§1.8 The tangent line approximation

Introduction

The tangent line

Let $g(x) = -x^2 + 3x + 2$. The limit definition gives which formula for $g'(x)$?

Which expression gives the tangent line approximation $L(x)$ to $f$ at the point $(a, f(a))$?

At the point of tangency, the graph of $f$ and its tangent line approximation $L$ share which data?

For $g(x) = -x^2 + 3x + 2$, compute $g'(1)$, the slope of the tangent line at $(1, g(1))$.

For $g(x) = -x^2 + 3x + 2$, compute $g(0)$, the value where a tangent line approximation at $0$ would start.

The tangent line approximation $L(x) = f(a) + f'(a)(x-a)$ is most accurate for values of $x$ close to $a$.