How Do Linear Approximations Differ from Tangent Lines in Calculus?

[sarahr]

what is the difference between a linear approximation and a tangent line?

my understanding is that the linear approximation is that it uses the tangent line at (a, f(a)) as an approximation to the curve y = f(x).

my question really is, then why in calculus I do they make an entirely separate section from equation of tangent lines for linear approximations? why call it something else?

 

[mathman]

Linear approximation is one application of the use of tangent line.

 

[tacman]

There are a few key differences between a linear approximation and a tangent line.

Firstly, a linear approximation is a method used to approximate a curve at a specific point using a tangent line, while a tangent line is a straight line that touches a curve at a specific point.

Secondly, a tangent line is a geometric concept, while a linear approximation is a mathematical concept. A tangent line can be visualized as the slope of the curve at a specific point, while a linear approximation is a mathematical tool used to estimate the value of a function at a specific point.

Additionally, a linear approximation is more accurate than a tangent line in terms of approximating a curve. A tangent line only gives a rough estimate of the curve at a specific point, while a linear approximation takes into account the behavior of the curve around the point of approximation.

Finally, the reason why calculus courses have a separate section on linear approximations is because it is a fundamental concept in the study of derivatives. Understanding linear approximations is crucial in understanding the concept of the derivative, as it allows us to approximate the slope of a curve at a specific point. So while linear approximations and tangent lines are related, they serve different purposes and have different levels of accuracy in approximating a curve.