Report 4

Maximilian Fernaldy - C2TB1702

Problem 1

First, we want to plot the curve of the function:

$$f(x) = sin^2(x) \cdot exp\left(-\cfrac{x}{2}\right) + 0.01x - 0.1$$

To do this, we first define the domain for x:

x = 0:0.1:10;

and define y as a function of x, i.e., $y = f(x)$:

y = sin(x).^2.*exp(-x/2)+0.01.*x-0.1;

With the arrays for x and y defined, we can now plot the curve:

plot(x,y)

Running the program, we find this graph, completing the first problem:

Problem 2

Next, we want to find the roots to the equation:

$$sin^2(x) \cdot exp\left(-\cfrac{x}{2}\right) + 0.01x - 0.1 = 0, (x\geq0)$$

It's obvious that this equation is the function in the first problem, but with $f(x)=0$. The roots of the equation are the $x$-coordinates of where the curve intersects the $x$-axis, or also called the $y = 0$ line. We can intuitively see that the curve crosses the $y = 0$ line several times. To make these intercepts clearer, we draw a line for $y = 0$. To do this, define a one-row array called intercept and filling it with zeros everywhere on x. We then change the plot line to also include (x,intercept). We make the curve and line different colors to better distinguish them.

intercept = zeros(1,length(x));
plot(x, y, "m", x , intercept)

However, we don't know if the curve keeps crossing over the line or not without calculating. To visually confirm if there are no more roots in the domain $x \geq0$, we can extend the array where we defined x to be longer. We can then see more of the curve.

x = 0:0.1:20; % extending x array to 20
y = sin (x).^2.*exp(-x/2)+0.01.*x-0.1;
intercept = zeros(1,length(x)); % Draw a line for y = 0 to find intercepts
plot(x, y, "m", x , intercept);

As we can see from this extended plot, the curve diverges from the $x$-axis, which means it won't cross the line again after the point around $x=10$.

At first glance, there seems to be 6 points where the curve intersects the x axis. However, upon closer inspection, there really only are 5.

Zooming in on the point close to $x=8$:

While it is close, the curve does not quite touch the line.

Finding initial guesses to put in x0

To get the actual values of the roots, we need to use fsolve, as the equation is nonlinear. However, fsolve requires us to make initial guesses about where the roots are, and these initial guesses have to be somewhat close to the actual point. Luckily, we have the plot as a tool.

Zooming in on the intersects:

The first intersect is around $x = 0.3$.

The second intersect is around $x = 2.6$.

The third intersect is around $x = 3.8$.

The fourth intersect is around $x = 5.3$.

The fifth and final intersect is around $x = 9.8$.

With these values, we construct an array x0:

x0 = [0.3; 2.6; 3.8; 5.3]

Now we declare a function y = f(x) for the left side of the equation.

function y = f(x)
  y = sin (x).^2.*exp(-x/2)+0.01.*x-0.1;
end

In order to find the roots of $f(x)$, we can use fsolve in this manner:

result = fsolve(@f, x0)

Which in turn outputs the following as answers.

>> CAPS_04_C2TB1702_roots

ans =

   0.3456
   2.5941
   3.8480
   5.3244
   9.8677

Checking validity of answers

To check the validity of our result, we can plug in the $x$-values we get and see if we get $f(x) = 0$.

x = result;
y = sin (x).^2.*exp(-x/2)+0.01.*x-0.1

Running this after getting the result from our fsolve script outputs:

>> CAPS_04_C2TB1702_check

y =

   1.6067e-06
   2.8132e-10
  -1.4310e-08
   4.1089e-09
   1.3504e-09

These values are very close to zero, validating our fsolve script.