Using Linear Approximation to Estimate Functions

Linear Approximation

Linear approximation, also known as tangent line approximation, is a method used to approximate the value of a function at a particular point using the equation of the tangent line at that point

Linear approximation, also known as tangent line approximation, is a method used to approximate the value of a function at a particular point using the equation of the tangent line at that point. It is based on the concept that for small variations around a point, a curve can be approximated by a straight line.

To understand the linear approximation method, let’s consider a function f(x) and a point (a, f(a)) on that function. The tangent line to the function at this point has the equation:

y = f(a) + f'(a)(x-a)

where f'(a) represents the derivative of the function evaluated at x = a.

Using this equation, we can approximate the value of the function at a nearby point x = b, where b is close to a. The linear approximation is given by:

f(b) ≈ f(a) + f'(a)(b – a)

The key idea is that for small changes in x around the point a, the value of the function can be approximated by the straight line defined by the tangent line.

To illustrate this, let’s consider a concrete example. Suppose we have the function f(x) = √x and we want to approximate the value of f(4.2).

First, we need to find the tangent line to the function at the point (a, f(a)). Let’s choose a = 4. The derivative of f(x) = √x is given by f'(x) = 1 / (2√x), so the derivative evaluated at x = 4 is f'(4) = 1 / (2√4) = 1/4.

Now, let’s plug these values into the linear approximation formula:

f(b) ≈ f(a) + f'(a)(b – a)
f(4.2) ≈ f(4) + f'(4)(4.2 – 4)
≈ √4 + (1/4)(4.2 – 4)
≈ 2 + (1/4)(0.2)
≈ 2 + 0.05
≈ 2.05

So, using linear approximation, we estimate that f(4.2) is approximately equal to 2.05.

It’s important to note that the accuracy of the linear approximation depends on how close the chosen point b is to the point a. The smaller the difference between a and b, the better the approximation. Additionally, linear approximation is most accurate for functions that are relatively smooth around the point a.

More Answers: