Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
MAST20009 Vector Calculus
Semester 2 2024
Assignment 1
1. This assignment is worth 5% of your final MAST20009 mark.
2. Assignments must be neatly handwritten in blue or black pen on A4 paper or may be typeset using LATEX. Diagrams can be drawn in pencil.
3. You agree to the plagiarism declaration on the LMS by submitting your assign- ment.
4. Full working must be shown in your solutions.
5. Marks will be deducted for incomplete working, insu伍cient justification of steps, incorrect mathematical notation and for messy presentation of solutions.
6. (a) If you printout the assignment template, write your solutions into the answer spaces and then scan your assignment to a PDF file using a scanning app on your mobile phone for upload;
(b) If you are unable to answer the whole question in the answer space provided then you can append additional pages to the end of your assignment. If you do this you MUST make a note in the correct answer space for the question, warning the marker that you have appended additional work at the end;
(c) You may handwrite/type your answers on blank paper and then scan for submission; in this case the first page must contain only your student number, subject code and subject name. Make sure your write-up follows the format of the template.
(d) Allow enough time to submit the PDF to Gradescope.
1. Show that
does not exist.
2. Consider the function
(a) Determine whether or not f (x, y) is continuous at (0, 0). Justify your answer.
(b) Determine whether 
exisits at (0, 0). If yes, evaluate (0, 0). If not, explain why.
(c) Determine where f (x, y) is C1 . Justify your answer.
3. Let f : R2 → R2 be the function
f (x, y) = (xsin y + x, ex2y).
(a) Find Df , the derivative matrix of f.
(b) Let g(x, y) be a diferentiable function such that g(-1, 1) = (-1, 0) and such that f (g(x, y)) = (x, y) for all (x, y) close to the point (-1, 1). Find Dg(-1, 1).
4. Consider the function
f(x, y) = ex+y2.
(a) Determine the second order Taylor polynomial p2 (x, y) for f near the point (0, 0).
(b) Approximate f (-0.1, 0.1) using the first order Taylor polynomial p1 (x, y) for f near the point (0, 0).
(c) Using Taylor’s remainder formula, find an upper bound for the error in your approximation of f (-0.1; 0.1) in part (b).