Applications of Derivatives, Objectives, Rate of Change of Quantities

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Objectives

After studying this lesson, you will be able to:

  • Find rate of change of quantities

  • Find approximate value of functions

  • Define tangent and normal to a curve (graph of a function) at a point;

  • Find equations of tangents and normal to a curve under given conditions;

  • Define monotonic (increasing and decreasing) functions;

  • Establish that in an interval for an increasing function and for a decreasing function;

  • Define the points of maximum and minimum values as well as local maxima and local minima of a function from the graph;

  • Establish the working rule for finding the maxima and minima of a function using the first and the second derivatives of the function; and

  • Work out simple problems on maxima and minima.

Rate of Change of Quantities:

Let be a function of and let there be a small change in , and the corresponding change in be .

Average change in per unit change in

As , the limiting value of the average rate of change of with respect to .

So the rate of change of per unit change in

Hence, represents the rate of change of with respect to .

Thus,

The value of at i.e.

represent the rate of change of with respect to at

Further, if two variables and are varying one with respect to another variable i.e. if and , then by chain rule.

Hence, the rate of change with respect to can be calculated by using the rate of change of and that of both with respect to .

Example:

A ladder long is leaning against a wall. The foot of the ladder is pulled along the ground, away from the wall, at the rate of ow fast is its height on the wall decreasing when the foot of ladder is 4m away from the wall?

Solution:

Let the foot of the ladder be at a distance metres from the wall and metres be the height of the ladder at any time , then

The foot of the ladder

The Foot of the Ladder

The foot of the ladder

Diff. w.r. to . We get

But iven)

When , from (1)

Putting and in (2), we get

Hence, the height of the ladder on the wall is decreasing at the rate of

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