Find the area of two functions in wolfram mathematica - function

I am having problems with drawing the area which is between two curves f(X)=-(x-2)^2+4 and g(x)=x/x+1 which says that the area is in the FIRST quadrant.
Then to calculate the area of the area.Here is my code in wolfram mathematica.
f[x_] = -(x - 2)^2 + 4;
g[x_] = x/x + 1;
Plot[f[x] - g[x], {x, , }]
Integrate[f[x]-g[x],{x,,}]
.Thank you

f[x_] := -(x - 2)^2 + 4
g[x_] := x/x + 1
Plot[{f[x], g[x]}, {x, -1 , 5}]
sol = NSolve[f[x] == g[x], x]
{{x -> 0.585786}, {x -> 3.41421}}
{a, b} = x /. sol
{0.585786, 3.41421}
Integrate[f[x] - g[x], {x, a, b}]
3.77124

First I typed:
-(x - 2)^2 + 4 = x/(x+1)
to find out where they meet.
Then:
area inside the curves y = -(x - 2)^2 + 4 and y = x/(x+1) for x from 0 to 3/2 + sqrt(21)/2

The g[x_] := x/x + 1 function looks odd.
Given the precedence rules, x/x will simplify to 1, and x/x + 1 will thus have constant value 2 (i.e. 1+1).
Did you mean
g[x_] := x/(x+1)
instead ?
To display the surface between the two curves, you can use the Filling option for the Plot function :
Plot[{f[x], g[x]}, {x, -1, 5}, Filling -> {1 -> {2}}]
With the modified g[x_] function above this gives
Area between two curves

In:
Clear[f, g, k]
f[x_] := -(x - 2)^2 + 4
g[x_] := x/x + 1
k[x_] := f[x] - g[x]
roots = x /. Solve[f[x] == g[x], {x}];
RegionMeasure[{x, k[x]}, {{x, First[roots], Last[roots]}}]
Out:
1/2 (6 Sqrt[2] + ArcSinh[2 Sqrt[2]])

Related

Plot surface in 3D where f(x,y,z) vanishes

I have a function
f(x,y,z) = 1 + 2xyz - xx - yy - z*z. My range of interest is [-1,1] in any variable.
Obviously, I could not construct a human visible 4D Plot, but I could plot the points in 3D where it vanishes, or fill with red the range where it is positive.
Having forgotten almost everything about matlab and octave, I searched examples and tried:
[x y z] = meshgrid(-1:0.1:1, -1:0.1:1, -1:0.1:1);
coords = [x(:) y(:) z(:)];
V = 1.0 + 2*x.*y.*z -y.*y - z.*z - x.*x;
for p = 1: 1:100
if V(p) >=0
c = 'red';
scatter3(x(:,p),y(:,p), z(:,p), 'c');
end
end
It produces a plot, but it must be bogus, because there are no red dots and 1,1,1 is a solution, which doesn’t show up in the plot. Also the z axis is messed up, showing only negative values.
Please help.
The Plot:
Try this
[x y z] = meshgrid( -1 : 0.1 : 1, -1 : 0.1 : 1, -1 : 0.1 : 1 );
V = 1.0 + 2 * x .* y .* z - y .* y - z .* z - x .* x;
scatter3( x(:), y(:), z(:), 100, V(:), 'filled' );

Why does the fibonacci recursive sequence work?

I am wondering why this Fibonacci recursive function works:
int fibRec(int n)
{
if ((n == 1) || (n == 0))
{
return n;
}
int i = fibRec(n - 1) + fibRec(n - 2);
return i;
}
I understand what the Fibonacci sequence is and I understand what a recursive function does and how this function is working. I'm just having troubles understanding why it works. I know that when you break it down, you are essentially adding a bunch of 0s and 1s, as this image depicts.
fibonacci recursive
But why is it that when I pass a 5 to the function and all the 0 and 1s are added that it will equal the 5th sequence number in the Fibonacci sequence? I've seen this question asked before but never really explained. The responses are all just "because recursion". Yes, I know what a recursive function is and how this one is working. But WHY does this recursive function give you the correct Fibonacci sequence number?
In the Fibonacci sequence the first two numbers are zero and one. Every number after these is the sum of the previous 2 numbers. So the first few numbers are
F(0) ≡ 0
F(1) ≡ 1
F(2) = F(1) + F(0) = 1 + 0 = 1
F(3) = F(2) + F(1) = 1 + 1 = 2
F(4) = F(3) + F(2) = 2 + 1 = 3
F(5) = F(4) + F(3) = 3 + 2 = 5
F(6) = F(5) + F(4) = 5 + 3 = 8
...
F(n) = F(n - 1) + F(n - 2) ∀ n > 1
Therefore when we calculate a Fibonacci number recursively we have to practice the following logical procedure (in pseudo-code out of respect to StackOverflow).
Integer NthFibonacci(Integer n) {
if (n < 0) {
return undefined;
} else if (n < 2) {
return n;
} else {
return NthFibonacci(n - 1) + NthFibonacci(n - 2);
}
}
I'm sure you know all this but I think it will help my explanation to have this part as a reference.
Where the Ones and Zeros Come In
The best way to explain this is probably with an example.
Imagine that, as above, we are trying to recursively calculate F(6). Try following the procedure given above. Remember that we will perform recursion only if n > 1.
First we start with F(6) = F(5) + F(4).
Then we find F(5) = F(4) + F(3).
Then we find F(4) = F(3) + F(2).
Then we find F(3) = F(2) + F(1).
Then we find F(2) = F(1) + F(0).
This is where things start to work out!
We have now gotten F(2) in terms of F(1) ≡ 1 and F(0) ≡ 0 (both of which are known), and so we are able to calculate an actual value instead of performing more recursion.
We can now find F(2) = F(1) + F(0) = 1 + 0 = 1.
NOTICE THE 1 AND 0 Those are what people are talking about when they say the whole thing comes down to ones and zeros. Every time we recurse down to find a base value we will end up finding F(2) = 1 + 0. This leads to more ones and zeros as we move back up our recursion tree being able to calculate higher and higher values, as follows.
F(3) = F(2) + F(1) = (1 + 0) + 1
F(4) = F(3) + F(2) = ((1 + 0) + 1) + (1 + 0)
F(5) = F(4) + F(3) = (((1 + 0) + 1) + (1 + 0)) + ((1 + 0) + 1)
F(6) = F(5) + F(4) = ((((1 + 0) + 1) + (1 + 0)) + ((1 + 0) + 1)) + (((1 + 0) + 1) + (1 + 0))
Now if you add up all the 1's you get a sum of 8, and so F(6) = 8, which is correct!
This is how it works, and this is how it breaks down to ones and zeros.
Remember, recursion works by breaking down the problem till we know what the answer is, and then building it up from there.
What do we know about the fibonacci sequence?
We know that when:
x = 1
and
x = 0
That that is the lowest it goes. That is an important key. Because when x = 0 we are really doing 0 + 0 and when x = 1 we are really doing 0 + 1. Now start at the top.
0,1,1,2,3,5,8,13...
If we are at 13. what is 13? Why simply 5 + 8 right? So That is where
int i = fibRec(n - 1) + fibRec(n - 2);
comes from. Because these are going to branch out lower and lower till we are at a base case for each one.
This is the recursive calling. Because now the method is going to go back to the stack and call fibRec again. You will notice that (n-1) and (n-2) are both added together and set to i. This is so that we don't lose the value. because of the + sign the stack then ends up returning more and more (n-1)s and (n-2)s until we are at the base case. I hope all of this makes sense. Thinking recursively can be very difficult. Here is a a visual representation from top to bottom of what it would look like.
In short. This just keeps adding the previous fibonacci sequences to the current one until it gets to the current loop.

Explain why x == ~(~x + 1) + 1 (two's complement and back!)

As we all know usually negative numbers in memory represents as two's complement numbers like that
from x to ~x + 1
and to get back we don't do the obvious thing like
~([~x + 1] - 1)
but instead we do
~[~x + 1] + 1
can someone explain why does it always work? I think I can proof it with 1-bit, 2-bit, 3-bit numbers and then use Mathematical induction but it doesn't help me understand how exactly that works.
Thanks!
That's the same thing anyway. That is, ~x + 1 == ~(x - 1). But let's put that aside for now.
f(x) = ~x + 1 is its own inverse. Proof:
~(~x + 1) + 1 =
(definition of subtraction: a - b = ~(~a + b))
x - 1 + 1 =
(you know this step)
x
Also, ~x + 1 == ~(x - 1). Why? Well,
~(x - 1) =
(definition of subtraction: a - b = ~(~a + b))
~(~(~x + 1)) =
(remove double negation)
~x + 1
And that (slightly unusual) definition of subtraction, a - b = ~(~a + b)?
~(~a + b) =
(use definition of two's complement, ~x = -x - 1)
-(~a + b) - 1 =
(move the 1)
-(~a + b + 1) =
(use definition of two's complement, ~x = -x - 1)
-(-a + b) =
(you know this step)
a - b
This is because if you increment ~x (assuming no overflow). Then converting it to back to x, you've incremented relative to ~x, but decremented relative to x. Same thing applies vice versa. Assuming your variable x has a specific value, every time you increment it, relative to ~x you'll notice it decrements.
From a programmer's point of view, this is what you'd essentially witness.
Let short int x = 1 (0x0001)
then ~x = 65534 (0xFFFE)
~x + 1 = 65534 + 1 (0xFFFF)
~(~x+1) = 0 (0x0000)
~(~x+1) + 1 = 0 + 1 (0x0001)

Maxima - Substitutions in Equations - let() and letsimp() without effect

My own equation is a bit longer, but the following example shows perfectly where I struggle at the moment.
So far I have been using the let() and letsimp() function
to substitute longer terms in an equation,
but in this example they have no effect:
(%i1) eq: ((2*u+a^2+d) * y+x)/2*a = x;
2
a ((2 u + d + a ) y + x)
(%o1) ------------------------ = x
2
(%i2) let(2*u+a^2+d, %beta);
2
(%o2) 2 u + d + a --> %beta
(%i3) letsimp(eq);
2
a ((2 u + d + a ) y + x)
(%o3) ------------------------ = x
2
What is the preferred way to replace 2*u+a^2+d with %beta in this sample equation?
And why has letsimp() no effect?
Thank you very much!
letsimp applies only to "*" expressions. You could try subst.

mathematica Sum of functions

I want to build a function in the following way:
f[x_] := 0;
For[i = 1, i <= n, i++,
g[x_] := 0;
For[k = 1, k <= i, k++,
g ^:= g[#]*(# - X[[k]]) &;
g[x_] = g;
]
f ^:= f[#] + Q[[1, i]]*g[#];
f[x_] = f;
So I get a polynomial
Q_11 * (x-x_1) + Q_12 * (x-x_1) * (x-2x_2) ...
This is the latest version I have, but it does not work. The problem is that I have to add functions (I found some posts where it is done) but I have to save the new function so I can use it again.
Does someone know what to do?
Thanks!
Marius
I cant make head or tail of your code but you can readily construct that result just like this:
qlist = Array[q, {10, 10}];
xlist = Array[x, 10];
poly[n_] := Sum[ qlist[[1, j]] Product[ (x - xlist[[i]] ), {i, j}] , {j, 3}]
poly[3]
q[1, 1] (x - x[1]) + q[1, 2] (x - x[1]) (x - x[2]) +
q[1, 3] (x - x[1]) (x - x[2]) (x - x[3])
If you really need help using UpSetDelayed you should pose a more simple / specific question ( ie without the For loops. )