![]() ![]() Thus, we define the function as f(x) = √x and a = 9. What Is √8 Using Linear Approximation Formula? The linear approximation of a function f(x) at a fixed value x = a is given by L(x) = f(a) + f '(a) (x - a). The linear approximation formula is based on the equation of the tangent line of a function at a fixed point. What Is Linear Approximation Formula Based On? The linear approximation is denoted by L(x) and is found using the formula L(x) = f(a) + f '(a) (x - a), where f '(a) is the derivative of f(x) at a x = a. We can use the linear approximation of a function f(x) to find the values of f(x) at the nearest values of a fixed number x = a. The linear approximation L(x) of a function f(x) at x = a is, L(x) = f(a) + f '(a) (x - a). The linear approximation formula, as its name suggests, is a function that is used to approximate the value of a function at the nearest values of a fixed value. (for more detailed information of finding this derivative, click here) We have to find the linear approximation of f(x) at a = 4.į ' (x) = d/dx (√x) = d/dx (x 1/2) = 1/2 . ![]() Using this, find the approximate value of √4.04. The linear approximation formula of f(x) is,Īnswer: The equation of linear approximation is, L(x) = -x + π/2.Įxample 2: Find the linear approximation of f(x) = √x at x = 4. We have to find the linear approximation of f(x) at a = π/2. Examples Using Linear Approximation FormulaĮxample 1: Find the equation of linear approximation of the function f(x) = cos x at x = π/2. We will see the linear approximation formula in the upcoming section. Math will no longer be a tough subject, especially when you understand the concepts through visualizations. f '(a) is the derivative of f(x) at x = a.L(x) is the linear approximation of f(x) at x = a.As we know the slope of this tangent is the derivative f '(a), its equation using the point-slope form is: ![]() Let us find the equation of a tangent line that is drawn to the curve y = f(x) at the point x = a (or) (a, f(a)). Linear Approximation FormulaĪs we discussed in the previous section, the linear approximation formula is nothing but the equation of a tangent line. This concept is known as the linear approximation and since it is done using the tangent line, it is also known as the tangent line approximation. If we find the equation of the tangent line at the given point, the value of the function at any point that is very close to the given point can be approximated using the equation of the tangent line. As soon as we see a curve (of a function) and a point on it, we remember the concept of the tangent line. The linear approximation of a function is nothing but approximating the value of the function at a point using a line. Let us learn more about this formula in the upcoming sections. Thus, the linear approximation formula is an application of derivatives. ![]() i.e., the slope of the tangent line is f'(a). We know that the slope of the tangent that is drawn to a curve y = f(x) at x = a is its derivative at that point. The concept behind the linear approximation formula is the equation of a tangent line. ![]()
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