# Instantaneous Velocity

## Key Questions

Instantaneous velocity is the derivative or slope of the distance function.

#### Explanation:

Let the distance function be $s \left(x\right)$

Then, the instantaneous velocity at x:

instantaneous velocity $= v \left(x\right) =$ ""_(hrarr0)^lim[s(x+h)-s(x)]/h

hope that helps

See explanation

#### Explanation:

If I remember correctly!!!

For this expression to be used you are talking about obtaining the velocity at any absolutely minute moment in time on a changing velocity.

The change in velocity is acceleration. The thing is; acceleration could also be changing giving you a double whammy!

If you were to plot the changing velocity against time then you would get a curve. The tangent to this curve at any point is the resulting acceleration.

Check this to make sure it is correct: the instantaneous velocity is a lead up to the process of differentiation and or integration in Calculus.

• The instantaneous velocity is found by taking the derivative of the curve and then substituting in a value of x.

Example:

$f \left(x\right) = {x}^{3}$

$f ' \left(x\right) = 3 {x}^{2}$

Below are the instantaneous velocities at various values of $x$ for the curve.

$x = - 3 \to f ' \left(- 3\right) = 3 {\left(- 3\right)}^{2} = 27$

$x = 0 \to f ' \left(0\right) = 3 {\left(0\right)}^{2} = 0$

$x = 1 \to f ' \left(1\right) = 3 {\left(1\right)}^{2} = 3$

$x = 5 \to f ' \left(5\right) = 3 {\left(5\right)}^{2} = 75$