Linear Approximation

Definition

Linear approximation is a method used to approximate the value of a function at a specific point by using the function's derivative and the value of the function at that point. It provides a way to estimate the behavior of a function near a particular point of interest.

5 Must Know Facts For Your Next Test

1. The linear approximation of a function $f(x)$ at a point $x_0$ is given by the formula: $L(x) = f(x_0) + f'(x_0)(x - x_0)$, where $f'(x_0)$ is the derivative of $f(x)$ evaluated at $x_0$.
2. Linear approximations are most accurate when evaluating the function near the point of approximation, $x_0$. The further away from $x_0$ the evaluation is made, the less accurate the linear approximation becomes.
3. The error in the linear approximation is given by the formula: $|f(x) - L(x)| \leq \frac{1}{2}|f''(c)||x - x_0|^2$, where $c$ is some value between $x$ and $x_0$, and $f''(c)$ is the second derivative of $f(x)$ evaluated at $c$.
4. Linear approximations are useful for quickly estimating the value of a function, especially when the function is difficult to evaluate directly. They are commonly used in numerical analysis, optimization, and other areas of applied mathematics.
5. The tangent plane to a surface $z = f(x, y)$ at the point $(x_0, y_0, f(x_0, y_0))$ is given by the equation: $z = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$, where $f_x$ and $f_y$ are the partial derivatives of $f$ with respect to $x$ and $y$, respectively.

Review Questions

• Explain how the linear approximation formula is derived and how it can be used to estimate the value of a function near a specific point.
  • The linear approximation formula is derived using the first-order Taylor polynomial of the function $f(x)$ around the point $x_0$. The formula $L(x) = f(x_0) + f'(x_0)(x - x_0)$ represents the linear function that best approximates the behavior of $f(x)$ near $x_0$. This formula can be used to quickly estimate the value of $f(x)$ for values of $x$ close to $x_0$ by evaluating the linear function $L(x)$ instead of the original function $f(x)$, which may be more difficult to compute. The linear approximation is most accurate when $x$ is close to $x_0$, and the error in the approximation can be bounded using the second derivative of $f(x)$.
• Describe the relationship between linear approximations and tangent planes, and explain how the tangent plane equation is derived.
  • The linear approximation of a function $f(x, y)$ at a point $(x_0, y_0)$ is closely related to the tangent plane to the surface $z = f(x, y)$ at that point. The tangent plane equation $z = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$ is derived by considering the linear approximation of $f(x, y)$ with respect to both $x$ and $y$ variables. The partial derivatives $f_x$ and $f_y$ represent the rates of change of the function in the $x$ and $y$ directions, respectively, and they define the orientation of the tangent plane. The tangent plane provides a local linear approximation of the surface near the point $(x_0, y_0, f(x_0, y_0))$.
• Analyze the accuracy of linear approximations and explain the factors that influence the error in the approximation. Discuss the practical implications of these error bounds.
  • The accuracy of linear approximations is determined by how close the value of $x$ is to the point of approximation, $x_0$. The error in the linear approximation is bounded by the formula $|f(x) - L(x)| \leq \frac{1}{2}|f''(c)||x - x_0|^2$, where $c$ is some value between $x$ and $x_0$, and $f''(c)$ is the second derivative of $f(x)$ evaluated at $c$. This means that the error in the linear approximation is proportional to the square of the distance between $x$ and $x_0$, as well as the magnitude of the second derivative of the function. The practical implication is that linear approximations are most useful when evaluating the function near the point of approximation, where the error is small. As the distance from $x_0$ increases, the linear approximation becomes less accurate, and higher-order approximations may be necessary. Understanding these error bounds is crucial for effectively using linear approximations in various applications, such as numerical analysis, optimization, and scientific computing.

Related terms

Tangent Plane: A tangent plane is the plane that best approximates the surface of a function at a given point. It is defined by the function's value and the function's partial derivatives at that point.

Differential: The differential of a function represents the change in the function's value due to small changes in its input variables. It is used to construct the linear approximation of the function.

Linearization: Linearization is the process of approximating a function by a linear function, which is the first-order Taylor polynomial of the function at a specific point.