Instructions: Please submit one pdf file for answers. Write ‘GRAD’ or ‘UNDERGRAD’ on the top of your submission. For problems which involve coding, please include answers or figures in the pdf file, and additionally upload the code as a separate file.
Make sure to show your work. Work is required for homework-quiz answers. Although use of various software tools is mostly encouraged for this class, in this homework one needs to show the computation of these values yourself.
- For the vector x, find the gradient ∇f(x) of the function f(x) = x3ex1+x2 and evaluate it at the point .
- For the previous function, find the Hessian matrix H(x).
- Find the 1-norm, the 2-norm, and the infinity norm of the vector x
- Find the linearization (the linear approximation) of f(x1,x2,x3) = around the point .
- Find the linear approximation and the quadratic approximation of
at the point x + δ for x.
- GRAD STUDENTS ONLY: Plot the original function, the linearization, and the quadratic approximation from the previous problem, all on one figure. In a different figure, plot the magnitude of the error as a function of delta. You can treat delta as a constant added identically to x1 and x2, allowing for a one-dimensional plot. Optionally, for a more interesting 2-variable plot, you can vary delta for each variable.
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