Section 3.9 « Difference Equations to Differential Equations

Section 3.9

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1. (a) $f$ is decreasing on $\displaystyle{\left(-\infty, \frac{1}{2}\right)}$ and increasing on $\displaystyle{\left(\frac{1}{2}, \infty\right)}$ . The graph of $f$ is concave up on $(-\infty, \infty)$ . $f$ has a local minimum of $-\dfrac{1}{4}$ at $x = \dfrac{1}{2}$ . The graph of $f$ does not have any inflection points or asymptotes.

(c) $g$ is increasing on $(-\infty, -2)$ and $(0, \infty)$ , and decreasing on $(-2, 0)$ . The graph of $g$ is concave down on $(-\infty, -1)$ and concave up on $(-1, \infty)$ . $g$ has a local maximum of $4$ at $x = -2$ and a local minimum of $0$ at $x = 0$ . $(-1, 2)$ is an inflection point. The graph of $g$ does not have any asymptotes.

(e) $f$ is increasing on $(-\infty, -1)$ and $(1, \infty)$ , and decreasing on $(-1, 1).$ The graph of $f$ is concave down on $(-\infty, 0)$ and concave up on $(0, \infty)$ . $f$ has a local maximum of $2$ at $x = -1$ and a local minimum of $-2$ at $x = 1$ . $(0, 0)$ is an inflection point. The graph of $f$ does not have any asymptotes.

(g) $h$ is increasing on $\displaystyle{\left(-\infty, -\sqrt{\frac{3}{5}}\right)}$ and $\displaystyle{\left(\sqrt{\frac{3}{5}}, \infty\right)}$ , and decreasing on $\displaystyle{\left(-\sqrt{\frac{3}{5}}, 0\right)}$ and $\displaystyle{\left(0, \sqrt{\frac{3}{5}}\right)}$ . The graph of $h$ is concave down on $\displaystyle{\left(-\infty, -\sqrt{\frac{3}{10}}\right)}$ and $\displaystyle{\left(0, \sqrt{\frac{3}{10}}\right)}$ , and concave up on $\displaystyle{\left(-\sqrt{\frac{3}{10}}, 0\right)}$ and $\displaystyle{\left(\sqrt{\frac{3}{10}}, \infty\right)}$ . $h$ has a local maximum of $\displaystyle{\frac{6}{25}\sqrt{\frac{3}{5}}}$ at $\displaystyle{x = -\sqrt{\frac{3}{5}}}$ and a local minimum of $\displaystyle{-\frac{6}{25}\sqrt{\frac{3}{5}}}$ at $\displaystyle{x = \sqrt{\frac{3}{5}}}$ . $\displaystyle{\left(-\sqrt{\frac{3}{10}}, \frac{21}{100}\sqrt{\frac{3}{10}}\right)}$ , $(0, 0)$ , and $\displaystyle{\left(\sqrt{\frac{3}{10}}, -\frac{21}{100}\sqrt{\frac{3}{10}}\right)}$ are inflection points. The graph of $h$ does not have any asymptotes.

(i) $g$ is decreasing on $(-\infty, 1)$ and $(1, \infty)$ . The graph of $g$ is concave down on $(-\infty, 1)$ and concave up on $(1, \infty)$ . The line $z = 1$ is a vertical asymptote and the line $y = 0$ (that is, the $z$ -axis) is a horizontal asymptote. $g$ does not have any local extreme values and the graph of $g$ does not have any inflection points.

(k) $f$ is decreasing on $(-\infty, 0)$ and $\displaystyle{\left(0, \frac{3}{2}\right)}$ , and increasing on $\displaystyle{\left(\frac{3}{2}, 0\right)}$ . The graph of $f$ is concave up on $(-\infty, 0)$ and $(1, \infty)$ , and concave down on $(0, 1)$ . $f$ has a local minimum of $-\dfrac{27}{16}$ at $x = \dfrac{3}{2}$ . $(0,0)$ and $(1, -1)$ are inflection points. The graph of $f$ does not have any asymptotes.

(m) $h$ is decreasing on $(-\infty, -2)$ , $(-2, 2)$ , and $(2, \infty)$ . The graph of $h$ is concave down on $(-\infty, -2)$ and $(0, 2)$ , and concave up on $(-2, 0)$ and $(2, \infty)$ . $(0, 0)$ is an inflection point. The lines $t = -2$ and $t = 2$ are vertical asymptotes and the line $y = 0$ (that is, the $t$ -axis) is a horizontal asymptote. $h$ does not have any local extreme values.

(o) $f$ is decreasing on $(-\infty, 0)$ and increasing on $(0, \infty)$ . The graph of $f$ is concave down on $\displaystyle{\left(-\infty, -\frac{1}{\sqrt{3}}\right)}$ and $\displaystyle{\left(\frac{1}{\sqrt{3}}, \infty\right)}$ , and concave up on $\displaystyle{\left(-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)}$ . $f$ has a local minimum of $0$ at $x = 0$ . $\displaystyle{\left(-\frac{1}{\sqrt{3}}, \frac{1}{4}\right)}$ and $\displaystyle{\left(\frac{1}{\sqrt{3}}, \frac{1}{4}\right)}$ are inflection points. The line $y = 1$ is a horizontal asymptote.

(q) $x$ is decreasing on $(-\infty, 1)$ and $(1, \infty)$ . The graph of $x$ is concave down on $(-\infty, 1)$ and concave up on $(1, \infty)$ . The line $t = 1$ is a vertical asymptote and the line $y = 2$ is a horizontal asymptote. $x$ does not have any local extreme values. The graph of $x$ does not have any inflection points.

3. (c) There is only one function which satisfies these conditions.

4. (c) There is only one function which satisfies these conditions.

5. (c) There is only one function which satisfies these conditions.

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Difference Equations to Differential Equations

Section 3.9

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