Don't need help with a, b, or ii. Only need help with iv and c. **Problem Statement:****(iv.)** Determine the population of fish in the lake at the end of the fourth year?**(c)** Evaluate the integral from 0 to 4 of the absolute value of the derivative of $ P(t) $, given by $ \int_{0}^{4} \left| P'(t) \right| dt $. **Problem 4: Fish Population Modeling in a Lake**Let $P(t)$ represent the population of fish (in thousands) in a lake. Suppose that the rate of change of population of a certain type of fish in a lake is given by $ P'(t) = t^3 - 2t^2 - 3t $, for $0 \le t \le 4$ years, and that the initial population of fish in the lake is 20 thousand.**(a) Graph $ P'(t) $ vs $ t $:**Carefully graph $ P'(t) $ vs $ t $, $ 0 \le t \le 4 $, by plotting points and then connecting the points with a smooth line.*Explanation of the Graph:*- The graph needs to plot $ P'(t) = t^3 - 2t^2 - 3t $ for the given interval.- Identify key points by substituting specific values of $ t $ (such as $ t = 0, 1, 2, 3, 4 $) into $ P'(t) $.- Plot these points on a coordinate system where the x-axis represents years ($ t $) from 0 to 4, and the y-axis represents the rate of change of the fish population ($ P'(t) $).- Once the points are plotted, connect them with a smooth line to represent the continuous nature of the function.**(b) Evaluate the following definite integrals and describe their biological meaning:**(i) $ \int_{0}^{1} P'(t) \, dt $.*Explanation:*- To evaluate this integral, compute the definite integral of $ P'(t) $ from $ t = 0 $ to $ t = 1 $.- The value obtained from this integral represents the total change in the fish population (in thousands) from year 0 to year 1. By evaluating this integral, we can understand the net effect of the fish population growth and decline over the first year considering the given rate of change.Note: The specific numeric evaluation of the integral is not provided in the text and requires performing the integration process. However, the integral calculation would typically involve finding the antiderivative of $ P'(t) $ and then applying the limits from 0 to 1.

Name: Don't need help with a, b, or ii. Only need help with iv and c. **Prob
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Description: Don't need help with a, b, or ii. Only need help with iv and c. **Problem Statement:****(iv.)** Determine the population of fish in the lake at the end of the fourth year?**(c)** Evaluate the integral from 0 to 4 of the absolute value of the derivati

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Answered: (iv.) Determine the population of fish…

Math

Calculus

(iv.) Determine the population of fish in the lake at the end of the fourth year? (c) Evaluate

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Don't need help with a, b, or ii. Only need help with iv and c.

**Problem Statement:**

**(iv.)** Determine the population of fish in the lake at the end of the fourth year?

**(c)** Evaluate the integral from 0 to 4 of the absolute value of the derivative of $ P(t) $, given by $ \int_{0}^{4} \left| P'(t) \right| dt $.

**Problem 4: Fish Population Modeling in a Lake**

Let $P(t)$ represent the population of fish (in thousands) in a lake. Suppose that the rate of change of population of a certain type of fish in a lake is given by $ P'(t) = t^3 - 2t^2 - 3t $, for $0 \le t \le 4$ years, and that the initial population of fish in the lake is 20 thousand.

**(a) Graph $ P'(t) $ vs $ t $:**
Carefully graph $ P'(t) $ vs $ t $, $ 0 \le t \le 4 $, by plotting points and then connecting the points with a smooth line.

*Explanation of the Graph:*
- The graph needs to plot $ P'(t) = t^3 - 2t^2 - 3t $ for the given interval.
- Identify key points by substituting specific values of $ t $ (such as $ t = 0, 1, 2, 3, 4 $) into $ P'(t) $.
- Plot these points on a coordinate system where the x-axis represents years ($ t $) from 0 to 4, and the y-axis represents the rate of change of the fish population ($ P'(t) $).
- Once the points are plotted, connect them with a smooth line to represent the continuous nature of the function.

**(b) Evaluate the following definite integrals and describe their biological meaning:**

(i) $ \int_{0}^{1} P'(t) \, dt $.

*Explanation:*
- To evaluate this integral, compute the definite integral of $ P'(t) $ from $ t = 0 $ to $ t = 1 $.
- The value obtained from this integral represents the total change in the fish population (in thousands) from year 0 to year 1.

By evaluating this integral, we can understand the net effect of the fish population growth and decline over the first year considering the given rate of change.

Note: The specific numeric evaluation of the integral is not provided in the text and requires performing the integration process. However, the integral calculation would typically involve finding the antiderivative of $ P'(t) $ and then applying the limits from 0 to 1.

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