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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
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All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 53a
Chapter 4, Problem 53a

Mean Value Theorem The population of a culture of cells grows according to the function P(t) = 100t / t+1, where t ≥ 0 is measured in weeks.






a. What is the average rate of change in the population over the interval [0, 8]?

Verified step by step guidance
1
Identify the function given: \( P(t) = \frac{100t}{t+1} \). This represents the population of cells as a function of time \( t \).
The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is given by the formula: \( \frac{f(b) - f(a)}{b - a} \).
Substitute the given interval \([0, 8]\) into the formula. Here, \( a = 0 \) and \( b = 8 \).
Calculate \( P(8) \) by substituting \( t = 8 \) into the function: \( P(8) = \frac{100 \times 8}{8 + 1} \).
Calculate \( P(0) \) by substituting \( t = 0 \) into the function: \( P(0) = \frac{100 \times 0}{0 + 1} \). Then, use these values to find the average rate of change: \( \frac{P(8) - P(0)}{8 - 0} \).

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Key Concepts

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

Average Rate of Change

The average rate of change of a function over an interval [a, b] is calculated as the difference in the function's values at the endpoints divided by the difference in the input values. Mathematically, it is expressed as (f(b) - f(a)) / (b - a). This concept is crucial for understanding how a function behaves over a specific interval, providing insight into its overall trend.
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Average Value of a Function

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For the given population function P(t) = 100t / (t + 1), evaluating it at t = 0 and t = 8 will yield the population sizes at these points. This step is essential for calculating the average rate of change over the specified interval.
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Continuous Functions

A continuous function is one that does not have any breaks, jumps, or holes in its graph over its domain. The function P(t) = 100t / (t + 1) is continuous for t ≥ 0, which ensures that the average rate of change can be accurately calculated over the interval [0, 8]. Understanding continuity is important for applying the Mean Value Theorem and other calculus concepts.
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Related Practice
Textbook Question

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs \(p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns \)w/hour.


e. Should the optimal speed be increased or decreased (compared with part (d)) if L is increased from 400 mi to 500 mi? Explain.

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Textbook Question

Mean Value Theorem The population of a culture of cells grows according to the function P(t) = 100t / t+1, where t ≥ 0 is measured in weeks.


b. At what point of the interval [0, 8] is the instantaneous rate of change equal to the average rate of change?

Textbook Question

Growth rate of bamboo Bamboo belongs to the grass family and is one of the fastest growing plants in the world.


a. A bamboo shoot was 500 cm tall at 10:00 A.M. and 515 cm tall at 3:00 P.M. Compute the average growth rate of the bamboo shoot in cm/hr over the period of time from 10:00 A.M. to 3:00 P.M.

Textbook Question

First Derivative Test


a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).


f(x) = x - 2 tan⁻¹ x on [-√3,√3)

Textbook Question

First Derivative Test


a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).


f(x) = x²/(x² - 1) on [-4,4]

1
views
Textbook Question

First Derivative Test


a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).


f(x) = 2x⁵ - 5x⁴ - 10x³ + 4 on [-2,4]