Linear Approximation Of F(x) = 4x² - X At A = 1 Explained

Introduction

In the realm of calculus, linear approximation, also known as tangent line approximation, is a fundamental technique used to estimate the value of a function at a specific point by using the equation of its tangent line at a nearby point. This method is particularly useful when dealing with complex functions where direct calculation is cumbersome or impossible. The core idea behind linear approximation is that, within a sufficiently small neighborhood of a point, the tangent line closely resembles the original function. This approximation simplifies calculations and provides valuable insights into the behavior of the function. For our specific problem, we aim to find the linear approximation L(x) of the function f(x) = 4x² - x at the point a = 1. This involves determining the equation of the tangent line to the function at x = 1, which will serve as our linear approximation. Understanding linear approximation is crucial for various applications in science and engineering, including optimization problems, numerical analysis, and error estimation. It provides a powerful tool for simplifying complex problems and obtaining accurate estimates.

To fully grasp the concept, it's essential to delve into the mathematical underpinnings of linear approximation. The tangent line to a function at a point represents the best linear estimate of the function's behavior near that point. The slope of this tangent line is given by the derivative of the function at that point, which quantifies the instantaneous rate of change of the function. The equation of the tangent line, and hence the linear approximation, is constructed using the point-slope form of a line. This form requires the slope (derivative) and a point on the line (the function value at the given point). By calculating these values and plugging them into the point-slope form, we can derive the linear approximation L(x). This approximation allows us to estimate the function's value at points close to the point of tangency with reasonable accuracy. The accuracy of the approximation depends on how close the point of estimation is to the point of tangency; the closer the points, the more accurate the approximation. Linear approximation serves as a cornerstone for more advanced numerical methods and provides a foundational understanding of how functions behave locally.

Steps to Find the Linear Approximation

To find the linear approximation L(x) of the function f(x) = 4x² - x at the point a = 1, we follow a systematic approach that involves several key steps. These steps ensure that we accurately determine the equation of the tangent line, which represents the linear approximation. First, we need to evaluate the function at the given point a. This provides us with the y-coordinate of the point on the function where the tangent line will touch. This value is crucial for defining the point through which the tangent line passes. Next, we must calculate the derivative of the function. The derivative, denoted as f'(x), represents the instantaneous rate of change of the function at any point x. It is essential for determining the slope of the tangent line at the specific point a. Once we have the derivative, we evaluate it at the point a to find the slope of the tangent line at that point. This slope, f'(a), is a critical component in the equation of the tangent line. With the slope and a point on the line, we can construct the equation of the tangent line using the point-slope form. This equation represents the linear approximation L(x) of the function near the point a. The final step is to express the linear approximation in the form L(x) = f(a) + f'(a)(x - a). This equation provides a simple and accurate way to estimate the value of the function near the point a. Understanding and meticulously following these steps is vital for correctly determining the linear approximation.

The process of finding the linear approximation involves a combination of algebraic manipulation and calculus concepts. We begin by substituting the value of a into the original function to find f(a). This gives us the y-coordinate of the point of tangency. Then, we apply the rules of differentiation to find the derivative f'(x). Differentiation is a fundamental operation in calculus that allows us to find the rate of change of a function. For polynomial functions like f(x) = 4x² - x, the power rule of differentiation is particularly useful. The power rule states that the derivative of x^n is nx^(n-1). Applying this rule to each term in the function, we can find the derivative f'(x). Once we have the derivative, we substitute the value of a into f'(x) to find the slope of the tangent line at that point. This slope is crucial for constructing the equation of the tangent line. Finally, we use the point-slope form of a line, which is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. Substituting f(a) for y₁, f'(a) for m, and a for x₁, we can rewrite the equation in the form L(x) = f(a) + f'(a)(x - a). This equation represents the linear approximation of the function near the point a. By understanding each step and the underlying calculus principles, we can confidently find the linear approximation of various functions.

Calculation of the Linear Approximation

To calculate the linear approximation L(x) for the function f(x) = 4x² - x at a = 1, we will meticulously follow the steps outlined earlier. First, we need to evaluate the function at the point a = 1. This means substituting x = 1 into the function f(x):

f(1) = 4(1)² - 1 = 4 - 1 = 3

So, f(1) = 3. This value represents the y-coordinate of the point on the function where the tangent line will touch. Next, we need to find the derivative of the function f(x). The derivative, denoted as f'(x), represents the instantaneous rate of change of the function. Using the power rule of differentiation, we differentiate each term in the function:

f'(x) = d/dx (4x²) - d/dx (x) = 8x - 1

Thus, the derivative of f(x) is f'(x) = 8x - 1. Now, we need to evaluate the derivative at the point a = 1. This will give us the slope of the tangent line at that point:

f'(1) = 8(1) - 1 = 8 - 1 = 7

So, f'(1) = 7. This is the slope of the tangent line to the function at x = 1. With the slope and the point (1, 3), we can now construct the equation of the tangent line using the point-slope form. The point-slope form is given by:

y - y₁ = m(x - x₁)

Substituting m = 7, x₁ = 1, and y₁ = 3, we get:

y - 3 = 7(x - 1)

Now, we can rewrite this equation in the form L(x) = f(a) + f'(a)(x - a). Adding 3 to both sides and simplifying, we get:

L(x) = 7(x - 1) + 3

L(x) = 7x - 7 + 3

L(x) = 7x - 4

Therefore, the linear approximation L(x) of the function f(x) = 4x² - x at the point a = 1 is L(x) = 7x - 4. This linear function provides a good approximation of the original function near x = 1. The accuracy of this approximation decreases as we move further away from x = 1, but within a small neighborhood of x = 1, it provides a reliable estimate of the function's behavior.

The Resulting Linear Approximation

Having meticulously followed the steps for calculating the linear approximation, we have arrived at the result: L(x) = 7x - 4. This equation represents the tangent line to the function f(x) = 4x² - x at the point a = 1. The linear approximation L(x) provides a simplified representation of the function's behavior in the vicinity of x = 1. It is a straight line that closely approximates the curve of the original function near this point. The accuracy of this approximation is highest when x is close to 1, and it gradually decreases as x moves further away. The linear approximation serves as a valuable tool for estimating function values without directly evaluating the original function, which can be particularly useful for complex functions or in situations where computational efficiency is crucial. The equation L(x) = 7x - 4 clearly shows the linear relationship, with a slope of 7 and a y-intercept of -4. This linear function can be easily graphed and used to visualize the approximation. Understanding the linear approximation allows us to make quick estimations and gain insights into the local behavior of the function around the point of tangency.

The result L(x) = 7x - 4 can be interpreted graphically as the equation of a straight line that is tangent to the parabola f(x) = 4x² - x at the point (1, 3). The slope of this tangent line is 7, which represents the instantaneous rate of change of the function at x = 1. The y-intercept of the line is -4, which is the point where the line intersects the y-axis. This linear approximation can be used to estimate the values of f(x) for x values close to 1. For example, if we want to estimate f(1.1), we can use L(1.1):

L(1.1) = 7(1.1) - 4 = 7.7 - 4 = 3.7

This estimate is close to the actual value of f(1.1), which can be calculated as:

f(1.1) = 4(1.1)² - 1.1 = 4(1.21) - 1.1 = 4.84 - 1.1 = 3.74

The difference between the estimated value and the actual value is small, illustrating the accuracy of the linear approximation near the point x = 1. However, as we move further away from x = 1, the difference between L(x) and f(x) will increase. This is a fundamental characteristic of linear approximations: they are most accurate near the point of tangency and become less accurate as we move away from that point. The linear approximation provides a powerful tool for simplifying calculations and gaining a local understanding of the function's behavior. It is a key concept in calculus and has numerous applications in various fields.

Conclusion

In conclusion, we have successfully found the linear approximation L(x) of the function f(x) = 4x² - x at the point a = 1. The linear approximation, calculated to be L(x) = 7x - 4, represents the equation of the tangent line to the function at the specified point. This linear function provides a close estimate of the original function's values in the immediate vicinity of x = 1. The process involved evaluating the function and its derivative at a = 1, and then using the point-slope form of a line to construct the equation of the tangent line. Linear approximation is a crucial concept in calculus, offering a simplified way to estimate function values and understand local behavior. It has wide-ranging applications in various fields, including engineering, physics, and economics, where complex functions often need to be approximated for practical calculations.

Throughout this process, we emphasized the importance of understanding the underlying principles and following a systematic approach. The steps involved—evaluating the function, finding the derivative, and constructing the tangent line equation—are fundamental to many calculus problems. The ability to find linear approximations is not only valuable for estimation purposes but also lays the foundation for more advanced techniques, such as numerical methods for solving differential equations and optimization problems. The linear approximation L(x) = 7x - 4 serves as a tangible example of how a complex function can be locally represented by a linear function, providing a clear and intuitive understanding of the function's behavior near a specific point. This concept is essential for anyone studying calculus and its applications, offering a powerful tool for simplifying complex problems and gaining insights into the behavior of functions.