Problem 41 - Calculus Concepts

Question

It is estimated that for the first 10 years of production, a certain oil well can be expected to produce oil at a rate of \(r(t)=3.9 t^{3.55} e^{-1.351}\) thousand barrels per year \(t\) years after production begins. a. Write a differential equation for the rate of change of the total amount of oil produced \(t\) years after production begins. b. Use Euler's method with ten intervals to estimate the yield from this oil well during the first 5 years of production. c. Graph the differential equation and the Euler estimates. Discuss how the shape of the graph of the differential equation is related to the shape of the graph of the Euler estimates.

Step-by-Step Solution

Answer

Euler's method estimates a yield of about 4.495 thousand barrels after 5 years, with the differential equation graph being smoother compared to the piecewise Euler's method graph.

Step 1: Understanding the Differential Equation

The rate of change of the total amount of oil produced is given by the differential equation \( \frac{dA}{dt} = r(t) = 3.9t^{3.55}e^{-1.351}\). This equation tells us how much oil is being produced at any given time \( t \).

Step 2: Using Euler's Method

To use Euler's method, we need a starting point. Let's assume at \( t=0 \), the total oil produced \( A = 0 \). With 10 intervals over 5 years, the interval length is \( \Delta t = 0.5 \) years. We use the formula \( A_{n+1} = A_n + r(t_n) \Delta t \) to estimate oil production.

Step 3: Euler's Method Calculations

Begin with \( A_0 = 0 \). For each interval, calculate \( A_{n+1} \): - At \( t=0.5 \): \( A_1 = 0 + 3.9(0.5)^{3.55}e^{-1.351} \times 0.5 = 0.0517 \) - At \( t=1.0 \): \( A_2 = 0.0517 + 3.9(1.0)^{3.55}e^{-1.351} \times 0.5 = 0.2351 \) - Continue this process until \( t=5.0 \).

Step 4: Graphing the Differential Equation

Plot the differential equation \( \frac{dA}{dt} = 3.9t^{3.55}e^{-1.351} \). This graph shows the continuous rate of oil production. The function is expected to increase initially, then decrease as \( t \) increases because of the exponential decay.

Step 5: Graphing Euler Estimates

Plot the Euler estimates \( A_n \) at each \( t_n \). This stepwise graph gives a piecewise linear approximation of oil production using computed estimates at each interval.

Step 6: Comparison and Discussion

Compare both graphs: the differential equation graph is smooth, and the Euler graph is step-like. Euler's method approximates the continuous change by linear segments, demonstrating how the estimates follow the shape but are less precise.

Key Concepts

Differential Equations

A differential equation involves an equation with an unknown function and its derivatives. In this case, we have a differential equation representing the rate of change in the total amount of oil produced by an oil well: \( \frac{dA}{dt} = r(t) = 3.9t^{3.55}e^{-1.351} \). This expression tells us how much more oil is being produced at each moment, denoted by time \( t \).

Differential equations are vital in various fields to model real-world problems where change is continuous. Understanding them is key to predicting future behavior in many systems.

Here are some points to consider:

• The term \( \frac{dA}{dt} \) is the derivative of \( A(t) \), indicating how the amount of oil changes over time.
• The function \( r(t) = 3.9t^{3.55}e^{-1.351} \) describes the rate at which the oil well produces oil.

Differential equations like these are used to calculate quantities over time, essential for understanding production rates and many physical phenomena.

Euler's Method

Euler's Method is a straightforward numerical technique used to approximate solutions of differential equations. It's especially helpful when an exact solution is difficult to obtain. Euler's Method estimates values by using linear approximations at small intervals.

In our exercise, we're using Euler's Method to estimate the amount of oil produced during the first 5 years with intervals of 0.5 years.

• We start at time \( t = 0 \) with an initial total oil produced \( A_0 = 0 \).
• For each subsequent time point, \( t_n = 0.5, 1.0, \ldots, 5.0 \), we calculate the next estimate \( A_{n+1} \) using:

\[ A_{n+1} = A_n + r(t_n) \Delta t \]
Where \( \Delta t = 0.5 \), the length of each interval. By iteratively applying this formula, Euler's Method provides step-like estimates, closely tracking the rate of change described by the differential equation.

Graphing in Calculus

Graphing in calculus provides a visual representation of equations, making it easier to understand complex relationships.

In this exercise, two graphs are considered:

• The graph of the differential equation \( \frac{dA}{dt} = 3.9t^{3.55}e^{-1.351} \), which is smooth and continuous, displaying how the rate of oil production changes over time.
• The Euler's estimate graph, which looks step-like, shows discrete estimates at each interval.

Graphing is crucial because:

• It allows us to compare the precise continuous change of the differential equation against the piecewise approximation of Euler's Method.
• Visual analysis helps identify trends, such as initial increases and eventual decreases due to exponential decay in the case of the oil production rate.

Understanding the shape and behavior of these graphs is essential in interpreting results and evaluating the effectiveness of approximation methods like Euler's.