Linearization Of F(x) = 5e^x / X^4 At X = 1 Calculation And Explanation

by ADMIN 72 views

In calculus, linearization is a method of approximating the value of a function at a specific point using a linear function. This linear function, often called the tangent line approximation, provides a close estimate of the original function's behavior in a small neighborhood around the point of tangency. Linearization is a powerful tool with applications in various fields, including physics, engineering, and economics, where it simplifies complex problems by providing manageable approximations. In this article, we will delve into the process of calculating the linearization L(x) of the function f(x) = 5e^x / x^4 at x = 1. We will explore the steps involved, from finding the derivative of the function to constructing the linear approximation, and express the final answer in terms of e and x.

Understanding Linearization

Before we dive into the specific problem, let's clarify the concept of linearization. The linearization of a function f(x) at a point x = a is a linear function L(x) that approximates f(x) near x = a. The formula for L(x) is given by:

L(x) = f(a) + f'(a)(x - a)

Where:

  • f(a) is the value of the function at x = a.
  • f'(a) is the derivative of the function evaluated at x = a, representing the slope of the tangent line to the graph of f(x) at x = a.
  • (x - a) is the difference between the input x and the point of linearization a.

The essence of linearization lies in using the tangent line at a particular point to approximate the function's behavior in its vicinity. This is based on the idea that a smooth function, when observed at a sufficiently small scale, resembles a straight line. The linearization provides a straightforward and computationally efficient way to estimate function values, especially when the function is complex or difficult to evaluate directly.

Problem Statement

Our task is to calculate the linearization L(x) of the function f(x) = 5e^x / x^4 at the point x = 1. This involves finding the value of the function at x = 1, calculating its derivative, evaluating the derivative at x = 1, and then plugging these values into the linearization formula. Let's break down the process step by step.

Step 1: Evaluate f(x) at x = 1

The first step is to find the value of the function f(x) at x = 1. This is a straightforward substitution:

f(1) = 5e^1 / 1^4 = 5e

So, the value of the function at x = 1 is 5e, where e is the base of the natural logarithm, approximately equal to 2.71828.

Step 2: Find the Derivative f'(x)

Next, we need to find the derivative of the function f(x) = 5e^x / x^4. This requires applying the quotient rule, which states that if f(x) = u(x) / v(x), then:

f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2

In our case, u(x) = 5e^x and v(x) = x^4. Let's find their derivatives:

  • u'(x) = d/dx (5e^x) = 5e^x (since the derivative of e^x is e^x)
  • v'(x) = d/dx (x^4) = 4x^3 (using the power rule)

Now, applying the quotient rule:

f'(x) = [(5ex)(x4) - (5ex)(4x3)] / (x4)2

Simplify the expression:

f'(x) = (5e^x * x^4 - 20e^x * x^3) / x^8

We can factor out 5e^x * x^3 from the numerator:

f'(x) = 5e^x * x^3 (x - 4) / x^8

And then simplify by canceling out x^3:

f'(x) = 5e^x (x - 4) / x^5

So, the derivative of f(x) is f'(x) = 5e^x (x - 4) / x^5.

Step 3: Evaluate f'(x) at x = 1

Now, we need to evaluate the derivative f'(x) at x = 1:

f'(1) = 5e^1 (1 - 4) / 1^5

f'(1) = 5e (-3) / 1

f'(1) = -15e

Thus, the value of the derivative at x = 1 is -15e.

Step 4: Construct the Linearization L(x)

Finally, we can construct the linearization L(x) using the formula:

L(x) = f(a) + f'(a)(x - a)

We have a = 1, f(1) = 5e, and f'(1) = -15e. Plugging these values into the formula:

L(x) = 5e + (-15e)(x - 1)

Simplify the expression:

L(x) = 5e - 15e(x - 1)

Distribute the -15e:

L(x) = 5e - 15ex + 15e

Combine like terms:

L(x) = 20e - 15ex

So, the linearization L(x) of f(x) = 5e^x / x^4 at x = 1 is L(x) = 20e - 15ex.

Expressing the Answer

The linearization L(x) is expressed in terms of e and x as:

L(x) = 20e - 15ex

This linear function provides a good approximation of the original function f(x) = 5e^x / x^4 near x = 1. To illustrate, we can substitute values of x close to 1 into both f(x) and L(x) and compare the results. The closer x is to 1, the more accurate the approximation will be.

For instance, let's consider x = 1.1:

  • f(1.1) = 5e^1.1 / (1.1)^4 ≈ 12.42
  • L(1.1) = 20e - 15e(1.1) = 20e - 16.5e = 3.5e ≈ 9.46

The values are somewhat close, but as we move further from x = 1, the approximation becomes less accurate.

Now, let's consider x = 1.01, a value much closer to 1:

  • f(1.01) = 5e^1.01 / (1.01)^4 ≈ 13.66
  • L(1.01) = 20e - 15e(1.01) = 20e - 15.15e = 4.85e ≈ 13.19

As expected, the approximation is significantly better when x is closer to 1. This highlights the local nature of linearization; it provides an accurate approximation only in a small neighborhood around the point of tangency.

Visualizing the Linearization

To gain a better understanding of linearization, it's helpful to visualize the function and its linear approximation. Imagine the graph of f(x) = 5e^x / x^4. At the point x = 1, we draw a tangent line to the curve. This tangent line represents the linearization L(x) = 20e - 15ex. Close to x = 1, the tangent line closely follows the curve of f(x), demonstrating the accuracy of the linear approximation in this region. However, as we move away from x = 1, the tangent line and the curve diverge, indicating that the linear approximation becomes less reliable.

Applications of Linearization

Linearization is not just a theoretical concept; it has numerous practical applications. Here are a few examples:

  • Approximating Function Values: As we've seen, linearization can be used to estimate the value of a function at a point close to a known value. This is particularly useful when the function is complex or difficult to evaluate directly.
  • Error Analysis: Linearization helps in estimating the error in a function's value due to small changes in the input. For example, in physics, it can be used to approximate the change in a quantity due to a slight variation in a measurement.
  • Optimization: Linearization is employed in optimization algorithms to approximate the objective function and constraints, making it easier to find optimal solutions.
  • Control Systems: In control theory, linearization is used to model nonlinear systems as linear systems, which are easier to analyze and control.

Conclusion

In this article, we successfully calculated the linearization L(x) of the function f(x) = 5e^x / x^4 at x = 1. We followed a step-by-step process, including evaluating the function and its derivative at the point of linearization, and constructing the linear approximation using the formula L(x) = f(a) + f'(a)(x - a). The final answer, expressed in terms of e and x, is:

L(x) = 20e - 15ex

We also discussed the concept of linearization, its limitations, and its applications in various fields. Linearization provides a valuable tool for approximating function values and simplifying complex problems, demonstrating the power of calculus in solving real-world challenges.