Problem 3 - Active Calculus

Question

Find a good numerical approximation to \(F(9)\) for the function with the properties that \(F^{\prime}(x)=e^{-x^{2} / 5}\) and \(F(0)=2\) \(F(9) \approx\) ___________.

Step-by-Step Solution

Answer

3.77

Step 1 - Understand Initial Conditions and Derivative

Identify the given conditions and the derivative function. The function has properties: the derivative is given by \( F'(x) = e^{-x^2 / 5} \) and the initial value \( F(0) = 2 \).

Step 2 - Set Up the Integral

To find \( F(x) \) from the derivative, integrate the function \( F'(x) \). Use the integral: \[ F(x) = F(0) + \int_{0}^{x} F'(t) \, dt \] where \( t \) is a dummy variable.

Step 3 - Plug in Given Values

Plug in the given values into the integral formula. For \( F(9) \), we get: \[ F(9) = 2 + \int_{0}^{9} e^{-t^2 / 5} \, dt \].

Step 4 - Approximate the Integral

Numerically approximate the integral \( \int_{0}^{9} e^{-t^2 / 5} \, dt \). This can be calculated using a numerical method such as Simpson's Rule or a numerical integration tool. The approximate value of the integral is around 1.77.

Step 5 - Calculate the Final Value

Add the initial value \( F(0) = 2 \) to the approximated integral value. Thus, \[ F(9) \approx 2 + 1.77 = 3.77 \].

Key Concepts

Initial Conditions

Initial conditions in calculus are the values given that help us solve differential equations uniquely. For example, in the exercise provided, you have the initial condition given as:

• \(F(0) = 2\)

This condition means that when the input using certain values (in this case, \(x = 0\)), the output of the function \(F\) equals 2. Initial conditions are crucial because they allow us to solve for constants that arise when integrating functions. Essentially, they anchor our solution to a specific point, providing one definite solution among many possible ones.

Integration of Functions

Integration in calculus is the method of finding the area under a curve represented by a function. In the provided problem, we have the derivative of a function, given by:

• \(F'(x) = e^{-x^2 / 5}\)

To find the original function \(F(x)\), we need to integrate this function. The integration process is expressed as:

• \[ F(x) = F(0) + \int_{0}^{x} F'(t) \, dt \]

In this integral, \(t\) is a dummy variable representing the input of the function within the integral. This formula effectively reverse-engineers the antiderivative of \(F'(x)\) to give us \(F(x)\). By recognizing the initial condition, we anchored our integral and used it to compute a specific value:

Numerical Approximation Methods

Numerical approximation methods are techniques used to estimate the values of integrals, given that some integrals are difficult or impossible to solve analytically. For this exercise, one possible technique is Simpson's Rule, which uses quadratic polynomials to approximate the integral value. Another approach might be a numerical integration tool that utilizes advanced algorithms. The integral in our case is:

• \[ \int_{0}^{9} e^{-t^2 / 5} \, dt \]

This integral does not have a simple closed-form solution. Therefore, we use numerical methods for an estimate. For instance, Simpson's Rule or a modern tool might give us an approximation of around\(1.77\). Finally, adding this value to the initial condition\( F(0) = 2\):

• \(F(9) \approx 2 + 1.77 = 3.77\)

Results from numerical methods give us highly accurate approximations, helping solve practical problems when exact values are unattainable.