Linearization Of F(x) = 7e^x / X^2 At X = 1 Calculation Guide

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In the realm of calculus, linearization is a powerful technique used to approximate the value of a function at a particular point using its tangent line. This method is especially valuable when dealing with complex functions or when an exact calculation is difficult or impossible. In this comprehensive guide, we will delve into the process of calculating the linearization L of the function f(x) = 7e^x / x^2 at x = 1. We will express our answer in terms of e and x, providing a step-by-step explanation to ensure a clear understanding of the concepts involved.

Understanding Linearization

Before we dive into the specific problem, let's first establish a solid understanding of what linearization entails. At its core, linearization involves approximating a function f(x) near a point x = a using a linear function, which is essentially the equation of the tangent line to the graph of f(x) at that point. This linear approximation, denoted as L(x), provides a close estimate of f(x) for values of x that are sufficiently close to a.

The formula for the linearization L(x) of a function f(x) at x = a 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
  • (x - a) represents the change in x from the point a

This formula essentially constructs the equation of the tangent line to the graph of f(x) at the point (a, f(a)) using the point-slope form of a line. The slope of the tangent line is given by the derivative f'(a), and the point (a, f(a)) lies on the line.

Step-by-Step Calculation of Linearization

Now that we have a firm grasp of the concept of linearization, let's apply it to our specific problem: finding the linearization L of the function f(x) = 7e^x / x^2 at x = 1. We will follow these steps:

1. Evaluate the function at x = 1

The first step is to determine the value of the function f(x) at the point x = 1. This will give us the y-coordinate of the point on the graph of f(x) where we want to find the tangent line. Substituting x = 1 into the function, we get:

f(1) = 7e^1 / 1^2 = 7e

Therefore, f(1) = 7e.

2. Find the derivative of the function

Next, we need to find the derivative of the function f(x). This will give us the slope of the tangent line at any point on the graph of f(x). To find the derivative of f(x) = 7e^x / x^2, we will use the quotient rule, which states that the derivative of a quotient of two functions u(x) and v(x) is given by:

(u(x) / v(x))' = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2

In our case, u(x) = 7e^x and v(x) = x^2. Let's find the derivatives of u(x) and v(x):

u'(x) = (7e^x)' = 7e^x v'(x) = (x^2)' = 2x

Now, applying the quotient rule, we get the derivative of f(x):

f'(x) = (7e^x * x^2 - 7e^x * 2x) / (x2)2 f'(x) = (7e^x * x^2 - 14xe^x) / x^4 f'(x) = 7e^x(x - 2) / x^3

3. Evaluate the derivative at x = 1

Now that we have the derivative f'(x), we need to evaluate it at x = 1. This will give us the slope of the tangent line specifically at the point x = 1. Substituting x = 1 into the derivative, we get:

f'(1) = 7e^1(1 - 2) / 1^3 f'(1) = 7e(-1) / 1 f'(1) = -7e

Therefore, f'(1) = -7e.

4. Plug the values into the linearization formula

We have now calculated all the necessary components to construct the linearization L(x). We have f(1) = 7e, f'(1) = -7e, and a = 1. Plugging these values into the linearization formula, we get:

L(x) = f(1) + f'(1)(x - 1) L(x) = 7e + (-7e)(x - 1) L(x) = 7e - 7ex + 7e L(x) = 14e - 7ex

Expressing the Answer

Therefore, the linearization L of the function f(x) = 7e^x / x^2 at x = 1 is:

L(x) = 14e - 7ex

This equation represents the tangent line to the graph of f(x) at the point (1, 7e). It provides a linear approximation of f(x) for values of x close to 1.

Importance of Linearization

Linearization is a fundamental concept in calculus with numerous applications in various fields, including physics, engineering, and economics. Here are some key reasons why linearization is important:

  • Approximating Complex Functions: Linearization allows us to approximate the behavior of complex functions with simpler linear functions. This is particularly useful when dealing with functions that are difficult to evaluate directly or when we need a quick estimate of the function's value near a specific point.
  • Simplifying Calculations: Linear approximations can significantly simplify calculations. For example, instead of evaluating a complex function at a specific point, we can evaluate its linear approximation, which is a much simpler linear expression.
  • Error Analysis: Linearization helps in understanding the error introduced when approximating a function. By comparing the actual value of the function with its linear approximation, we can estimate the magnitude of the error and determine the range of values for which the approximation is accurate.
  • Optimization Problems: Linearization plays a crucial role in optimization problems, where we seek to find the maximum or minimum value of a function. By linearizing the function around a critical point, we can analyze its behavior and determine whether it is a local maximum, local minimum, or a saddle point.
  • Numerical Methods: Linearization is a fundamental building block in many numerical methods for solving equations, approximating integrals, and solving differential equations. These methods often rely on approximating functions with linear functions over small intervals.

Conclusion

In this guide, we have explored the concept of linearization and its application to the function f(x) = 7e^x / x^2 at x = 1. We have demonstrated a step-by-step approach to calculating the linearization L(x) = 14e - 7ex, expressing the answer in terms of e and x. Furthermore, we have highlighted the importance of linearization as a powerful tool for approximating functions, simplifying calculations, and solving various problems in mathematics and other disciplines. Understanding linearization provides a valuable foundation for further exploration of calculus and its applications.

This method is especially valuable when dealing with complex functions or when an exact calculation is difficult or impossible. The linearization provides a close estimate of f(x) for values of x that are sufficiently close to a. The formula for the linearization L(x) of a function f(x) at x = a 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. (x - a) represents the change in x from the point a.

To find the derivative of f(x) = 7e^x / x^2, we will use the quotient rule, which states that the derivative of a quotient of two functions u(x) and v(x) is given by: (u(x) / v(x))' = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2. In our case, u(x) = 7e^x and v(x) = x^2. Let's find the derivatives of u(x) and v(x): u'(x) = (7e^x)' = 7e^x. v'(x) = (x^2)' = 2x. Now, applying the quotient rule, we get the derivative of f(x): f'(x) = (7e^x * x^2 - 7e^x * 2x) / (x2)2.

Linearization is a fundamental concept in calculus with numerous applications in various fields, including physics, engineering, and economics. Linearization allows us to approximate the behavior of complex functions with simpler linear functions. This is particularly useful when dealing with functions that are difficult to evaluate directly or when we need a quick estimate of the function's value near a specific point.