$\"\"$

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A tank holds $\"\"$ gallons of water.

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(a)

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The point is $\"\"$ .

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Find the slopes of the secant lines $\"\"$ .

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Consider the point $\"\"$ .

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At $\"\"$ and the corresponding the value of $\"\"$ is $\"\"$ .

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So the point is $\"\"$ .

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Slope of the two points is $\"\"$ .

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The slope of the secant line $\"\"$ is $\"\"$ .

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At $\"\"$ and the corresponding the value of $\"\"$ is $\"\"$ .

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So the point is $\"\"$ .

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The slope of the secant line $\"\"$ is $\"\"$ .

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At $\"\"$ and the corresponding the value of $\"\"$ is $\"\"$ .

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So the point is $\"\"$ .

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The slope of the secant line $\"\"$ is $\"\"$ .

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At $\"\"$ and the corresponding the value of $\"\"$ is $\"\"$ .

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So the point is $\"\"$ .

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The slope of the secant line $\"\"$ is $\"\"$ .

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At $\"\"$ and the corresponding the value of $\"\"$ is $\"\"$ .

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So the point is $\"\"$ .

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The slope of the secant line $\"\"$ is $\"\"$ .

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The slopes of the secant lines $\"\"$ are $\"\"$ and $\"\"$ .

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$\"\"$

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(b)

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Find the average of the slopes of the secant lines near to $\"\"$ .

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Consider the points are near to $\"\"$ .

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Points are $\"\"$ and $\"\"$ .

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The slopes of the secant lines $\"\"$ are formed from the points $\"\"$ and $\"\"$ is $\"\"$ and $\"\"$ .

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The average of the slopes is $\"\"$

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Therefore the slope of the tangent line at $\"\"$ is $\"\"$ .

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$\"\"$

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(c)

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Graph :

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Use the values from the table and graph the function.

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(1) Draw the coordinate plane.

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(2) Plot the points from the table.

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(3) Connect the plotted points to a smooth curve.

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(4) Draw a approximate tangent line at $\"\"$ .

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$\"\"$

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From the graph, the green line represents the approximate tangent line at $\"\"$ .

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So the slope of the tangent line is $\"\"$ .

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$\"\"$

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(a) The slopes of the secant lines $\"\"$ are $\"\"$ and $\"\"$ .

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(b) The slope of the tangent line at $\"\"$ is $\"\"$ .

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(c) The slope of the tangent line after $\"\"$ minutes is $\"\"$ .