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Chapter 2: Q. 3 (page 236)

Give precise mathematical definitions or descriptions of each

of the concepts that follow. Then illustrate the definition with

a graph or algebraic example, if possible:

the real–world interpretations of average rate of change and

instantaneous rate of change.

Short Answer

Expert verified

The formal definition of the derivative of a function f at point x=c is

f'(x)=limh→0f(x+h)-f(x)handf'(x)=limz→xf(z)-f(x)z-xboth are same and it depends on limit.

Step by step solution

01

Step 1. Concept used

First derivative and average rate of change formula is sameravg=f(b)-f(a)b-a.

02

Step 2. Calculation

The average rate of change is defined as

ravg=f(b)-f(a)b-aover some interval [a,b]

Average velocity can tell only average not particular things whereas to have particular point.

The instantaneous rate of change is used here it’s defined as the slope of fevaluated at a single point, example f'(c)where cis predefined point.

If a slope is constant average and instantaneous is same.

Hence, ravg=f(b)-f(a)b-awhereas instantaneous rate of change is particular rate of change.

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