Intersection between two functions

Hi, fairly new to Matlab. Struggling with a question requiring me to find an x value for which:

2.2/sqrt(2*9.81*x) = tanh((3.5/(2*4.5))*sqrt(2*9.81*x)

Tried to use finding an intersection between two functions in accordance with another answer on this website, but I get multiple errors, both in graphing the function to see roughly where the correct solution should be and in finding a solution at all for the intersection. I also tried using symbolic variables (not sure what the difference is to be honest), but couldn't get it to work.

I also get this error at the very start:

Warning: Function behaves unexpectedly on array inputs. To improve performance, properly vectorize

your function to return an output with the same size and shape as the input arguments.

> In matlab.graphics.function.FunctionLine>getFunction

In matlab.graphics.function/FunctionLine/updateFunction

In matlab.graphics.function/FunctionLine/set.Function_I

In matlab.graphics.function/FunctionLine/set.Function

In matlab.graphics.function.FunctionLine

In fplot>singleFplot (line 245)

In fplot>@(f)singleFplot(cax,{f},limits,extraOpts,args) (line 200)

In fplot>vectorizeFplot (line 200)

In fplot (line 166)

In question (line 7)

Error using /

Matrix dimensions must agree.

Error in question (line 15)

f1a= 2.2/(sqrt(2*9.81*x));

This is my code at the moment:

%input array

x=linspace(0,1,1000);

f1=@(x) 2.2/sqrt(2*9.81*x);

f2=@(x) tanh((3.5/(2*4.5))*sqrt(2*9.81*x));

% Graph functions

fplot(f1,'b')

hold on

fplot(f2,'r')

grid on

title('Finding Intersections of Functions')

xlabel('Input Values (x)')

ylabel('Ouput Values (f)')

% Find the x-cordinates of intersecting points

f1a= 2.2/(sqrt(2*9.81*x));

f2a= tanh((3.5/(2*4.5))*sqrt(2*9.81*x));

Intersections=find(abs(f1a-f2a)<=(0.0001));

X_Values= x(Intersections)

Would appreciate any help.

Answer 1

[Star Strider]

Try this slightly changed version of the existing code —

f1=@(x) 2.2./sqrt(2*9.81*x);
f2=@(x) tanh((3.5/(2*4.5))*sqrt(2*9.81*x));

% Graph functions
figure
fplot(f1,'b')
hold on
fplot(f2,'r')
grid on
title('Finding Intersections of Functions')
xlabel('Input Values (x)')
ylabel('Ouput Values (f)')
% Find the x-cordinates of intersecting points
Intersections = fzero(@(x) f1(x)-f2(x), 1)

Intersections = 0.3924

X_Values= Intersections

X_Values = 0.3924

Y_Values = f1(Intersections)                                % Can Use Either Function, since this is common to both

Y_Values = 0.7929

plot(X_Values, Y_Values, 'sg', 'MarkerSize',10)
hold off

Experiment to get different results.

.

Answer 2

[Chien Poon]

All you need is the element wise division (adding ./) on variable f1a

%input array
x=linspace(0,1,1000);
f1=@(x) 2.2/sqrt(2*9.81*x);
f2=@(x) tanh((3.5/(2*4.5))*sqrt(2*9.81*x));
% Graph functions
fplot(f1,'b')
hold on
fplot(f2,'r')
grid on
title('Finding Intersections of Functions')
xlabel('Input Values (x)')
ylabel('Ouput Values (f)')
% Find the x-cordinates of intersecting points
f1a= 2.2./(sqrt(2*9.81*x));
f2a= tanh((3.5/(2*4.5))*sqrt(2*9.81*x));
Intersections=find(abs(f1a-f2a)<=(0.0001));
X_Values= x(Intersections)

[Elina Nikolopoulou]

how do i find the y-coordinates of the intersecting points ?

[Star Strider]

@Elina Nikolopoulou — By using the approach in my Answer!

Answer 3

[William Rose]

@Behbod Izadi,

syms x
S=solve(2.2/sqrt(2*9.81*x) == tanh((3.5/(2*4.5))*sqrt(2*9.81*x)),x)

Warning: Unable to solve symbolically. Returning a numeric solution using vpasolve.

S = 

gives an answer. Is it reasonable? The solution x=-212 means you are taking the square root of a negative number, which has an imaginary result. Is that acceptable to you? vpasolve returns the first solution it finds. I wonder if there is a solution involivng a non-negative value of x, which vpasolve did not find. Let's plot to find out.

x=0:.1:10;
figure;
plot(x,2.2./sqrt(2*9.81*x),'.r-',x,tanh((3.5/(2*4.5))*sqrt(2*9.81*x)),'.b-');
xlabel('x'); ylabel('f(x)'); legend('sqrt','tanh');

Note that in the plot command, I had to use './sqrt()' rather than '/sqrt()', to avoid an error. Note also that the red plot had a value of y=Inf when x=0, and that was not a problem; the plot function just skips that point. The plot shows that there IS a non-negative x which is a solution, and it appears to be between x=0.3 and 0.4, very close to x=0.4. We can tell vpasolve() to look for a solution in the range [0 10], as follows:

syms x
S=vpasolve(2.2/sqrt(2*9.81*x) == tanh((3.5/(2*4.5))*sqrt(2*9.81*x)),x,[0 10])

S = 

0.39242469611262814472271476457847

This value is consistent with the plot.