Calculus – Tangent of a function in the 1st quadrant forms a triangle. Find the area.

So, I have this specific question, which is based on calculus.
Here is the question below:

Consider the graph of f (x) = exp (-x) for x> = 0. The tangent to the graph of f (x) at x = a and intersects the x-axis at point A and the y-axis at Point B. Determine the area of ​​the triangle AOB with respect to a.

Therefore, the coordinate on the x-axis (A, 0) and the coordinate on the y-axis is (0, B).

I worked on that.
The slope of the tangent would be: m = -exp (-a)
So my equation would be the tangent: y = [-exp(-a)]x + B,
Next I insert the point: (A, 0)
Then you get the relationship B = A * exp (-a), so the full equation of the tangent line is:
y = -exp (-a)[x-A]

Now the surface is a right triangle with vertices AOB,
This area is:
Area = AB / 2.
Insert the relationship B = A
exp (-a).
Then area = [(A^2)*exp(-a)]/ 2

BUT my formula depends on the x-axis coordinate A.
SO not sure if this A can be replaced by being related to & # 39; a & # 39 ;, the tangent point.

Hope someone can contribute to this.

Research on user behavior in social media in the area of ​​HCI

I know that HCI is a diversified area, especially for graduates. I am planning to find college graduates and looking for a good research topic in the HCI area. However, since everyone is talking about the design aspect of HCI, I was wondering if social media and user behavior are being investigated. Social media could be related to HCI?

List manipulation – Find the 3D area with the origin bounded by certain levels

I am writing a code to generate the Wigner-Seitz cell of the reciprocal lattice for a given set of lattice translation vectors. For example, consider the Body Centered Cubic (BCC) grid, whose base translation vectors are given by

a1 = {-1, 1, 1} / 2;
a2 = {1, -1, 1} / 2;
a3 = {1, 1, -1} / 2;

The reciprocal basis vectors are then defined according to

d = 2 Pi;
v = a1. (a2 [Cross]a3);
b1 = d / v (a2 [Cross]a3);
b2 = d / v (a3 [Cross]a1);
b3 = d / v (a1 [Cross]a2);

The reciprocal lattice is then defined by the set of reciprocal lattice vectors, the set of all linear combinations of integer multiples of reciprocal base vectors, i.

$$ vec {G} = n_1 vec {b} _1 + n_2 vec {b} _2 + n_3 vec {b} _3, qquad n_i in mathbb {Z} $$

The Wigner-Seitz cell (in this case the first Brillouin zone) is defined as the region containing the origin delimited by the vertical bisector levels of the reciprocal lattice vectors. We can generally accomplish this by considering only the first, second, and perhaps third, closest mutual grid points to the origin. For example, in the case of BCC, the following vectors suffice:

recipvecs =
    Table[n1 b1 + n2 b2 + n3 b3, {n1, -1, 1}, {n2, -1, 1}, {n3, -1, 1}], 2].
standard[#] <= 2 d &];

Question: How can I construct the Wigner-Seitz cell with these vectors?

For example, one possibility is to construct the equations for all levels

plan = ({x, y, z} - (# / 2)). # == 0 & / @ reciplattice

(Note that there is a redundancy for the origin that only exists Truecan be removed). Now it is about rewriting each of these equations as inequalities, so that the half space defined by the inequality contains the origin. I do not think that would be too difficult, but not every one of the equations can be solved for one of the coordinates, eg. we can not solve every equation for $ z $, to like

To solve[#, z] & / @ Planes

Some of the equations have to be solved $ x $ or $ y $ before they are transformed into inequalities. I think I could find a brute force solution, but I hope there is something more elegant.

Finally, I want to get the inequalities that define the region so that I can imagine it RegionPlot3D and use it Choose Points from a network.

To update:

So far I have done a simple one If construct to solve the equations for both $ z $. $ x $, or $ y $.

sols = flattening[
If[Sz=release[Sz=Solve[sz=Löse[sz=Solve[#, z]; sz! = {}, sz,
If[sx = Solve[#, x]; sx! = {}, sx, solve[#, y]]]& / @ Airplanes]

This prints out the following list of rules

{z -> -1, z -> -1 - y, z -> -1 + x, y -> -1, x -> 1 + y, x -> 1, z -> -1 - x, z -> -1 + y, x -> -1 - y, x -> 1 - y, z -> 1 + y, z -> 1 - x, x -> -1, x -> -1 + y, y -> 1, z -> 1 + x, z -> 1 - y, z -> 1}

Now I just have to convert each of these into $ LHS <RHS $ and $ LHS> RHS $ and test, which contains the origin, but I do not yet know how to achieve that.

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Taylor-like expansion for a holomorphic function in the not easily connected area

Accept $ f $ is a holomorphic function in an easily connected open set $ U $and we know it's a Taylor expansion at some point $ p in U $, We can then find a holomorphic map $ g $ from $ U $ to the device disk that sends $ p $ to 0 and get Taylor's series of $ f (g ^ {- 1} (z)) $ Near $ z = 0 $, Mapping back to $ U $ this then gives an extension of $ f $ in a series of functions that converges in all $ U $, It is important that if only we knew first $ N $ Terms, if the Taylor expansion, we can still get first $ N $ Terms of the final extension.

Can we find a similar extension for? $ f $ if $ U $ is not connected easily? That is writing $ f (z) = sum_n a_n h_n (z) $, converging in all $ U $With $ h_n $ Single rated in $ U $, and $ a_n $ a linear combination of first $ n $ regarding the Taylor extension of $ f $ at the $ p $? Of course the functions $ h_n $ should not depend on the choice $ f $,

Javascript – Replace characters in a specific area

Previous question:

I have to replace characters in a certain border, in this example it would be preserved by the characters 0-4, then it would be the other way round, -4

var transaction = & # 39; 1234567890987651907 & # 39 ;;
// result 1234 **** 1907


var str = "1234567895555556541";
str.replace (/ ^ ( d {0,4}) ( d {4}) (. *) / gi, "$ 1 **** $ 2");
// Result "1234 **** 6541"

LF Postcode Codes in Trinidad

LF Postcode Codes in Trinidad

+ Answer to thread

  1. LF Postcode Codes in Trinidad

    Where do I find primaries for Trinidad? Codes used for postal services. Thank you in advance.

  2. Thanks for the answers here. I use this site Here we can find every region, every country and every city. Where do you get your data from? Which website I would like to compare. After all, it is better to rely on other resources.

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Complex analysis – Möbius transformation: area outside of two circles, which are imaged on the inside of a circular ring

In my current homework there is the following task:

"The area outside these two circular discs

$ K_1 = {z { in} mathbb {C}: | z- frac {5} {2} | le frac {3} {2} } $, and

$ K_2 = {z { in} mathbb {C}: | z + frac {5} {2} | le frac {3} {2} } $

is to map the inside of a circular ring around zero. Find a suitable conversion. "

The only idea I have so far is to find a transformation in which both circles are mapped onto the aforementioned annulus (which I can not really know), but I do not know if that is so the area outside these two circles is in this new ring … Anyway, I'm completely confused and really do not know how to do that. Any help would be appreciated!

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