Fundamentals of Differentiation

At the endpoint, $c = a$ or $c = b$.

The specific value of $c$ cannot be determined without additional information about the function.

For all points such that $a < c < b$.

At the midpoint, $c = \frac{a + b}{2}$.

$5$

Undefined

Depends on the value of x

$0$

Constant at that point

Continuous on most of the domain

Discontinuous at that point

Continuous at that point

A function is differentiable at a point if its integral exists at that point

A function is differentiable at a point if its derivative exists at that point

A function is differentiable at a point if its limit exists at that point

A function is differentiable at a point if its limit is equal to zero at that point

The derivative $f'(a) = 0$.

The graph of $f(x)$ has a corner or cusp at $x = a$.

The function has a maximum or minimum at $x = a$.

$f(x)$ is continuous at $x = a$.

s(t) has a local minimum at t=2.

There is a gap in the graph of s(t) at t=2.

The graph of s(t) has a vertical tangent at t=2.

s(t) is discontinuous at t=2.

This condition only ensures continuity of f but not necessarily differentiability.

Existence of this limit implies there are no critical points for f near $x=a$.

This limit guarantees that the second derivative of f exists and is finite.

This limit defines the derivative of $f$ at $x=a$ and confirms its differentiability there.

The graph of the function has an asymptote at $x=c$.

The graph of the function has either a corner, cusp, or vertical tangent line at $x=c$.

The function cannot be integrated in an interval containing $x=c$.

The derivative of the function exists and is zero at $x=c$.

$f(x)$ is continuous at $x = a$.

$f'(a) = 0$.

The limit as $x$ approaches $a$ of $\frac{f(x) - f(a)}{x - a}$ does not exist.

The graph of $f(x)$ has a tangent line parallel to the x-axis at $x = a$.

A function is continuous at a point if the limit of the function as x approaches that point is equal to the value of the function at that point

A function is continuous at a point if the integral exists at that point

A function is continuous at a point if the derivative exists at that point

A function is continuous at a point if the limit of the function as x approaches that point is equal to zero