Second Derivatives: Concavity, Approximation, and Multivariable Extensions

All flashcards

Second derivative

The derivative of the first derivative, written $f''=(f')'$, measuring how the rate of change of a function changes.

Symmetric second-difference quotient

A limit using $f(x+h)$, $f(x)$, and $f(x-h)$ to estimate a second derivative.

Second-order Taylor polynomial

A local quadratic approximation using the function value, first derivative, and second derivative at its center.

Hessian

The matrix of second-order partial derivatives of a scalar-valued function.

Laplacian

The sum of pure second partial derivatives; equivalently, the divergence of the gradient.

Quiz questions

1. What condition characterizes an inflection point of a graph?Answer: The graph changes concavity at the point.
2. At a stationary point $c$, what does $f''(c)<0$ imply under the second derivative test?Answer: The function has a local maximum at $c$.
3. At a stationary point, what does $f''(c)=0$ establish about the classification?Answer: The second derivative test is inconclusive.
4. What does a finite symmetric second-difference limit establish by itself?Answer: It does not necessarily establish that the ordinary second derivative exists.
5. When are the mixed partial derivatives in a Hessian guaranteed to be equal?Answer: When the relevant mixed partial derivatives are continuous.