English · PDF 1107
中文 · PDF 1107
62
| $a$ | $b$ | $c$ | $d$ | $e$ | $f$ | |
|---|---|---|---|---|---|---|
| $\sin x$ | $\frac{1}{2}$ | $\pm\frac{1}{2}$ | $\pm\sqrt{\frac{1}{2}}$ | $\pm\sqrt{\frac{1}{2}}$ | $-\frac{1}{2}$ | $\pm\frac{1}{2}$ |
| $\cos x$ | $\pm\frac{1}{2}\sqrt{3}$ | $-\frac{1}{2}\sqrt{3}$ | $\pm\sqrt{\frac{1}{2}}$ | $\sqrt{\frac{1}{2}}$ | $\pm\frac{2}{\sqrt{3}}$ | $\pm\frac{1}{2}\sqrt{3}$ |
| $\tan x$ | $\pm\sqrt{\frac{1}{3}}$ | $\pm\sqrt{\frac{1}{3}}$ | $-1$ | $\pm 1$ | $\pm\sqrt{\frac{1}{3}}$ | $\sqrt{\frac{1}{3}}$ |
| $\operatorname{cosec} x$ | $2$ | $\pm 2$ | $\pm\sqrt{2}$ | $\pm\sqrt{2}$ | $-2$ | $\pm 2$ |
| $\sec x$ | $\pm 2\sqrt{\frac{1}{3}}$ | $-2\sqrt{\frac{1}{3}}$ | $\pm\sqrt{2}$ | $\sqrt{2}$ | $\pm\frac{2}{3}$ | $\pm 2\sqrt{\frac{1}{3}}$ |
| $\cot x$ | $\pm\sqrt{3}$ | $\pm\sqrt{3}$ | $-1$ | $\pm 1$ | $\pm\sqrt{3}$ | $\sqrt{3}$ |
63 (a) $2 \sin 2\theta \cos \theta$ (b) $2 \sin \frac{3}{2} \theta \sin \frac{1}{2} \theta$
(c) $2 \cos \frac{7}{2} \theta \cos \frac{3}{2} \theta$ (d) $-2 \cos \frac{3}{2} \theta \sin \frac{1}{2} \theta$
64 (a) $\frac{1}{2}(\cos 2\theta - \cos 4\theta)$ (b) $\frac{1}{2}(\sin 4\theta + \sin 2\theta)$
(c) $\frac{1}{2}(\sin 4\theta - \sin 2\theta)$ (d) $\frac{1}{2}(\cos 4\theta + \cos 2\theta)$
65 (a) $2 \cos(\theta - \frac{2}{3}\pi)$ , $2 \sin(\theta - \frac{1}{6}\pi)$
(b) $\sqrt{2} \cos(\theta - \frac{3}{4}\pi)$ , $\sqrt{2} \sin(\theta - \frac{1}{4}\pi)$
(c) $\sqrt{2} \cos(\theta - \frac{1}{4}\pi)$ , $\sqrt{2} \sin(\theta - \frac{7}{4}\pi)$
(d) $\sqrt{13} \cos(\theta - 0.9828)$ , $\sqrt{13} \sin(\theta - 5.6952)$
66 $x = 2n\pi, 2n\pi \pm \frac{2}{3}\pi$ ( $n = 0, \pm 1, \pm 2, \dots$ )
67 (a) $\pi/6$ (b) $-\pi/6$ (c) $\pi/3$
(d) $2\pi/3$ (e) $\pi/3$ (f) $-\pi/3$
70
71
72 (a) $(2e + 1)e^5$ (b) $e^{4x}$ (c) $e^6$
(d) $e^9$ (e) $e^{x/2}$
74 (a) 3 (b) $-2$ (c) $-\frac{1}{2}$
(d) 4 (e) $\frac{1}{2}$ (f) $-\frac{1}{2}$
75 (a) $2 \ln x + \ln y$ (b) $\frac{1}{2} \ln x + \frac{1}{2} \ln y$
(c) $5 \ln x - 2 \ln y$
76 (a) $\ln 4$ (b) $\ln 3.2$ (c) $\ln 0.75$ (d) $\ln 0.5$
77 (a) $\sqrt{[(1-x)/(1+x)]}$ (b) $x^2$
79 $\ln(20 \pm 6\sqrt{10}) = 3.6629, 0.02599$
80 $\frac{3}{2} \ln(x^2 + 1) - \frac{1}{3} \ln(x^4 + 1) - \frac{1}{5} \ln(x^4 + 4)$
81
| $a$ | $b$ | $c$ | $d$ | $e$ | $f$ | |
|---|---|---|---|---|---|---|
| $\sinh x$ | $\pm\frac{3}{4}$ | $\frac{8}{15}$ | $\frac{7}{24}$ | $\pm\frac{12}{5}$ | $-\frac{4}{3}$ | $\frac{12}{5}$ |
| $\cosh x$ | $\frac{5}{4}$ | $\frac{17}{15}$ | $\frac{25}{24}$ | $\frac{13}{5}$ | $\frac{5}{3}$ | $\frac{13}{5}$ |
| $\tanh x$ | $\pm\frac{3}{5}$ | $\frac{8}{17}$ | $-\frac{7}{25}$ | $\pm\frac{12}{13}$ | $-\frac{4}{5}$ | $\frac{12}{13}$ |
| $\operatorname{cosech} x$ | $\pm\frac{4}{3}$ | $\frac{15}{8}$ | $\pm\frac{24}{7}$ | $\pm\frac{5}{12}$ | $-\frac{3}{4}$ | $\frac{5}{12}$ |
| $\operatorname{sech} x$ | $\frac{4}{5}$ | $\frac{15}{17}$ | $\frac{24}{25}$ | $\frac{5}{13}$ | $\frac{3}{5}$ | $\frac{5}{13}$ |
| $\operatorname{coth} x$ | $\pm\frac{5}{3}$ | $\pm\frac{17}{8}$ | $-\frac{25}{7}$ | $\pm\frac{13}{12}$ | $-\frac{3}{4}$ | $\frac{13}{12}$ |
82 (a) $\tanh 3x = \frac{(3 + \tanh^2 x) \tanh x}{1 + 3 \tanh^2 x}$
(b) $\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y$
(c) $\cos 2x = 1 - 2 \sin^2 x$
(d) $\sinh x - \sinh y = 2 \sinh \frac{1}{2}(x - y) \cosh \frac{1}{2}(x + y)$
84 (a) 0.7327 (b) 1.3170 (c) 0.5493
85 17.1383 (4dp)
86 1.0074 (4dp)
88 $A = 250, B = -273.26$
91 (a) 尖点位于 $x = 0$ , 在……处取得最大值 $x = 4$ , asymptote $y = -x$
(b) 最小值位于 $x = 2$ , asymptotes $y = \pm\sqrt{x - 1}, x = 1$
94 $\frac{ax}{l} H(x) + \frac{2a}{l} (l - x) H(x - l) - \frac{a}{l} (2l - x) H(x - 2l)$
95 $x[1 - H(x)] - (x - 1)H(x - 1)$
96 $\operatorname{INTPT}(x + \frac{1}{2})$
99 0.9401, 0.005, 0.9425
100 0.04, 0.16, 0.01, 0.00625
101 0.3081, 0.2829, 16.79
102 0.2954, 0.2688, 17.10
103 $f(x) = -\frac{1}{84}x^3 + \frac{85}{84}x$
| $x$ | 3045 | 3051 | 3058 | 3064 | 3070 | 3077 | 3083 |
|---|---|---|---|---|---|---|---|
| $y$ | 14.50 | 14.51 | 14.52 | 14.53 | 14.54 | 14.55 |