Thomas’ Calculus Early Transcendentals 13ed

Homework Assignment for Chapter 03

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

2. Read about second and higher order derivatives at end of section 3.3.

3. Section 3.3: problems 47, 55, 67, 70, 75(c).

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 are the corresponding formulae for 3 by 3 determinants, 4 by 4 determinants, etc.?

5. Apply the product rule repeatedly to get \frac{d^{n}}{dx^{n}} (u(x)v(x)) in terms of derivatives of u(x) and v(x). Start with n = 2, then n = 3, ⋯, to find the formula.

6. Section 3.5: problems 17, 33(a), 34(a), 49 (Hint: what is \frac{d sin θ}{dθ}|_{θ=c}?) , 57, 58.

7. Section 3.6: Do as many as time permits from problems 51, 53, ⋯, 77.

8. 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?

9. Section 3.7: problems 27, 31, 48, 51(a).

10. Section 3.8: problems 8, 9, 37, 39, 51 (Hint: take ln on both sides first), 65, 77, 89, 91, 93, 95, 98.

11. Section 3.9: problems 9, 11, 21, 23, 25, 33, 35, 39, 53, 55.

12. Section 3.11: problems 9, 11, 16(c,d), 17, 53, 55, 65(a,b,f(for f(x) only)), 66.

13. A key point in section 3.11 is that the error of linear approximation, f(x) − L(x),

satisfies

lim_{x→a} \frac{f(x) − L(x)}{x − a} = 0  (1)

provided f is differentiable at x = a.

The following statement gives more details about the error f(x) − L(x) and will be introduced in the near future. \underline{Take this statement for granted for now}:

If f is twice differentiable near x = a, then

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

for some c between x and a. From (2), we have an error bound

|f(x) − L(x)| ≤ \frac{1}{2} ( max_{c between x and a} |f′′(c)|) (x − a)^{2}  (3)

Use (3) to estimate the error of linear approximation (i.e. find out |f(x) − L(x)| ≤???) for problem 17 (b) of Section 3.11.

14. Chapter 3, additional and advanced problems: problems 16, 21, 22(d), 23.