Editing Acceleration (section)

===Instantaneous acceleration===
[[File:1-D kinematics.svg|thumb|right|'''From bottom to top''': {{bulleted list
 | an acceleration function {{math|''a''(''t'')}};
 | the integral of the acceleration is the velocity function {{math|''v''(''t'')}};
 | and the integral of the velocity is the distance function {{math|''s''(''t'')}}.
}}]]
Instantaneous acceleration, meanwhile, is the [[limit of a function|limit]] of the average acceleration over an [[infinitesimal]] interval of time. In the terms of [[calculus]], instantaneous acceleration is the [[derivative]] of the velocity vector with respect to time:
<math display="block">\mathbf{a} = \lim_{{\Delta t} \to 0} \frac{\Delta \mathbf{v}}{\Delta t} = \frac{d\mathbf{v}}{dt}.</math>
As acceleration is defined as the derivative of velocity, {{math|'''v'''}}, with respect to time {{mvar|t}} and velocity is defined as the derivative of position, {{math|'''x'''}}, with respect to time, acceleration can be thought of as the [[second derivative]] of {{math|'''x'''}} with respect to {{mvar|t}}:
<math display="block">\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{x}}{dt^2}.</math>

(Here and elsewhere, if [[Rectilinear motion|motion is in a straight line]], [[Euclidean vector|vector]] quantities can be substituted by [[Scalar (physics)|scalars]] in the equations.)

By the [[fundamental theorem of calculus]], it can be seen that the [[integral]] of the acceleration function {{math|''a''(''t'')}} is the velocity function {{math|''v''(''t'')}}; that is, the area under the curve of an acceleration vs. time ({{mvar|a}} vs. {{mvar|t}}) graph corresponds to the change of velocity.
<math display="block" qid=Q11465>\mathbf{\Delta v} =  \int \mathbf{a} \, dt.</math>

Likewise, the integral of the [[Jerk (physics)|jerk]] function {{math|''j''(''t'')}}, the derivative of the acceleration function, can be used to find  the change of acceleration at a certain time:
<math display="block">\mathbf{\Delta a} =  \int \mathbf{j} \, dt.</math>