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Section 7.3

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2. (a) f'(t) = -2\sin(2t) + 2i\cos(2t)

(c) \displaystyle{g'(t) = \frac{1}{2}\cos\left(\frac{t}{2}\right) - \frac{1}{2}i\sin\left(\frac{t}{2}\right)}

(d) z'(t) = -2\mathrm{sech}(2t)\tanh(2t) + 2i\mathrm{sech}^2(2t)

(e) f'(t) = 2 + 2it

(f) g'(t) = 2t + 4it^3

(g) \displaystyle{z'(t) = 3ie^{it}}

(h) \displaystyle{h'(t) = 3ite^{it} + 3e^{it}}

(i) \displaystyle{z'(t) = \frac{6}{t}ie^{2it} - \frac{3}{t^2}e^{2it}}

3. (a) Velocity = -\sqrt{3} + i
Speed = 2
Acceleration = -2 - 2\sqrt{3}i

(c) Velocity = \mathrm{sech}^2(3) - i\mathrm{sech}(3)\tanh(3) = 0.009866 - 0.098837i ,
rounded to six decimal places
Speed = 0.099328 , rounded to six decimal places
Acceleration = -2\mathrm{sech}^2(3)\tanh(3) + i(\mathrm{sech}(3)\tanh^2(3) - \mathrm{sech}^3(3))
= -0.019634 + 0.097368i , rounded to six decimal places

(e) Velocity = -5
Speed = 5
Acceleration = -5i

4. (a) \displaystyle{\int_0^4 (2t + it)dt = 16 + 8i}

(c) \displaystyle{\int_0^{\frac{\pi}{2}}(-3\sin(2t) + it^3)dt = -3 + \frac{\pi^4}{64}i}

(e) \displaystyle{\int_0^\pi 2te^{3it}dt = -\frac{4}{9} + \frac{2\pi}{3}i}

5. z(t) = (1 + \sin(t)) + i(2 - \cos(t))

6. (a) \displaystyle{t = \frac{1}{16}s_0\sin(\alpha)}

(b) \displaystyle{R = \frac{s_0^2\sin(\alpha)\cos(\alpha)}{16} = \frac{s_0^2\sin(2\alpha)}{32}}

7. (a) Maximum range = 703.125 feet

(b) If \alpha = \dfrac{\pi}{6} , the range is \dfrac{5625\sqrt{3}}{16} feet and the projectile strikes the ground after 4.6875 seconds.
If \alpha = \dfrac{\pi}{3} , the range is \dfrac{5625\sqrt{3}}{16} feet and the projectile strikes the ground after \dfrac{75\sqrt{3}}{16} seconds.

10. (b) \displaystyle{\int_0^\infty e^{-t}dt = \varphi(0) = 1}
\displaystyle{\int_0^\infty te^{-t}dt = \frac{\varphi'(0)}{i} = 1}
\displaystyle{\int_0^\infty t^2e^{-t}dt = \frac{\varphi''(0)}{i^2} = 2}
\displaystyle{\int_0^\infty t^3e^{-t}dt = \frac{\varphi'''(0)}{i^3} = 6}
\displaystyle{\int_0^\infty t^4e^{-t}dt = \frac{\varphi''''(0)}{i^4} = 24}

11. \displaystyle{\varphi(\lambda) = \sqrt{2\pi}e^{-\frac{\lambda^2}{2}}}
\displaystyle{\int_{-\infty}^\infty e^{-\frac{t^2}{2}}dt = \varphi(0) = \sqrt{2\pi}}
\displaystyle{\int_{-\infty}^\infty te^{-\frac{t^2}{2}}dt = \frac{\varphi'(0)}{i} = 0}
\displaystyle{\int_{-\infty}^\infty t^2e^{-\frac{t^2}{2}}dt = \frac{\varphi''(0)}{i^2} = \sqrt{2\pi}}
\displaystyle{\int_{-\infty}^\infty t^3e^{-\frac{t^2}{2}}dt = \frac{\varphi'''(0)}{i^3} = 0}
\displaystyle{\int_{-\infty}^\infty t^4e^{-\frac{t^2}{2}}dt = \frac{\varphi''''(0)}{i^4} = 3\sqrt{2\pi}}

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