Question

"Linear component" of a fn: Geometric intuition?

During the linear and quadratic approximations lecture, Dr. Jerison talks about these components. I get what he was referring to in the equations, but what do these components /really/ mean geometrically? Can you think of a function as a linear and quadratic component (like you could decompose a vector into it's x and y projections)?

Answer 1

For now we can say that the linear approximation is the line the most closely approximates the curve at a particular point, and the quadratic approximation is the quadratic equation that most closely mimics the curve at that point. Late in the course you'll see that these are foreshadowing the Taylor series, for which the linear and quadratic terms are just the first of a potentially infinite series of polynomial terms that more and more closely approximate any well-behaved function.