Question

The growth rate of a certain tree (in feet) is given by \(y=\frac{2}{t+1}+e^{-t^{2} / 2}\), where \(t\) is time in years. Estimate the growth of the tree through the end of the second year by using Simpson's rule, using two subintervals. (Round the answer to the nearest hundredth.)

Short answer

The estimated growth of the tree through the end of the second year is approximately 3.41 feet.

Step-by-step solution

Understanding Simpson’s Rule

Simpson’s rule is a method for approximating the integral of a function using quadratic polynomials. It approximates the area under a curve by dividing it into an even number of subintervals and then fitting a parabola to each pair of intervals. The formula for Simpson’s rule is \( \int_a^b f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + \, \ldots \, + 4f(x_{n-1}) + f(x_n)] \), where \( h = \frac{b-a}{n} \). Here, \( n \) must be even.

Setting Up the Problem

We are tasked with estimating the integral of the function \( y = \frac{2}{t+1} + e^{-t^2 / 2} \) from \( t = 0 \) to \( t = 2 \) using Simpson’s rule with two subintervals. Given \( n = 2 \), the endpoints are \( a = 0 \) and \( b = 2 \). Therefore, \( h = \frac{b-a}{n} = \frac{2-0}{2} = 1 \). This means we evaluate the function at \( t = 0, 1, \) and \( 2 \).

Calculating Function Values at Specific Points

First, calculate \( y(0) = \frac{2}{0+1} + e^{-0^2/2} = 2 + 1 = 3 \). Then, calculate \( y(1) = \frac{2}{1+1} + e^{-1^2/2} = 1 + e^{-0.5} \). Approximate \( e^{-0.5} \) using a calculator to get \( y(1) \approx 1 + 0.60653 \approx 1.60653 \). Next, compute \( y(2) = \frac{2}{2+1} + e^{-2^2/2} = \frac{2}{3} + e^{-2} \). With approximation \( e^{-2} \approx 0.1353 \), we find \( y(2) \approx 0.66667 + 0.1353 \approx 0.80197 \).

Applying Simpson’s Rule

Using the function values from Step 3 and Simpson's Rule, we compute the integral: \[ \int_0^2 y(t) \, dt \approx \frac{1}{3} [y(0) + 4 \cdot y(1) + y(2)] = \frac{1}{3} [3 + 4 \cdot 1.60653 + 0.80197] \]. Calculate, \[ = \frac{1}{3} \times (3 + 6.42612 + 0.80197) = \frac{1}{3} \times 10.22809 \approx 3.40936 \]. Hence, the estimated growth through the end of the second year is approximately 3.41 feet.

Key concepts

Numerical Integration

Numerical integration is an essential concept in mathematics. It helps in approximating the integral of a function when finding the exact solution analytically is challenging or impossible.
Simpson's Rule is a popular method for numerical integration. Here's how it works:

• Simpson's Rule divides the integration interval into an even number of subintervals.
• It approximates the area under the curve using parabolas fitted to these subintervals.
• The Rule can provide a more accurate result compared to other methods like the Trapezoidal Rule, due to its fitting of quadratic polynomials.

When using Simpson's Rule, it's crucial to decide the number of subintervals (denoted as \( n \)). An even \( n \) ensures each set of three points forms a parabola.
In a real-world context, numerical integration and Simpson's Rule are widely used in physics, engineering, and other sciences to estimate solutions where precise calculation of integrals is pivotal.

Growth Rate Calculation

In the context of the given exercise, we are tasked with estimating the growth of a tree. Growth rate calculation is pivotal in understanding how quickly a process changes over time, especially in natural sciences like biology and environmental science.
For this exercise, we use the function \( y=\frac{2}{t+1}+e^{-t^{2}/2} \). Here's how we convert it into practical predictions:

• Calculate the values of the function at the chosen points \( t = 0, 1, \) and \( 2 \).
• Use approximations for exponential parts where necessary, such as using a calculator for \( e^{-0.5} \apprx 0.60653 \).

With these values, Simpson's Rule is applied to estimate the integral of the function, translating into the total growth through the specified period.
Such calculations are essential in assessing biological growth, providing insights into the expected size or outcome at the end of a time frame, which is crucial for planning and analysis in botany and ecology.

Mathematics Education

Mathematics education plays a fundamental role in equipping students with essential problem-solving skills. Concepts like numerical integration and growth rate calculations are vital in various advanced fields.
Here's why Simpson's Rule and foundational math skills matter:

• They offer students a toolset for approaching seemingly complex real-world problems.
• Understanding these concepts enables critical thinking and analytical reasoning.
• Educational exercises like this promote active learning, encouraging students to apply mathematical theories practically.

By integrating these concepts into the curriculum, educators can better prepare students for future challenges in academia and industry alike.