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MATH 237 Online Calculus 3 for Honours Mathematics
Spring 2024
Mini-midterm 1
Due date: 11pm, May 31 2024
1.(40 points) Evaluate the limit or show the limit does not exist.
(i) (15 points)
(ii) (15 points)
(iii) (10 points) Suppose a, b, c, d are positive numbers and a 
Find
2.(30 points) Suppose
(i) Find the gradient of f.
(ii) Find the linear approximation at the point (0, 25, 1)
(iii) Use the linear approximation above to estimate 
3.(20 points) Suppose
(i) (5 points) Is fpx, yq continuous at p0, 0q?
(ii) (10 points) Calculate fxyp0, 0q and fyxp0, 0q.
(iii) (5 points) State Clairaut’s Theorem. Discuss why (ii) is not contradictive to Clairaut’s Theorem.
4.(10 points) Suppose fpx, yq is bounded on the disk
Moreover, f is homogeneous with order 
i.e.,
Find
