lsqcurvefit for exponential regression - objective function

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I read online about fitting an exponential curve to my data via least squares and I have understood it. In matlab however, I am not sure what algorithm is being used if I pass in my arguments below. Can someone explain to me the theory behind matlab's least squares fit ?
1. Rather than compute the sum of squares, lsqcurvefit requires the user-defined function to compute the vector-valued function. What does this mean and how is this different from computing least squares ?
2. What is the default algorithm being used in my code below ? Least squares ? Or the trust region algorithm (I have no read what that is about)
3. What is the difference between x and xdata ?
F = @(p,xdata) p(1)*exp(p(2)*(xdata+p(3)));
x0 = [5 0 -10];
[p,resnorm] = lsqcurvefit(F,x0,xdata,ydata);

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