How should I plot the function (sqrt(1 + x) - 1) / x using the values x = 0.1, 0.01, 0.001, ..., 10^-20?

Below is my attempt to write a program for part of an assignment that does what's mentioned in the title. I cannot seem to get it to work and it just returns a blank graph. I'm still very new to Matlab so I'm not sure what I'm specifcally doing wrong. Any help is greatly appreciated!

x = -1:-20
f = @(x) (sqrt(1 + (10^x)) - 1) / (10^x);
plot(f(x));

Answer 1

[Paul]

Hi Jackson,

If you want x to from -1 to -20 you have to put the stride of -1 in the colon operator, otherwise it uses the default stride of +1, and since we can't step from -1 to -20 with stride of +1, x is an empty row vector.

x = -1:-20

x =

  1×0 empty double row vector

That empty row vector causes an error when evaluating f(x).

Instead, we could do

x = -1:-1:-20

x = 1×20

-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20

But these aren't really the values we want to use to evaluate the function.

I'd suggest defining the function as it's defined in the question.

It's important to use ./ (note the . (dot)) for element-wise division

f = @(x) (sqrt(1+x)-1)./x;

Now we want to evaluate at the indicated values. Again, we use element-wise power operator .^

format short e
vals = 10.^(-1:-1:-20)

vals = 1×20

1.0000e-01 1.0000e-02 1.0000e-03 1.0000e-04 1.0000e-05 1.0000e-06 1.0000e-07 1.0000e-08 1.0000e-09 1.0000e-10 1.0000e-11 1.0000e-12 1.0000e-13 1.0000e-14 1.0000e-15 1.0000e-16 1.0000e-17 1.0000e-18 1.0000e-19 1.0000e-20

The rest is up to you.

Suggest reading this doc page and the links at the bottom to learn more about Malab operators and operations..

Answer 2

[Star Strider]

Using the logspace function to create the independent variable ‘x’ vector makes this easier.

Plot it as a function of the independent variable.

It returns a blank plot because you need to use element-wise array division (./) instead of matrix division (/) in order tor the numerator vector to be divided element-wise by the denominator vector. (This is probably the most frequent problem I see here on Answers, so you are not alone in not considering it. See Array vs. Matrix Operations for details.)

If you want it to go from to , it will be necessary to reverse the x-axis direction. MATLAB plots from the lowest (here ) to the highest () by default.

Try this —

format shortE
x = logspace(-20, -1, 20)

x = 1×20

1.0000e-20 1.0000e-19 1.0000e-18 1.0000e-17 1.0000e-16 1.0000e-15 1.0000e-14 1.0000e-13 1.0000e-12 1.0000e-11 1.0000e-10 1.0000e-09 1.0000e-08 1.0000e-07 1.0000e-06 1.0000e-05 1.0000e-04 1.0000e-03 1.0000e-02 1.0000e-01

f = @(x) (sqrt(1 + (x)) - 1) ./ (x);

figure
plot(x,f(x),'.-')
grid
Ax = gca;
Ax.XScale = 'log';

figure
plot(x,f(x),'.-')
grid
Ax = gca;
Ax.XScale = 'log';
Ax.XDir = 'reverse';

I log-scaled the x-axis and added the grid lines for clarity. Delete those lines if you do not want the plot to show them.

.