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The size of a population of fish in a pond ismodeled by the function P, where P(t) givesthe number of fish and t gives the numberyears after the first year of introduction of thefish to the pond for 0 < t < 10. The graphof the function P and the line tangent to P att= 4 are shown above. Which of thefollowing gives the best estimate for theinstantaneous rate of change of P at t=4?

The size of a population of fish in a pond ismodeled by the function P where Pt givesthe number of fish and t gives the numberyears after the first year of intr class=

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Answer:

The slope of the line joining (3.9, P(3.9)) and (4.1, P(4.1)). Option D is correct.

Explanations:

Instantaneous rate of change is the measure of slope of a curve at a given instant.

From the given diagram, we can see that there is a line drawn tangential to the curve. The x-axis (time) for the slope of the curve must lie within the range of the tangential line.

Sine the range of time for the tangential line is between around t = 3 and t = 5, hence the x-axis for the slope must lie within this range.

From the given option, the only coordinate points that lie within the range is (3.9, P(3.9)) and (4.1, P(4.1))

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