Linear Approximation
Linear approximation, also known as tangent line approximation, is a method used in calculus to estimate the value of a function near a particular point
Linear approximation, also known as tangent line approximation, is a method used in calculus to estimate the value of a function near a particular point. It involves determining the equation of the tangent line to the graph of the function at that point. By using this tangent line, we can obtain an approximate value for the function at a nearby point.
To perform a linear approximation, we start by selecting a point on the graph of the function that we want to approximate. Let’s say this point is (a, f(a)), where a is the x-coordinate and f(a) is the corresponding y-coordinate. To find the equation of the tangent line, we need to determine its slope. The slope is given by the derivative of the function evaluated at the point (a, f(a)). If the derivative is denoted as f'(a), then the slope of the tangent line is f'(a).
Once we have the slope, we can use the point-slope form of a linear equation (y – y1) = m(x – x1) to obtain the equation of the tangent line. Substituting the values of (a, f(a)) and f'(a) into the equation, we get y – f(a) = f'(a)(x – a).
Now that we have the equation of the tangent line, we can use it to estimate the value of the function at a nearby point. Let’s say we want to find the approximate value of the function at x = b, where b is close to a. We substitute b into the equation of the tangent line, which gives us an equation of the form y = f(a) + f'(a)(b – a). By evaluating this equation, we obtain our linear approximation for the function at x = b.
It is important to note that the accuracy of the linear approximation depends on how close the point of approximation is to the original point (a, f(a)). The smaller the interval between a and b, the more accurate the estimate will be. Additionally, this method assumes that the function is differentiable near the point of approximation.
More Answers:
Understanding the Inverse Sine Function and Evaluating y = sin⁻¹(u/a)Derivative of a Function with Power of u | Using the Chain Rule and Power Rule
The Chain Rule | Finding the Derivative of ln(u) with respect to x and the Importance of Knowing the Derivative of u.