Problem 3 - Calculus Early Transcendentals

Question

How is linear approximation used to approximate the value of a function \(f\) near a point at which \(f\) and \(f^{\prime}\) are easily evaluated?

Step-by-Step Solution

Answer

Answer: The main idea of linear approximation is to use the tangent line at a point (a, f(a)) to approximate the function's behavior near that point. To estimate the value of a function near a point where the function and its derivative can be easily evaluated, we can apply the equation \(f(x) \approx f(a) + f^{\prime}(a)(x - a)\) using the values of \(f(a)\) and \(f^{\prime}(a)\). The approximation is more accurate for values of x close to a and may become less accurate as x gets farther away from a.

Step 1: Understand linear approximation

Linear approximation is a method used to estimate the value of a function \(f(x)\) near a point \((a, f(a))\) where the function and its derivative are easily evaluated. The idea is to use the tangent line (the line which passes through \((a, f(a))\) and best approximates \(f(x)\) in the neighborhood of \(a\)) as an approximation of the function's behavior in the vicinity of the point.

Step 2: Find the equation of the tangent line

The tangent line at the point \((a, f(a))\) will have the same slope as the derivative of the function, \(f^{\prime}(a)\), at that point. So, the slope of the tangent line is given by \(f^{\prime}(a)\). To find the equation of the tangent line, we can use the point-slope form of a line, which is given by: \(y - f(a) = f^{\prime}(a)(x - a)\)

Step 3: Use the tangent line to approximate the function

Now that we have the equation of the tangent line, we can use it to approximate the value of the function \(f(x)\) close to the point \((a, f(a))\). \(f(x) \approx f(a) + f^{\prime}(a)(x - a)\) So, to approximate the value of a function \(f\) near a point where \(f\) and \(f^{\prime}\) are easily evaluated, we can simply plug in the values for \(f(a)\) and \(f^{\prime}(a)\) into the equation above, and then we can use the tangent line as an approximation for the function. Remember that this approximation is more accurate for values of x close to a. As x gets farther away from a, the approximation error may increase.