{Use of Tech} Spring runoff The flow of a small stream is monitored for 90 days between May 1 and August 1. The total water that flows past a gauging station is given by v(t) = where V is measured in cubic feet and t is measured in days, with t=0 corresponding to May 1.c. Describe the flow of the stream over the 3-month period. Specifically, when is the flow rate a maximum?

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.6.60c

Chapter 3, Problem 3.6.60c

{Use of Tech} Spring runoff The flow of a small stream is monitored for 90 days between May 1 and August 1. The total water that flows past a gauging station is given by v(t) = <matrix 2x2> where V is measured in cubic feet and t is measured in days, with t=0 corresponding to May 1.
c. Describe the flow of the stream over the 3-month period. Specifically, when is the flow rate a maximum?

Verified step by step guidance

To describe the flow of the stream over the 3-month period, we need to analyze the function v(t) that represents the total volume of water that flows past the gauging station. The flow rate of the stream is given by the derivative of v(t), denoted as v'(t).

Calculate the derivative v'(t) to find the flow rate function. This involves differentiating the given function v(t) with respect to time t.

Once you have v'(t), identify the critical points by setting v'(t) = 0 and solving for t. These critical points are potential candidates for maximum or minimum flow rates.

To determine whether each critical point is a maximum or minimum, use the second derivative test. Calculate the second derivative v''(t) and evaluate it at each critical point. If v''(t) is negative at a critical point, the flow rate is at a maximum there.

Finally, evaluate the flow rate at the endpoints of the interval (t = 0 and t = 90) and compare these values with the flow rates at the critical points to determine when the flow rate is at its maximum over the entire 90-day period.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Flow Rate

Flow rate refers to the volume of fluid that passes through a given surface per unit of time. In this context, it is essential to understand how the flow rate of the stream varies over the specified period. The flow rate can be represented mathematically as the derivative of the volume function, indicating how quickly water is flowing at any given time.

Integration

Integration is a fundamental concept in calculus that allows us to calculate the total accumulation of quantities, such as the total volume of water flowing over time. In this scenario, integrating the flow rate function over the 90-day period will provide the total volume of water that has flowed past the gauging station, which is crucial for understanding the stream's behavior.

Critical Points

Critical points are values of the independent variable where the derivative of a function is zero or undefined, indicating potential maximum or minimum values. To determine when the flow rate is at its maximum, one must find the critical points of the flow rate function and evaluate them to identify the maximum flow rate during the monitoring period.

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Critical Points

Related Practice

Textbook Question

Another way to approximate derivatives is to use the centered difference quotient: f' (a) ≈ f(a+h) - f(a- h) / 2h. Again, consider f(x) = √x.

c. Explain why it is not necessary to use negative values of h in the table of part (b).

Textbook Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 64 ft/s from a height of 32 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+64t+32.

c. What is the height of the stone at the highest point?

Textbook Question

Derivatives from a graph If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>

c. (f^-1)'(f(2))

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Textbook Question

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

c. Suppose the population density of the city remains constant from year to year at 1000 people mi². Determine the growth rate of the population in 2030.

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Textbook Question

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x) = (x−1) sin^−1 x on [−1,1]

Textbook Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 19.6 m/s from a height of 24.5 m above the ground. The height (in meters) of the stone above the ground t seconds after it is thrown is s(t) = -4.9t²+19.6t+24.5.

c. What is the height of the stone at the highest point?