The Local Linear Approximation Calculator allows you to find the linear approximation to a curve using the equation of the tangent line. To use it, you just have to follow these simple steps:
Linear approximation is a method for approximating the value of a function at a given point by using a straight line that is tangent to the function at that point. This is often used when the function is too complex to be evaluated exactly, or when we only need a rough estimate of its value.
Linear approximation can be a useful tool for understanding the behavior of a function near a given point, and can be especially useful in applications such as optimization, where we may need to find the local minimum or maximum of a function.
To achieve linearization of the function f(x), the equation of the tangent line to the curve at x=a is calculated.
To find the linear approximation of a function f(x) at a point x=a, we first find the slope of the function at that point, which is given by the derivative f'(a). We then use this slope to construct a straight line that passes through the point (a, f(a)) and has the same slope as the function. This line is known as the tangent line to the function at x=a.
The formula for linear approximation is:
f(x) ≈ f(a) + f'(a)(x – a)
where f(x) is the function being approximated, a is the point at which we are approximating the function, and f'(a) is the derivative of the function at that point.
This formula tells us that the value of the function at any point x can be approximated by evaluating the derivative at x=a and using it to construct a straight line that passes through the point (a, f(a)). We can then use this line to approximate the value of the function at other points.
For example, suppose we have a function f(x) and we want to find an approximation for its value at x=a+h. We can use the formula above to find an approximate value for f(a+h):
f(a+h) ≈ f(a) + f'(a)(a+h – a)
≈ f(a) + f'(a)h
This approximation will be more accurate the closer the value of h is to zero, since the tangent line becomes a better approximation of the function as it gets closer to the point x=a.