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Consider a forest where the population of a particular plant species grows exponentially. In a real-world scenario, we often deal with systems where the analytical function describing the phenomenon is not available. In such cases, numerical methods come in handy. For the sake of this task, however, you are provided with an analytical function so that you can compare the results of the numerical methods to some ground truth. The population P(t) of the plants at time t (in years) is given by the equation: P(t) = 200 0.03 t You are tasked with estimating the rate of change of the plant population at t = 5 years using numerical differentiation methods. First, compute the value of P'(t) at t = 5 analytically. Then, estimate P'(t) at t = 5 years using the following numerical differentiation methods: ⚫ forward difference method (2nd-order accurate) 3 ⚫ backward difference method (2nd-order accurate) ⚫ central difference method (2nd-order accurate) Use h = 0.5 as the step size and round all intermediate calculations to a precision of 10-6. Compare the results with the analytical solution. Which method provides the best approximation? When should you use the backward, forward, and central difference methods? Next, use a smaller step size, h = 0.1, for the 2nd-order central difference and compare the result with the 2nd-order central difference for h = 0.5. What is the impact of decreasing the step size? How does this change compare to shrinking h by a factor of 5? Afterwards, use the 4th-order central difference method with step size h = 0.5. Compare the result to the 2nd-order central difference method with h = 0.5. How does this change compare to doubling the order? Finally, consider how numerical differentiation methods can help to assess the impact of forest manage- ment interventions on the plant growth.

Consider a forest where the population of a particular plant species grows exponentially. In a real-world scenario, we often deal with systems where the analytical function describing the phenomenon is not available. In such cases, numerical methods come in handy. For the sake of this task, however, you are provided with an analytical function so that you can compare the results of the numerical methods to some ground truth. The population P(t) of the plants at time t (in years) is given by the equation: P(t) = 200 0.03 t You are tasked with estimating the rate of change of the plant population at t = 5 years using numerical differentiation methods. First, compute the value of P'(t) at t = 5 analytically. Then, estimate P'(t) at t = 5 years using the following numerical differentiation methods: ⚫ forward difference method (2nd-order accurate) 3 ⚫ backward difference method (2nd-order accurate) ⚫ central difference method (2nd-order accurate) Use h = 0.5 as the step size and round all intermediate calculations to a precision of 10-6. Compare the results with the analytical solution. Which method provides the best approximation? When should you use the backward, forward, and central difference methods? Next, use a smaller step size, h = 0.1, for the 2nd-order central difference and compare the result with the 2nd-order central difference for h = 0.5. What is the impact of decreasing the step size? How does this change compare to shrinking h by a factor of 5? Afterwards, use the 4th-order central difference method with step size h = 0.5. Compare the result to the 2nd-order central difference method with h = 0.5. How does this change compare to doubling the order? Finally, consider how numerical differentiation methods can help to assess the impact of forest manage- ment interventions on the plant growth.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 9T
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Consider a forest where the population of a particular plant species grows exponentially. In a real-world scenario, we often deal with systems where the analytical function describing the phenomenon is not available. In such cases, numerical methods come in handy. For the sake of this task, however, you are provided with an analytical function so that you can compare the results of the numerical methods to some ground truth. The population P(t) of the plants at time t (in years) is given by the equation: P(t) = 200 0.03 t You are tasked with estimating the rate of change of the plant population at t = 5 years using numerical differentiation methods. First, compute the value of P'(t) at t = 5 analytically. Then, estimate P'(t) at t = 5 years using the following numerical differentiation methods: ⚫ forward difference method (2nd-order accurate) 3 ⚫ backward difference method (2nd-order accurate) ⚫ central difference method (2nd-order accurate) Use h = 0.5 as the step size and round all intermediate calculations to a precision of 10-6. Compare the results with the analytical solution. Which method provides the best approximation? When should you use the backward, forward, and central difference methods? Next, use a smaller step size, h = 0.1, for the 2nd-order central difference and compare the result with the 2nd-order central difference for h = 0.5. What is the impact of decreasing the step size? How does this change compare to shrinking h by a factor of 5? Afterwards, use the 4th-order central difference method with step size h = 0.5. Compare the result to the 2nd-order central difference method with h = 0.5. How does this change compare to doubling the order? Finally, consider how numerical differentiation methods can help to assess the impact of forest manage- ment interventions on the plant growth.
Transcribed Image Text:Consider a forest where the population of a particular plant species grows exponentially. In a real-world scenario, we often deal with systems where the analytical function describing the phenomenon is not available. In such cases, numerical methods come in handy. For the sake of this task, however, you are provided with an analytical function so that you can compare the results of the numerical methods to some ground truth. The population P(t) of the plants at time t (in years) is given by the equation: P(t) = 200 0.03 t You are tasked with estimating the rate of change of the plant population at t = 5 years using numerical differentiation methods. First, compute the value of P'(t) at t = 5 analytically. Then, estimate P'(t) at t = 5 years using the following numerical differentiation methods: ⚫ forward difference method (2nd-order accurate) 3 ⚫ backward difference method (2nd-order accurate) ⚫ central difference method (2nd-order accurate) Use h = 0.5 as the step size and round all intermediate calculations to a precision of 10-6. Compare the results with the analytical solution. Which method provides the best approximation? When should you use the backward, forward, and central difference methods? Next, use a smaller step size, h = 0.1, for the 2nd-order central difference and compare the result with the 2nd-order central difference for h = 0.5. What is the impact of decreasing the step size? How does this change compare to shrinking h by a factor of 5? Afterwards, use the 4th-order central difference method with step size h = 0.5. Compare the result to the 2nd-order central difference method with h = 0.5. How does this change compare to doubling the order? Finally, consider how numerical differentiation methods can help to assess the impact of forest manage- ment interventions on the plant growth.
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