Instantaneous rate of change

Today's Lesson is Instantaneous rate of change:

Instantaneous rate of change is the rate of change that is measured at a single point on a continuous curve. Instantaneous rate of change corresponds to the slope of the tangent line at that point.

There are three ways to find the instantaneous rate of change.

Estimate an Instantaneous Rate of Change from a graph:

Method 1: Use the slope of a Secant

We use the slope of a secant passing through the point and another point on the curve that is very close to it to find the instantaneous rate of change.

Example 1:

Point S(4,22) and point Q (8,50)
The slope of the secant SQ is
m SQ= delta d/delta t
=50-22/8-4
=28/4
= 7

Method 2: Use Two Point on an Approximate Tangent Line
Another Method to find an instantaneous rate of change from a graph is sketch an approximate tangent line through that point on a second point on the tangent line.

Example 2:

Point S(4,22) and Point R(10,63)
m RS= delta d / delta t
=63-22 / 10-4
=41/6
=6.83333333

Estimating Instantaneous rate of change from a table of values:

To find the estimating Instantaneous rate of change from a table, you first need to calculate the average rate of change over a short interval.

Estimating Instantaneous rate of change from an equation:
To find the estimating Instantaneous rate of change from an equation by using a very short interval between the tangent point and a second point found using the equation.

Example1 :
Find the instantaneous rate of change at 5.5 modelled by the function h(t)=-4.2t^2+10t+2.

h(t)=-4.2t^2+10t+2
h(5.5)=-4.2(5.5)^2+10(5.5)+2
h(5.5)=-70.05

t delta t
5.5 -70.05
5.6 -73.71
5.51 -70.41
5.501 -70.09

AROC(5.5,5.6)
=(-73.71--70.05) / (5.6-5.5)
=36.6

AROC(5.5,5.51)
=(-70.41--70.05) / (5.51-5.5)
=-36

AROC(5.5,5.501)
=(-70.09--70.05) / (5.501-5.5)
=-40

Therefore I would approximate the slope of the tangent at 5.5 to be about -40.

In Conclusion, You can find the Instantaneous rate of change through three methods:
1) from a graph,determine the slope of a secant passing through a point and another point on the curve that is very close to it
2) table of values, calculate the average rate of change over a short interval by using points in the table that are closest to the tangent point
3) from an equation, determine the average rate over shorter and shorter intervals (0.1,0.01, 0.001, etc)

Hope this helps! =)