Here is an example of MATLAB code that takes as input a positive parameter k and a desired error threshold err, and returns the first value n so that the nth-order Taylor Polynomial for f(x)=e^(kx) about x=0 is within err of e^(kx) for all x in the interval [-1,1]:

function n = taylor_polynomial_order(k, err)

% This function takes as input a positive parameter k and a desired error

% threshold err, and returns the first value n so that the nth-order

% Taylor Polynomial for f(x) = e^(kx) about x = 0 is within err of e^(kx)

% for all x in the interval [-1,1].

k=1

err=0.01

% Initialize n to 0

n = 0;

% Calculate the first n terms of the Taylor series for e^(kx) about x = 0

taylor_series = 1;

% Initialize the error to be greater than the desired threshold

error = err + 1;

while error > err

% Calculate the next term in the Taylor series

n = n + 1;

term = (k^n)/factorial(n);

taylor_series = taylor_series + term;

% Calculate the error

error = max(abs(taylor_series - exp(k*(-1:0.01:1))));

end

% Return the value of n

n

end

the problem I am having with this is that for example when testing this function by calling it with specific values for k and err, such as taylor_polynomial_order(1, 0.01). This should return the value 13, which is the first value of n that satisfies the condition in the problem statement.

I am not getting thirteen and when I change k and err and I keep getting these errors should please help fix this so that it runs and doesn't give me issues.