Tangent line is p1 Let f be differentiable at x=a. a. Find...

Tangent line is $p_{1}$ Let $f$ be differentiable at $x=a$.
a. Find the equation of the line tangent to the curve $y=f(x)$ at $(a, f(a))$.
b. Verify that the Taylor polynomial $p_{1}$ centered at $a$ describes the tangent line found in part (a).

Key Concepts

Differentiability

Differentiability of a function at a point indicates that the function has a well-defined derivative at that point, meaning that the function is locally linear and smooth. This concept is crucial because it guarantees the existence of a unique tangent line at the point, which is a fundamental property used in local approximations and analysis.

Tangent Line

The tangent line to a curve at a given point is the straight line that just touches the curve at that point and has the same slope as the curve. It represents the best linear approximation to the function at that point, and its equation is derived using the point of tangency and the derivative (slope) of the function at that point.

Linear Approximation

Linear approximation involves using the tangent line to estimate the value of a differentiable function near a specific point. This method relies on the idea that, if a function is differentiable, its behavior near a point can be approximated by a linear function with the same value and derivative at that point.

Taylor Polynomial

The Taylor polynomial is a series expansion used to approximate a function around a specific point. The first order Taylor polynomial, which is equivalent to the tangent line, consists of the function's value and the first derivative at that point. This polynomial provides an approximation that is locally accurate and is a key concept in both differential calculus and numerical analysis.

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