« Section 8.1
Section 8.3 »

Section 8.2

Please post comments if you find any errors, or if you wish to contribute answers for additional problems. You may find information on using \LaTeX here.

1. (a) \displaystyle{x = 75e^{-0.9t}}

(c) \displaystyle{y^2 = t^2 + 25}, or \displaystyle{y = \sqrt{t^2 + 25}}

(e) \displaystyle{x^2 = 2t - 2\log(1 + t) + 16}, or \displaystyle{x = \sqrt{2t - 2\log(1 + t) + 16}}

(g) \displaystyle{x = \frac{1}{1 + 4e^{-t}}}

2. (a) \displaystyle{x = \frac{2x_0}{2 + x_0t^2}}

(c) If x_0 > 0 , domain = (-\infty, \infty) .
If x_0 < 0 , domain \displaystyle{= \left\{t : t \ne \sqrt{-\frac{2}{x_0}}\right\}}.

3. (a) The equation of the curve is ax^2 + by^2 = c , where c is a constant. This is the equation of an ellipse, which becomes a circle if a = b .

4. (a) \displaystyle{x = \frac{x_0}{1 - kx_0t}}

(c) \displaystyle{x = \left(\frac{1}{2}kt + \sqrt{x_0}\right)^2 = \frac{1}{4}k^2t^2 + k\sqrt{x_0}t + x_0}

(e) The slowest population growth occurs when 0 < b < 1 , the most rapid when b > 1 . The case b > 1 is called the doomsday model because the population goes to infinity in a finite amount of time.

5. (b) \displaystyle{v = \sqrt{\frac{mg}{k}}\left(\frac{1 - e^{2\sqrt{\frac{kg}{m}}t}}{1+e^{2\sqrt{\frac{kg}{m}}t}}\right) = -\sqrt{\frac{mg}{k}}\tanh\left(\sqrt{\frac{kg}{m}}t\right)}

This entry was posted on 20 August 20 2007 at 8:19 pm and is filed under Section 8.2. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Leave a Reply