Calculus II, Spring 2024 (http://www.math.nthu.edu.tw/˜ wangwc/) Thomas’ Calculus Early Transcendentals 13ed

Study guide for quiz 07

Quiz problems include both the lecture contents and homework problems.

1. Section 14.3:

Study how to evaluate \frac{∂f}{∂x}, \frac{∂f}{∂y} at (x_{0}, y_{0}) when f(x, y) is given explicitly, and \frac{∂z}{∂x}, \frac{∂z}{∂y}

at (x_{0}, y_{0}, z_{0}) when z(x, y) is given implicitly by F (x, y, z) = 0.

2. Section 14.3:

Study the definition of differentiability for functions of two or more variables in section 14.3. Study why the four statements (1)-(4) on page 1 of Lecture 15 are identical (see also problem 2 of homework 07).

3. Section 14.3:

Study why ”existence of partial derivatives at a point” does not imply ”differentiability at a point” (find a counter example). Does this contradict Theorem 3?

4. Section 14.3:

Study why differentiability at a point implies continuity at that point (study the proof of Theorem 4). Try to find a function of two variables that is continuous at a point but not differentiable at that point.

5. Section 14.3:

The proofs of Theorem 2 and Theorem 3 are in Appendix 9. They will not appear in any exam in this course.