1.2 Basic Concepts
- 1
(a) \(3\) (b) \(2\) (c) \(1\) (d) \(2\)
- 3
(a) \(y=-\dst{\frac{x^{2}}{2}}+c\) (b) \(y=x\cos x-\sin x+c\)
(c) \(y=\dst{\frac{x^{2}}{2}\ln x-\frac{x^{2}}{4}+c}\) (d) \(y=-x\cos x+2\sin x+c_{1}+c_{2}x\)
(e) \(y=(2x-4)e^{x}+c_{1}+c_{2}x\) (f) \(y=\dst{\frac{x^{3}}{3}}-\sin x+e^{x}+c_{1}+c_{2}x\)
(g) \(y=\sin x+c_{1}+c_{2}x+c_{3}x^{2}\) (h) \(y=-\dst{\frac{x^{5}}{60}}+e^{x}+c_{1}+c_{2}x+c_{3}x^{2}\)
(i) \(y=\dst{\frac{7}{64}}e^{4x}+c_{1}+c_{2}x+c_{3}x^{2}\)
- 4
(a) \(y=-(x-1)e^{x}\) (b) \(y=1-\dst{\frac{1}{2}}\cos x^{2}\) (c) \(y=3-\ln(\sqrt{2}\cos x)\)
(d) \(y=\dst{-\frac{47}{15}-\frac{37}{5}(x-2)+\frac{x^{5}}{30}}\) (e) \(y=\dst{\frac{1}{4}xe^{2x}-\frac{1}{4}e^{2x}+\frac{29}{4}}\)
(f) \(y=x\sin x+2\cos x-3x-1\) (g) \(y=(x^{2}-6x+12)e^{x}+\dst{\frac{x^{2}}{2}}-8x-11\)
(h) \(y=\dst{\frac{x^{3}}{3}+\frac{\cos 2x}{6}+\frac{7}{4}x^{2}-6x+\frac{7}{8}}\) (i) \(y=\dst{\frac{x^{4}}{12}+\frac{x^{3}}{6}+\frac{1}{2}(x-2)^{2} -\frac{26}{3}(x-2)-\frac{5}{3}}\)
- 7
(a) \(576\) ft (b) \(10\) s
- 8
(b) \(y=0\)
- 10
(a) \((-2c-2,\infty)\) \((-\infty,\infty)\)