Calculus I, Fall 2015 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 03

1. Section 3.2: problems 17, 48, 54, 57.

2. Section 3.3: problems 47, 55, 67, 70.

3. \frac{d^{n}}{dx^{n}} (f(x)g(x)) =? Try n = 2, 3, ⋯ and conclude the general formula. Memorize the result.

4. Use product rule to show (and memorize) that

\begin{aligned}\frac{d}{dx} | \begin{matrix}f_{11}(x) & f_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{matrix} | & = | \begin{matrix}f′_{11}(x) & f_{12}(x) \\ f′_{21}(x) & f_{22}(x) \end{matrix} | + | \begin{matrix}f_{11}(x) & f′_{12}(x) \\ f_{21}(x) & f′_{22}(x) \end{matrix} | \\ & = | \begin{matrix}f′_{11}(x) & f′_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{matrix} | + | \begin{matrix}f_{11}(x) & f_{12}(x) \\ f′_{21}(x) & f′_{22}(x) \end{matrix} | \end{aligned}

What is the corresponding formula for a 3 by 3 determinant? How about 4 by 4, etc?