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This document discusses average and instantaneous rates of change. It defines dependent and independent variables, and explains that the average rate of change is the change in the dependent variable divided by the change in the independent variable over an interval. The instantaneous rate of change is the limit of the average rate of change as the change in the independent variable approaches zero. Several examples are provided to demonstrate calculating average and instantaneous rates of change for functions of distance, area of a circle, and other variables.

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531 views5 pages

Contoh Soalan

This document discusses average and instantaneous rates of change. It defines dependent and independent variables, and explains that the average rate of change is the change in the dependent variable divided by the change in the independent variable over an interval. The instantaneous rate of change is the limit of the average rate of change as the change in the independent variable approaches zero. Several examples are provided to demonstrate calculating average and instantaneous rates of change for functions of distance, area of a circle, and other variables.

Uploaded by

Pa Manja
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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http://www.emathzone.com/tutorials/calculus/average-and-instantaneous-rate-ofchange.

html

Average and Instantaneous Rate of Change

Dependent and Independent Variables: A variable which can assign any value independently is called independent variable and the variable which depends on some independent variable is called the dependent variable. For Example:

If

etc, then

We see that as or

behaves independently, so we call it the independent variable. But the behavior of . So we call it dependent variable.

depends on the variable

Increment: Literally the word increment means on increase, but in mathematics, this word covers both increase as well as decrease; for the increment may be positive or negative. Briefly and simply, the word increment, in mathematics means, the difference between two values of variables. i.e., the final value minus the initial value is called an increment in the variable. The increment in is denoted by the symbols or (read as delta ) If value , and changes from an initial value . to the final value , then changes from an initial

to the final value Thus, the increment in Produces a corresponding increment in

Average Rate of Change:

If change of

is real valued continuous function in the interval with respect to over this interval is

, then the average rate of

But

Instantaneous Rate of Change: If rate of change of is real valued continuous function in the interval with respect to over this interval is , then the average

But

This shows that

, as

Average or Instantaneous Rate of Change of Distance: OR Average or Instantaneous Velocity: Suppose a particle (or an object) is moving in a straight line and its positions (from some fixed point) after times velocity is and are given by and , then the average rate of change or the average

Also, the instantaneous rate of change of distance or instantaneous velocity is

Examples of Average and Instantaneous Rate of Change

Example: Let (a) Find the average rate of change of with respect to over the interval (b) Find the instantaneous rate of change of with respect to at the point Solution: (a) For Average Rate of Change: We have . .

Put

Again Put

The average rate of change over the interval

is

(b) For Instantaneous Rate of Change: We have

Put

Now, putting

then

The instantaneous rate of change at the point

is

. Example: A particle moves on a line away from its initial position so that after seconds it is

feet from its initial position. (a) Find the average velocity of the particle over the interval (b) Find the instantaneous velocity at . Solution: (a) For Average Velocity: We have .

Put

Again Put

The average velocity over the interval

is

(b) For Instantaneous Velocity: We have

Put

Now putting

The instantaneous velocity at

is

Example: Use the formula for the area of a circle to find, (a) the average rate of which the area of a circle changes with as the radius increases form . to

(b) the instantaneous rate at which the area of a circle changes with Solution: (a) For Average Velocity: We have Put Again Put The average rate of change from to is

when

(b) For Instantaneous Rate of Change: We have Put

Now putting The instantaneous rate of change at the point is

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