Thomas’ Calculus Early Transcendentals 12ed

Homework Assignment for Chapter 03

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

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

3. \frac{d^{n}}{dx^{n}} (f(x)g(x)) =? 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?

5. Section 3.5: problems 17, 49, 57, 58.

6. Section 3.6: Do as many as time permits from problems 51, 53, ⋯, 75.

7. Assume g(2) = 3, g′(2) = 0.1, f′(2) = 3, f′(3) = 4 and f′(4) = 5. What is \frac{d}{dx}f(g(x)) at x = 2?

8. Section 3.7: problems 27, 31, 42, 48.

9. Section 3.8: problems 7, 9, 37, 39, 51 (Hint: take ln on both sides first), 65, 77, 89, 91, 98. 93, 95.

10. Section 3.9: problems 21, 23, 25, 33, 35, 39, 55.

11. Start with domain and range for csc and csc^{−1}, derive the formula for the derivative of csc^{−1}.

12. Section 3.11: problems 9, 11, 17, 63, 64.

13. The error formula for linear approximation L(x, x_{0}) (also denoted as L(x) sometimes) is not mentioned explicitly in the textbook till a later Chapter. Just take it for granted and memorize it for now:

f(x) − L(x, x_{0}) = \frac{1}{2}f′′(c)(x − x_{0})^{2}

where c lies between x and x_{0}. As a consequence, we have an error bound

|f(x) − L(x, x_{0})| ≤ \frac{1}{2} ( max_{c between x and x_{0}} |f′′(c)|) (x − x_{0})^{2}

Use this formula to give an estimate on the error of linear approximation for problem 17 (b).