$\"\"$

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(a)

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Logistic differential equation with carrying capacity $\"\"$ is $\"\"$ .

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Here carrying capacity is $\"\"$ and $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$ .

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Logistic differential equation is $\"\"$ .

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$\"\"$

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(b)

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Graph the direction field for differential equation $\"\"$ .

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$\"\"$

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Observe the graph:

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Slope is independent of $\"\"$ .

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Thus, $\"\"$ and $\"\"$ are equilibrium solutions.

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Slope is positive for $\"\"$ and negative for $\"\"$ .

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$\"\"$

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(c)

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Graph the directional field to graph the solutions of $\"\"$ and $\"\"$ .

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$\"\"$

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Observe the direction field in the graph:

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The slope field values are in

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Some solutions are increasing and some are decreasing.

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Solutions of $\"\"$ and $\"\"$ have inflection point at $\"\"$ .

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$\"\"$

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(d)

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Slopes are close to zero when $\"\"$ or $\"\"$ .

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Thus, $\"\"$ and $\"\"$ are equilibrium solutions.

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Other solutions are differ from the above as they are moving away from $\"\"$ towards $\"\"$ .

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$\"\"$

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(a) $\"\"$ and $\"\"$ .

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(b)

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Slopes are close to zero when $\"\"$ or $\"\"$ .

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Slopes are largest on the line $\"\"$ .

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Solutions are increasing in the interval $\"\"$ .

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Solutions are decreasing in the imterval $\"\"$ .

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(c)

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$\"\"$

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(d)

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$\"\"$ and $\"\"$ are equilibrium solutions.