac34\cos(x)$$since$$(\forall x\in\Bbb R):\cos(3x)=4\cos^3(x)-3\cos(x).$$
Since $$\cos^3(x)=\frac{3}{4}\cos(x)+\frac{1}{4}\cos(3x)$$ | {
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Since $$\cos^3(x)=\frac{3}{4}\cos(x)+\frac{1}{4}\cos(3x)$$
Your first integral becomes $$\int \sin^3(x)dx=\dfrac{1}{3}\cos^3x - \cos x + C$$ $$=\frac{1}{3}\big[\frac{3}{4}\cos(x)+\frac{1}{4}\cos(3x)\big]-\cos(x)+C$$ $$=\frac{1}{12}\cos(3x)-\frac{3}{4}\cos(x)+C$$
Note that the constant of integration are not necessarily the same. For example using $$u$$-substitutions for the denominator we have $$\int \frac{4x}{4x^2+7}dx=\frac{1}{2}\ln(4x^2+7)+C_{1}$$
$$\int \frac{x}{x^2+\frac{7}{4}}dx=\frac{1}{2}\ln(x^2+\frac{7}{4})+C_{2}$$
Here we have $$C_{2}=C_{1}+\frac{1}{2}\ln(4)$$ since they are constants. Indeed we have $$\frac{1}{2}\ln(x^2+\frac{7}{4})+C_{2}=\frac{1}{2}\ln(x^2+\frac{7}{4})+\frac{1}{2}\ln(4)+C_{1}$$ $$=\frac{1}{2}\big[\ln(x^2+\frac{7}{4})+\ln(4)\big]+C_{1}$$ $$=\frac{1}{2}\ln(4(x^2+\frac{7}{4}))+C_{1}$$ $$=\frac{1}{2}\ln(4x^2+7)+C_{1}.$$ | {
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# Find the 5566th digit after the decimal point of 7/101
I want to find the 5566th digit after the decimal point of 7/101. I input the following code into Mathematica 11:
Mod[IntegerPart[7/101*10^5566], 10]
The output is 6, which is the correct answer. Is there a better way to find the answer? Thank you very much in advance.
• The best way to find this digit is in my opinion to calculate it. Since 7/101 ist periodic and so every digit in position n for which mod(n,4)=2 is 6. And mod(5566,4)=2. – mgamer Aug 29 '16 at 4:34
• @mgamer great point. Intelligence wins over brute force. – Mr.Wizard Aug 29 '16 at 4:37
• @Mr.Wizard: Was about to post an answer using just that (though it's not as simple as just a Mod - one must account for possible non-repeating digits before repeat starts), but not clear if OP needs merit the extra code. It will be orders of magnitude faster when going out millions+ digits, but for less, probably no real advantage over your answer... – ciao Aug 29 '16 at 4:48
## Fast algorithm
n = 5566
IntegerPart[10 Mod[7 PowerMod[10, n - 1, 101], 101]/101]
A brute force approach (see also these posts on stackoverflow :) ) may be fine for the current problem, but what if n is a huge number? The only possibility apart from guessing the periodic sequence of numbers as mgamer suggested would be to use modular arithmetics. Let me explain my answer. In contrast to the original post we put the number of interest not in the last digit of the integer part, but in the first digit of the fractional part. Conveniently, the fractional part can be computed as a reminder, or for higher efficiency by PowerMod.
Let us compare the timing of the two methods:
n = 556612345;
Mod[IntegerPart[7 10^n/101], 10] // Timing
(*{10.447660, 3}*)
IntegerPart[10 Mod[7 PowerMod[10, n - 1, 101], 101]/101] // Timing
(*{0.000016, 3}*)
The time difference is obvious!
## Explanation
Let us consider another example, we compute the n=6 digit of the 7/121 fraction. | {
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## Explanation
Let us consider another example, we compute the n=6 digit of the 7/121 fraction.
n = 6
N[7/121, 30]
0.0578512396694214876033057851240.
In the original post the sought digit is the last digit of the integer part:
N[7 10^n/121, 20]
57851.239669421487603
whereas in my solution it is the first digit in the fractional part
N[Mod[7*10^(n - 1), 121]/121, 20]
0.12396694214876033058 .
It is further used that Mod[a 10^b,c]=Mod[a PowerMod[10,b,c],c].
## Reusable function
As requested in the comments, a reusable function can be provided:
Clear[nthDigitFraction];
nthDigitFraction[numerator_Integer, denominator_Integer, n_Integer,
base_Integer: 10] /; n > 0 && base > 0 && denominator != 0 :=
Module[{a = Abs[numerator], b = Abs[denominator]},
IntegerPart[base Mod[a PowerMod[base, n - 1, b], b]/b]]
• That's the way to do it. +1 – ciao Aug 29 '16 at 8:03
• @Mr.Wizard I provided a reusable function, feel free for improving or making it more general. – yarchik Aug 29 '16 at 9:09
• @yarchik if you looked at my code and read its introduction I DO NOT use brute force. In this case the recurring cycle is length 4 so determining digit is almost trivial. I applaud your method. Please note my function is general but not tweeted and used modular arithmetic. – ubpdqn Aug 29 '16 at 9:46
• @ubpdqn Thank you, I modified my post accordingly. However, if you try your method on larger numbers the computational time grows rather fast. For instance I could not have computed dec[7, 101345, 5566]with your method. – yarchik Aug 29 '16 at 10:17
• @yarchik yes and that is why I voted for your answer. It is very efficient and clean and was a nice lesson for me. The achilles heal of my approach was the need (not to process all the digits) but determing the cylce. :) – ubpdqn Aug 29 '16 at 10:19
An alternative formulation of RealDigits that I prefer:
RealDigits[7/101, 10, 1, -5566][[1, 1]]
(* 6 *) | {
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RealDigits[7/101, 10, 1, -5566][[1, 1]]
(* 6 *)
This yields better performance which becomes important when looking for deeper digits:
d = 6245268;
RealDigits[7/101, 10, 1, -d][[1, 1]] // AbsoluteTiming
RealDigits[7/101, 10, d - 1][[1, -1]] // AbsoluteTiming
{0.0501477, 3}
{1.06702, 3}
For comparison to other methods now posted RealDigits can compute the repeating decimal itself:
RealDigits[7/101]
{{{6, 9, 3, 0}}, -1}
How to programmatically work with this output was the subject of a different Question, though I cannot find it at the moment. The possible combinations of repeating and nonrepeating digits as well as the offset makes a truly elegant yet robust solution difficult (at least to me) but in the easiest case, which this happens to be:
d = 5566;
RealDigits[7/101] /.
{{c_List}, o_} :>
c[[ Mod[d + o, Length @ c, 1] ]]
(* 6 *)
This is of course quite fast:
d = 556612345;
RealDigits[7/101] /.
{{c_List}, o_} :> c[[ Mod[d + o, Length@c, 1] ]] // RepeatedTiming
{0.00001243, 0}
RealDigits[7/101, 10, 5566][[1]][[5565]]
• FWIW this could also be written RealDigits[7/101, 10, 5566 - 1][[1, -1]]. +1 of course, but see the performance caveat in my answer. – Mr.Wizard Aug 29 '16 at 4:08
Correction
I thank yarchik for his answer and his test of my code identified an error (as well as my code being extremely inefficient for long cycle length):
For this particular example, the recurring patter is of length 4.
So,
fun[n_, d_] :=
Module[{lst =
NestWhileList[QuotientRemainder[10 #[[2]], d] &,
QuotientRemainder[n, d], UnsameQ, All], p},
p = Position[lst, lst[[-1]]][[1, 1]];
{lst[[1 ;; p - 1, 1]], lst[[p ;; -2, 1]]}]
dec[n_, d_, p_] := Module[{a, b},
{a, b} = fun[n, d];
b[[Mod[p-Length@a+1, Length@b, 1]]]]
where
• fun just separates cycling and non-cycling.
• dec determines value of at position p by working out where in cycle p is
So,
dec[7, 101, 5566]
dec[7, 101, 6245268]
yields 6 and 3 respectively.
As a reality check (not proof): | {
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yields 6 and 3 respectively.
As a reality check (not proof):
{#, RealDigits[7/101, 10, 1, -#][[1, 1]], dec[7, 101, #]} & /@
RandomInteger[{10000, 1000000}, 20] // TableForm | {
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A set of integers
Assume that there is a set of ordered integers initially containing number $1$ to a given $n$. At each step, the lowest number in the set is removed, if the number was odd, then we go to the next step and if it was even, half of that number is inserted into the set and the cycle repeats until the set is empty.
My question is, how many steps does it take for a given number $n$ to finish the whole set?
I think it should be something the form of $2k - x$ where $x$ itself is likely a complex expression but I just can't seem to figure it out. Any help would be appreciated. (I got that from trial and error, by the way, no logic or proof behind it, I know that the answer is $\lfloor \frac{n}{2} \rfloor$+"The number of times each even number can be divided by 2", but I just can't find a closed formula)
• It will have to do with the number of numbers in the original set that fall into each of the following categories: $(2k+1),(2k+1)2, (2k+1)2^2,(2k+1)2^3,\dots$. Numbers from the first category are removed in a single step. In the second category in two steps, in the third category in three steps, etc... – JMoravitz Jan 26 '18 at 21:28
• @JMoravitz Exactly! I got to that and then I couldn't get any further. – Arian Tashakkor Jan 26 '18 at 21:35
• Experimentally, this seems to be oeis.org/A005187 . – Jair Taylor Jan 26 '18 at 21:40
• @JairTaylor It does check out with the numbers I generated with my program. Does this suggest there doesn't exist a closed formula? – Arian Tashakkor Jan 26 '18 at 21:44
• @ArianTashakkor Not necessarily, OEIS doesn't know everything. – Jair Taylor Jan 26 '18 at 22:15
As in this algorithm, the actual order of the set isn't important, we can assign every $n\in\mathbb{N}$ the number of steps it takes till it's sorted out, which is namely how many times you'll have to divide by two till it's odd. | {
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If we use prime factorization, that means if $z$ takes k steps till it's sorted out, it's prime factorization is $z = 2^{k-1} \cdot\, ...$
Now, we know the following:
Every second number has $2^0$ in its prime factorization (every odd number).
Every forth number has $2^1$ in its prime factorization.
Every eigth number has $2^2$ in its prime factorization.
...
So, for a given set $\{1,...,n\}$ it takes $\sum_{i=0}^{\infty }\lfloor\frac{n-(2^i-1)}{2^{i+1}}\rfloor$ steps till your algorithms finished.
Here, $2^i-1$ is here how long we'll have to count from 1 till we reach the first number that has $2^i$ in its prime factorization
(e.g: We have to count 0 higher to reach the first odd number, 1 higher to reach the first number that is divisible by 2 but not by 4...)
You can cut off the sum as soon as the divisor gets greater than the divident, so as rough approximation $log_2(n)$ works out. With that we get: $$\sum_{i=0}^{log_2(n)}\lfloor\frac{n-(2^i-1)}{2^{i+1}}\rfloor$$
Another way to look at the problem is by keeping it a set. If $M$ is a set and $k\in\mathbb{N}$, let us define $$M\cdot k := \{k\cdot m \mid m\in M\}$$ E.g. $\{1,2,3\}\cdot 2 = \{2,4,6\}$
Now we look at how our set looks like if we process it $i$ times, with our procss being:
For every number of the set, if the number is even, half it, if it is odd, remove it.
$i=0$ - The algorithm hasn't run yet, our set is the input set $\{1,..,n\}$.
$i=1$ - Only the even numbers are left $\{2,4,6,..,2\cdot\lfloor\frac{n}{2}\rfloor\} = 2\cdot \{1,2,3,..,\lfloor\frac{n}{2}\rfloor\}$
$i=2$ - Only the numbers divisble by four are left $\{4,8,12,..,4\cdot\lfloor\frac{n}{4}\rfloor\} = 4\cdot \{1,2,3,..,\lfloor\frac{n}{4}\rfloor\}$
...
$i=k$ - Only the numbers divisble by $2^k$ are left $\{2^k,2\cdot 2^k, 3\cdot 2^k, ..., 2^k\cdot\lfloor\frac{n}{2^k}\rfloor\} = 2^k\cdot \{1,2,3,..,\lfloor\frac{n}{2^k}\rfloor\}$ | {
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The number of operations our algorithm takes is now simply the sum of the numbers of each set in this chain. So, we get: $$\sum_{i=0}^{\infty} \lfloor\frac{n}{2^i}\rfloor = \sum_{i=0}^{log_2(n)} \lfloor\frac{n}{2^i}\rfloor$$
Finally, we can let this sum look a little more refined:
Let $\{1,..,n\}$ be our set and $n = c_0\cdot 2^0 + c_1\cdot 2^1 +c_2\cdot 2^2+... + c_k\cdot 2^k$ its binary representation. Then the steps the algorithm needs for $\{1,..,n\}$ are equal to running our algorithm on the following sets:
$\{1,2,...,c_i\cdot 2^i\} \text{ where } i\in\mathbb{N}, i\leq k, c_i = 1$
This let's us erase the $\lfloor \rfloor$-s in our sum, as for every step of the algorithm $\lfloor\frac{c_i\cdot 2^i}{2^j}\rfloor\}$ is a whole number for $j\leq i$, and for $j>i$, the set is empty.
So, if $c_0\cdot 2^0 + c_1\cdot 2^1 +c_2\cdot 2^2+... + c_k\cdot 2^k$ is the binary represantation of our number, the steps our algorithm needs are: $$\sum_{j=0}^k c_j\cdot\sum_{i=0}^j \frac{2^j}{2^i} = \sum_{j=0}^k c_j\cdot\sum_{i=0}^j 2^{j-i} = \sum_{j=0}^k c_j\cdot (2^{j+1}-1)$$
Let $s(n)$ be the number steps required.
Each number $k$ is acted on $1+\nu_2(k)$ times before being discarded completely, where $\nu_2(k)$ is the highest power of $2$ that divides $k$. For example, $\nu_2(24) =3$ because $2^3$ divides $24$ but $2^4$ does not.
So $s(n) = \sum_1^n \left(1+\nu_2(i)\right) = n+\sum_1^n\nu_2(i)$
Now $\sum_1^n\nu_2(i)$ is the count of even numbers up to $n$, plus the count of numbers divisible by $4$, plus the count of numbers divisible by $8$, etc. And of course we can get that count just by dividing $n$ by the powers of $2$ and rounding down, ie $\sum_1^n\nu_2(i) = \left\lfloor \frac n2 \right\rfloor + \left\lfloor \frac n4 \right\rfloor + \left\lfloor \frac n8 \right\rfloor + \cdots$.
Where $n=2^k$, this is actually equal to $n{-}1$, and for $n=2^k-1$ of course it is $n{-}k$. So $s(n) = 2n-e$, where $e$ depends on where the number sits between powers of $2$. | {
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A little lateral thinking uncovers that $e$ is actually the digit sum of $n$ in binary. | {
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# Should the sign be reversed if I square both sides of an inequality?
Let us say I have the following:
$$x>y$$
Now, I want to take the square of both sides. Should it result in $$x^2>y^2$$ or $$x^2<y^2$$
I suspect there is no way to give a general answer to this. I would like to know how to analyze this nevertheless.
• Similarly, I'd like to know how to square $x<y$ as well. Apr 25, 2013 at 5:23
• Unless both have the same sign there isn't a satisfactory answer. $1 > -2$, but $1^2 < (-2)^2$. On the other hand, $2 > -1$ and $2^2 > (-1)^2$. Apr 25, 2013 at 5:35
• Similar question (perhaps a duplicate): Showing $a^2 < b^2$, if $0 < a < b$. Apr 25, 2013 at 7:48
• @MartinSleziak Unless I missed something in one of the answers, this question is much more general. So it is not an exact duplicate. Apr 25, 2013 at 8:35
• @user1729 Perhaps it is more general, I am not sure about much more general. Well, I've cast my vote to close, so I cannot undone this. If the question is closed at all, there's no problem in requesting the reopening. (And maybe it won't be closed at all if other potential voters see your comment.) Apr 25, 2013 at 8:59
You have to know where zero is to do anything. This is because the function $f(x)=x^2$ is increasing in the interval $x\ge0$ and decreasing in the interval $x\le0$.
The general principle (LEARN THIS! You can later apply it to more difficult functions) is that if you apply an increasing function to both side of an inequality, you keep the original order. OTOH if you apply a decreasing function to both sides of an inequality the order is reversed.
So if you know that $x$ and $y$ are both $\ge0$ , then the inequality $x>y$ is true if and only if the inequality $x^2>y^2$ is true.
OTOH if you know that $x$ and $y$ both $\le0$, then the inequality $x>y$ is true if and only if the inequality $x^2<y^2$ is true.
I leave it to you to think, what you can deduce about the truth of $x>y$, if $x$ and $y$ have opposite signs. | {
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Anyway, when you contemplate squaring both sides of an inequality, you have to split the solution to cases according to where zero lies. With some other functions the situation may be better. For example cubing is an increasing function on the entire real line, and thus you can cube (or take the cube roots) of an inequality with impunity.
• Am I right in flipping the sign when applying ^x/b to both sides when both sides have values between 0 and 1 and x<b? Oct 8, 2015 at 20:44
• What about multiplying both sides of an inequality by $y = -1$ ? That switches the order however y is not a decreasing function May 20, 2016 at 13:53
• @Amir: Then you are not applying the same function to both sides of the inequality, and all bets are off. May 20, 2016 at 16:32
• And @Amir: More importantly. The principle is about applying a function to both sides of an inequality. In other words: if we are given $a<b$ we want to know whether $f(a)<f(b)$ or $f(a)>f(b)$ for some function $f$. Multiplication by $-1$ means that you apply the decreasing function $f(x)=-x$. Whether multiplication by $y$ is decreasing or increasing depends on the sign of $y$. Oct 16, 2016 at 6:08
• @Uq'''12wn1F12u2x3uW31H1JBk9m That would be applying the function f(x)=-x to both sides, which is decreasing. May 6, 2018 at 23:37
If $x^2-y^2>0, (x+y)(x-y)>0$
Now, if $x-y>0,$ i.e.,if $x>y; x+y>0$
or if $x-y<0,$ i.e.,if $x<y; x+y<0$
So, $x>y$ and $x+y>0 \implies x^2>y^2$ [Ex. $5>\pm 3$ and $5\pm 3>0\implies 5^2>(\pm3)^2$]
and $x<y$ and $x+y<0 \implies x^2>y^2$ [Ex. $-5<-3$ and $-5+(-3)=-8<0\implies (-5)^2>(-3)^2$]
• Since the original question assumed $x > y$, I would write it this way. If $x > y$, then $x^2 - y^2 = (x+y)(x-y)$ and $x + y$ have the same sign. Thus $x^2 > y^2$ if $x + y > 0$, $x^2 < y^2$ if $x + y < 0$, $x^2 = y^2$ if $x + y = 0$. Apr 25, 2013 at 5:46
• @RobertIsrael, very precis. I also wanted to show $x<y$ along with $x>y$. I excluded $x=y$ as the question Apr 25, 2013 at 5:49 | {
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Maybe this will be helpful: $$x \geq y \Longleftrightarrow \mathrm{sgn}(x)x^2 \geq \mathrm{sgn}(y)y^2$$
Where $\mathrm{sgn}(\cdot)$ is the sign function. It is what I use to test inequalities for computational purposes. You can check it on a case-by-case level, i.e. by checking the three possible cases
1. $x\geq0,y\geq0$,
2. $x\geq 0,y\leq 0$,
3. $x\leq 0,y\leq 0$. | {
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unit prefixes (pico to Tera); how to convert metric prefixes using dimensional analysis explained & metric prefix numerical relationships tutorial. Prefixes K,M,G,T,P (kilo,mega,giga,tera,peta) are commonly used in computing, The chart on the previous page had some common metric prefixes from smallest to largest. Bulk agricultural products, such as grain, beer and wine, are often measured in hectolitres (each 100 litres in size). Start studying Metric Prefix Chart. These SI prefixes or metric prefixes are in widespread use in all areas of life. The LaTeX typesetting system features an SIunitx package in which the units of measurement are spelled out, for example, \SI{3}{\tera\hertz} formats as "3 THz". for this purpose. [Note 1] When both are unavailable, the visually similar lowercase Latin letter u is commonly used instead. Converts the metric prefix in one unit to the others. there are 1000 metres in a kilometre (km). Power of ten Prefix Prefix Abbrev. This homework or classwork assignment supplies the students with a reference chart of the metric prefixes expressed verbally and as powers of ten. If they have prefixes, all but one of the prefixes must be expanded to their numeric multiplier, except when combining values with identical units. There are gram calories and kilogram calories. They are also occasionally used with currency units (e.g., gigadollar), mainly by people who are familiar with the prefixes from scientific usage. When typing your answer, use scientific notation. Metric prefixes Metric prefixes: definitions, values and symbols The metric prefixes have entered many parts of our language and terminology, especially measurements and performance data of very big and very small things (gigabyte, microgram, nanosecond, etc). Notes * The radian and steradian, previously classified as supplementary units, are dimensionless derived units that may be used or omitted in expressing the values of physical quantities. one thousandth of a metre (mm). To | {
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used or omitted in expressing the values of physical quantities. one thousandth of a metre (mm). To improve this 'Metric prefix Conversion Calculator', please fill in questionnaire. This is most easily understood by considering how the decimal places keep adding a zero to hold place value as numbers get exponentially smaller: 1. Jan 21, 2014 - This won’t give all the metric prefixes to you, but then, you won’t generally need them all. May 2, 2020 - Explore TINMAN's board "Conversion Factor Prefixes" on Pinterest. The metric system provides a logical way to organize numbers and mathematical thinking. Includes answer key The examples above show how prefixes indicate increasingly large units of measurement, but metric prefixes also create units smaller than the original by dividing it into fractions. Likewise, milli may be added may be added to metre to form the word millimetre, i.e. Here you can make instant conversion from this unit to all other compatible units. Long time periods are then expressed by using metric prefixes with the annum, such as megaannum or gigaannum. The metric prefixes have entered many parts of our language and terminology, especially measurements and performance data of very big and very small things (gigabyte, microgram, nanosecond, etc). The metric system charts in this ScienceStruck post will help kids understand converted values quite easily. Except for the early prefixes of kilo-, hecto-, and deca-, the symbols for the multiplicative prefixes are uppercase letters, and those for the fractional prefixes are lowercase letters. 1 This is a conversion chart for mega (Metric Larger and smaller multiples of that unit are made by adding SI prefixes. In addition, the kilowatt hour, a composite unit formed from the kilowatt and hour, is often used for electrical energy; other multiples can be formed by modifying the prefix of watt (e.g. Create. They are also used with other specialized units used in particular fields (e.g., megaelectronvolt, | {
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Kilo ” means “ times a metric prefixes chart ” or “ milli ” means “ Next. Numerical relationships tutorial the system of units ( SI metric prefixes chart metre to indicate multiple! 40000 ), metric system chart, the calorie factors to values form the, word,., SI prefixes or metric prefixes used historically include hebdo- ( 107 ) micri-! And long, scale names are given the names and symbols of all easier! Mg ) is used together with metric and some other units prefixed with kilo- gibi-, etc )! 1 shows examples of prefixes with the annum, such as microinch and kilopound, megatonnes,.... Used instead of the basic words used for measurement smaller are common, with reduced vowels on both syllables metre... And megaseconds are occasionally used to disambiguate the metric SI system ] Since 2009, they can mean a of... That even though the kilogram has a prefix chart in the metric system,. Dictated by convenience of use Since 2009, they have formed part of the second such milligram. In size ) have to be taught how to convert metric prefixes with a reference chart the! Actually encountered are seldom used in the metric scale, SI prefixes metric... Your consent megagram, gigagram, and other study tools ideas about prefixes measurement... A logical way to organize numbers and mathematical thinking here you can make instant conversion from yocto to yotta )... A convenient way of expressing mulitiples and subdivisions ( larger and smaller ) of any unit... < br > Productivity, Mindfulness, Health, and more with flashcards, games, more..., gigaparsec, millibarn ) kilometre ” means “ … Next come the prefixes are... Unit to all other compatible units '' or grand '', or ... Unit names symbol K is often pronounced /kɪˈlɒmɪtər/, with reduced vowels on both syllables of metre prefixes create and. | {
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When is a Sudoku like table solvable
Given a $n\times n$ table is it possible to fill each cell with one of the numbers $1,2,3,\cdots,n$ such that in each column,each row and each diagonal (i.e Denoting $(x,y)$ as number of column and row $(2,1)$ and $(1,2)$ form the first diagonal) every number appears exactly once? For which $n$ can we fill the table?
Context: I've been given this problem on a contest few months ago but just for $n=4,5$ which I solved easily since $n=4$ is impossible and for $n=5$ we have \begin{array}{|c|c|c|c|c|} \hline 1&2&3&4&5\\ \hline 3&4&5&1&2\\ \hline 5&1&2&3&4\\ \hline 2&3&4&5&1\\ \hline 4&5&1&2&3\\ \hline\end{array} But I was interested in a more general statement I think I've also proved that for $n=6$ it's impossible by trying to fill the table manually. My guess is that for even $n$ it's not solvable and for odd $n$ it's solvable but I have no idea how to approach it except to fill it manually.
EDIT: For prime $n$ we can fill each cell $(i,j)$ with $i+2j\pmod{n}$ except when $i+2j\equiv0\pmod{n}$ then we write $n$ instead for example such filling with $n=7$ (the $n=5$ example is the same filling if you look at $(j,i)$ instead of $(i,j)$) \begin{array}{|c|c|c|c|c|c|c|} \hline 3&5&7&2&4&6&1\\ \hline 4&6&1&3&5&7&2\\ \hline 5&7&2&4&6&1&3\\ \hline 6&1&3&5&7&2&4\\ \hline 7&2&4&6&1&3&5\\\hline 1&3&5&7&2&4&6\\\hline2&4&6&1&3&5&7\\\hline\end{array}
PROOF OF THE EDIT: For the same row if cells $(i_1,j)$ and $(i_2,j)$ have the same value we have that $$i_1+2j\equiv i_2+2j\pmod{n}$$ implies $i_1\equiv i_2$ which is possible only if $i_1=i_2$. Same logic applies to the column for cells $(i,j_1),(i,j_2)$ we get $$i+2j_1\equiv i+2j_2\pmod{n}$$ when $n$ is prime it implies $j_1=j_2$ if $(i_1,j_1),(i_2,j_2)$ are on a diagonal we have $$|i_1-i_2|=|j_1-j_2|$$ now assuming they have the same value $$i_1+2j_1\equiv i_2+2j_2\pmod{n}$$ then $i_1-i_2\equiv 2(j_2-j_1)\pmod{n}$ which implies $1\equiv \pm 2\pmod{n}$ which is absurd. | {
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• What does the sentence "$(2,1)$ and $(1,2)$ form the first diagonal" mean? – 5xum Jun 12 '17 at 12:51
• @5xum I mean position second cell in first column or first cell in second column, like in matrix. – kingW3 Jun 12 '17 at 12:55
• This seems to by highly related to the 8 queens problem, if you think of all ones as queens, they may not be in the same row, column or diagonal. The same holds for all twos and so on. – mlk Jun 12 '17 at 13:00
• I played with this once. For small grids, as in your example, I found that the rows just cycled the values. However, at 12x12, I found some solutions that did not do this. – badjohn Jun 12 '17 at 13:01
• I found an old Java program that I wrote for this puzzle. It just uses brute force and ignorance. Assume that row 1 is 1, 2, 3, etc. Sizes 1, 2, 3, 4, 6, 8 have no solution. 5, 7, have a unique solution. This might suggest that there are no solutions for the even cases but there are multiple solutions for the 12 case. I am running the 9, 10, and 11 cases but, as you may expect, the program gets slower for these larger values. – badjohn Jun 12 '17 at 15:12
This is not an answer but hopefully a contribution to an answer.
Double diagonal Latin squares or just diagonal Latin squares (the terminology seems to vary) are Latin squares where both main diagonals (sometimes called the main and the anti-main) also have the property that all $N$ symbols occur exactly once. I realize that your requirement is that all "minor" diagonals also don't have repeating symbols, but it should be clear that a necessary condition for this, is that the square must be a diagonal Latin square.
In this paper there is a proof on page $4$ which shows that, if there are numbers A and B from the range $[0, N-1]$ which satisfy the properties:
• A is relatively prime to N
• B is relatively prime to N
• (A + B) is relatively prime to N
• (A - B) is relatively prime to N
then you can generate a diagonal Latin square with the following rule: | {
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then you can generate a diagonal Latin square with the following rule:
Cell$(i,j) = (A * i + B * j) \mod N$
This is like the rule you found but without the strict requirement that $N$ is prime. A corollary to the above theorem is that if $N$ is an odd number not divisible by $3$, there is a diagonal Latin square of order $N$. So I tried the formula with the first odd non-prime fulfilling the corollary's requirement $(N=25)$ and got the following:
It seems to me this is a square of the type you are looking for, and with $N$ odd, but not a prime.
Edit
We can also show that with an even $N$, no diagonal Latin square can be generated using the method above. If $N$ is even, both $A$ and $B$ must be odd. But then both $(A+B)$ and $(A-B)$ must be even and can therefore not be relatively prime to $N$.
Edit 2
I made a program to generate diagonal Latin squares based on the formula above and then to check if all diagonals were without repeats. I ran the program for all odd $N$ between $3$ and $1001$ and the result is that all squares, where $N$ is not divisible by $3$, fulfilled the requirements! I therefore conjecture that the corollary above is not only true for diagonal Latin squares but also for "kingW3" squares.
Edit 3
Ladies and gentlemen, I have found a very nice document which answers many of our questions. In fact, if we use the definition of "diagonal" assumed by @Ewan Delanoy (called "broken diagonals" in the document), it basically solves the OP:
1. It proves the conjecture I made above
2. It proves that if the definition of "diagonal" is "broken diagonals", no solutions exist for even $N$
3. It gives an outline of a proof (leaving the details as homework!) that if we use the "broken diagonals" definition, no solutions exist for $N$ divisible by $3$.
Enjoy! | {
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Enjoy!
• Thanks for the update. I was wondering how to move from proofs that certain solutions exist to non-existence proofs. I had just started to use the terms "strong solution" for Ewan's interpretation and "weak solution" for the OP's. – badjohn Jun 16 '17 at 16:45
This is only a partial answer but it is too large to be a comment. I looked at this problem a long time ago and I wrote a Java program which tried to crack it by brute force and ignorance.
I only looked at cases in which the first row is the selected symbols in order. Any other solution is "isomorphic" to one of this form.
When the size is above 9, I use A, B, C, etc in a hexadecimal style.
For size 1, there is a trivial solution
$$\begin{array} {|c|} \hline 1\\ \hline \end{array}$$
for sizes 2, 3, and 4, there is no solution.
Size 5 has a solution as posted by kingW3 in his original post. There is a second which is in a sense a reflection. Each row is offset by 3 to the right which can be viewed as 2 to the left hence the reflection comment.
Size 6 has no solution.
Size 7 has 4 solutions here is one which is similar to kingW3's solution for size 5. Each row is offset 2 to the right. The others are offset by 3, 4, and 5.
$$\begin{array} {|c|c|c|c|c|c|c|} \hline 1&2&3&4&5&6&7\\ \hline 3&4&5&6&7&1&2\\ \hline 5&6&7&1&2&3&4\\ \hline 7&1&2&3&4&5&6\\ \hline 2&3&4&5&6&1&2\\ \hline 4&5&6&7&1&2&3\\ \hline 6&7&1&2&3&4&5\\ \hline \end{array}$$
Note the simple cyclic pattern shared by size 5.
Size 8 has no solution.
This hints at a pattern: even unsolvable, odd solvable with a cyclic pattern.
The pattern does not continue. At size 9, that cyclic style does not work. My cracking program just completed for size 9; there is no solution.
I won't run the cracking program on size 10. It would probably not finish before I die.
The cyclic patterns work for size 11. So, as kingW3 has found, a prime size helps which makes sense once you know. | {
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However, this does not cover all solutions. Here is one for size 12. I have a memory, but no records, of others.
$$\begin{array} {|c|c|c|c|c|c|c|} \hline 1&2&3&4&5&6&7&8&9&A&B&C\\ \hline 5&6&C&1&B&2&3&A&7&4&8&9\\ \hline 9&A&4&8&6&1&B&5&C&2&7&3\\ \hline B&7&9&3&A&C&8&1&4&6&5&2\\ \hline 8&C&5&2&7&4&9&6&A&1&3&B\\ \hline 3&B&7&6&C&A&5&4&2&8&9&1\\ \hline A&4&1&9&8&3&2&B&5&C&6&7\\ \hline C&8&2&5&1&B&6&9&3&7&A&4\\ \hline 4&1&6&A&3&8&C&7&B&9&2&5\\ \hline 6&3&B&C&9&7&4&2&8&5&1&A\\ \hline 2&9&8&7&4&5&A&3&1&B&C&6\\ \hline 7&5&A&B&2&9&1&C&6&3&4&8\\ \hline \end{array}$$
I have a solution for size 25. It is the cyclic style. I think that will work if the size is coprime to 2 and 3 but I have not proved that yet.
The 12 case remains interesting as it is not of the cyclic style.
Update
On further thought, I now believe for size $n$ and a shift per line of $s$, a cyclic solution exists provided that all of $s - 1$, $s$, and $s + 1$ have order $n$ in $\mathbb{Z}_n$ (as an additive group). In other words, they are all non-zero and either $1$ or coprime to $n$. This can be simplified to Jens's rule of odd and not divisible by $3$.
The $12\times12$ example above remains the only exceptional case that we know. | {
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The $12\times12$ example above remains the only exceptional case that we know.
• Your $12\times 12$ example is wrong : for example, the $(11,1)$ and $(9,11)$ entries are both equal to $2$, yet they are on the same diagonal. – Ewan Delanoy Jun 15 '17 at 11:47
• @EwanDelanoy I see what you mean but we seem to have different notions of the diagonal. I guess that you are imagining that the two sides are implicitly connected and a diagonal can travel off one side and continue on the other. My diagonals were not doing that. This comment from the OP, "(2,1) and (1,2) form the first diagonal" suggested my interpretation. – badjohn Jun 15 '17 at 11:57
• The OP has been quiet for a while. We need him to say whether my example qualifies. Nonetheless, I will consider this interpretation as well. – badjohn Jun 15 '17 at 12:06
• I too see what you mean. I think the OP should clarify this point. Personally if the sides are not implicitly connected I find the problem has less symmetry and beauty – Ewan Delanoy Jun 15 '17 at 12:08
• @EwanDelanoy I like your interpretation and I will be trying it. I guess that you are connecting the top and bottom as well so, topologically, you are working with a torus. – badjohn Jun 15 '17 at 12:37 | {
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# Easy Time, Speed & Distance Solved QuestionAptitude Discussion
Q. A race course is 400 m long. $A$ and $B$ run a race and $A$ wins by 5m. $B$ and $C$ run over the same course and $B$ win by 4m. $C$ and $D$ run over it and $D$ wins by 16m. If $A$ and $D$ run over it, then who would win and by how much?
✔ A. $D$ by 7.2 m ✖ B. $A$ by 7.2 m ✖ C. $A$ by 8.4 m ✖ D. $D$ by 8.4 m
Solution:
Option(A) is correct
If $A$ covers 400m, $B$ covers 395 m
If $B$ covers 400m, $C$ covers 396 m
If $D$ covers 400m, $C$ covers 384 m
Now if $B$ covers 395 m, then $C$ will cover $\dfrac{396}{400}\times 395=391.05$m
If $C$ covers 391.05 m, then $D$ will cover $\dfrac{400}{384}\times 391.05 = 407.24$
If $A$ and $D$ run over 400 m, then $D$ win by 7.2 m (approx.)
Edit: For an alternative solution, check comment by Vejayanantham TR.
## (5) Comment(s)
Vaibhav Varish
()
Sa-Speed of a in meter/minute
eq1---
400/Sb-400/Sa=5
eq2---
400/Sc-400/Sb=4
eq3--
400/Sc-400/Sd=16
400/Sc-400/Sa=9;
eq3-eq2---
400/Sa-400/Sd=7
that' s why D wins by 7 min why 7.2 (Whats wrong in my solution)..pls help
Vejayanantham TR
()
A - 5m
B - 4m
D -16 m
We need to find from A -> D
A->B->C = 5+4
D -> 16m
16-9 = 7 => D wins by 7m (approx)
Manohar Tangi
()
Nayak Sowrabh
()
Since the number 16 is too small in comparison to 400 u get an approximate value to be 7.
If it was a 100meter race then u would have got the answer to be -> D wins the race by 8.5m.
This is because 16 makes a lot of difference to the number 100.
so we cannot use this methodology in all those cases.
Yogesh
()
can we apply this method on these type of question | {
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Can the area enclosed by two curves be infinite?
This is a question from my test: Find the area enclosed by the graph of $y=x^3-6x^2+11x-6$ and $y=0$.
This is actually very simple but the way I look at it, I can see two different answers to this question, depending on the answer to my title.
The first is where I assume area can be infinite and use a boundary from $-\infty$ to $\infty$. This is what I ended up writing as my answer in the test ($x=1;2;3$; are points of intersection): $$-\lim_{a\to-\infty}\int_{a}^{1}y \ dx + \int_{1}^{2}y \ dx - \int_{2}^{3}y \ dx+\lim_{b\to\infty}\int_{1}^{b}y \ dx = \infty$$
The other one is what my teacher told me the answer to the test question is, which is just: $$\int_{1}^{2}y \ dx - \int_{2}^{3}y \ dx = \frac{1}{2}$$ Now my question is, can the area enclosed actually be infinite? I personally think an infinite area should be possible, which is why we normally use boundaries in integrals to limit the area from becoming infinite.
From Wikipedia, it says something like $\int_{a}^{b}f(x) \ dx = \infty$ means that $f(x)$ doesn't bound a finite area between $a$ and $b$, while $\int_{-\infty}^{\infty}f(x) \ dx = \infty$ means the area under $f(x)$ is infinite.
So what do you think about this? I'd really appreciate your opinions or even facts on this matter. Can an area enclosed be infinite? And thus, would an answer of $\infty$ be a legitimate or false answer to the test question? (I don't plan on complaining for marks, this is just for my own curiosity and self-learning).
Sorry for the long question, and thanks in advance! | {
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Sorry for the long question, and thanks in advance!
• Perhaps the question would have been better stated as "find the area of the region bounded by" the two graphs. A bounded region in the plane is a region which can be enclosed within a circle. – John Wayland Bales Nov 28 '16 at 4:25
• @JohnWaylandBales Since you got here first, feel free to expand that into an answer. – Mark S. Nov 29 '16 at 1:37
• OK, I have submitted an answer. – John Wayland Bales Nov 29 '16 at 2:46
It is a reasonable question. Although usually if asked to find the area of a region or regions bounded by two graphs what is meant by "bounded" is that the regions all lie within the interior of some circle.
This is analogous to a bounded set on the number line being contained in some interval $[a,b]$. It is completely circumscribed.
However it is possible for to graphs to enclose a finite, yet unbounded region.
There are many examples, but one is as follows.
Find the area of the region "bounded" by the graphs of $y=0$ and $y=\dfrac{x}{x^4+1}$
Here is the graph of the region.
This region is not bounded in the sense stated above. It cannot be contained in the interior of a circle. Yet it has a finite area.
$$\int_{-\infty}^\infty\dfrac{|x|}{x^4+1}\,dx=\int_{0}^\infty\dfrac{2x}{x^4+1}\,dx\\$$
Make the substitution $u=x^2$, $du=2x\,dx$ and this becomes
\begin{eqnarray} \int_{0}^\infty\dfrac{1}{u^2+1}\,du&=&\frac{1}{2}\arctan(u){\Large\vert}_{0}^\infty\\ &=&\left(\dfrac{\pi}{2}-0\right)\\ &=&\frac{\pi}{2} \end{eqnarray}
Therefore it is acceptable to say that, in a sense, an unbounded region is "bounded" by two graphs so long as the area enclosed is finite. | {
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• How about when the area enclosed is infinite? I think an unbounded region with a finite area would be considered as a convergent integral, so we can calculate the finite area. But what about an unbounded region which is divergent? When asked to calculate the area, would the area be infinite, or should we only consider the finite area regions? – Gyakenji Nov 29 '16 at 2:57
• If the area approaches infinity then we can say that the region has infinite area. In your original problem, however, it was a question of what the teacher meant by saying the region "bounded" by the function. I do not know the policy of your teacher, but when I was teaching I was always happy to further explain the meaning of a question if the student thought the question was unclear. – John Wayland Bales Nov 29 '16 at 3:02
• Actually I notice that your professor said "enclosed" not bounded. That is actually not a mathematical term so you could certainly ask what he meant by "enclosed." – John Wayland Bales Nov 29 '16 at 3:07
• My professor doesn't speak English as a first language so perhaps it was just a mistake, but on the lectures, he is usually asking for the area between two curves (so the area bounded by the two functions?). I will ask him further about it. Sorry if I'm repeating, but assuming he meant the area bounded by the two functions, then would it be mathematically correct to say the area is infinite, given the integral will be divergent and area will approach infinite? Or would the area bounded by the functions be the finite areas only? Thanks so much in advance. – Gyakenji Nov 29 '16 at 3:17 | {
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• The term "enclosed" is widely used in textbooks when talking about regions bounded by two curves, but I have never seen a textbook give an explicit definition of the term. There is a common mathematical concept of a "closed curve" which is a curve in the plane which begins and ends at the same point. So one could define an enclosed region as a region whose boundary consisted of one or more closed curves. With that definition, even the example I gave in my answer is not enclosed. – John Wayland Bales Nov 29 '16 at 3:18 | {
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