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Rate of Change II

The document discusses rates of change and how calculus can be used to analyze them. It defines the rate of change as the slope, or derivative, of a function. It provides examples of how average and instantaneous rates of change are calculated. Finally, it discusses how related rates problems can be solved using implicit differentiation with respect to time to determine the appropriate rate of change between two or more variables that depend on a common variable, like time.

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Nasrul Naim
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46 views2 pages

Rate of Change II

The document discusses rates of change and how calculus can be used to analyze them. It defines the rate of change as the slope, or derivative, of a function. It provides examples of how average and instantaneous rates of change are calculated. Finally, it discusses how related rates problems can be solved using implicit differentiation with respect to time to determine the appropriate rate of change between two or more variables that depend on a common variable, like time.

Uploaded by

Nasrul Naim
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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Rate of Change

By using the derivative, one can not only find the slope, but also the rate of change. These
should be no surprise to any well taught Calc student. Here's why: if we were to find the rate
of change of a graph wouldn't we find the slope. Therefore the rate of change is some what
like the slope. However, this slope is called the rate. Which is expressed as
                     Rate = Distance = f(d) - f(c)
                                    Time           d - c
Many applications of rates of change are for describing the motion of an object moving in a
straight line. In Calculus, one can find two types of rates, an average rate and
aninstantaneous rate.

Notes on the Position Function


Many rates of change problems and related rates problems will utilize the position function
and its derivatives, which are seen below. Note that g is gravity if s is a function of a falling
object. v0 is the initial velocity and s0 is the initial height or position.
Position function = s(t) = 1/2gt2 + v0t + s0           {for falling objects}
s '(t) = v(t)     {The velocity function}
s ''(t) = a(t)     {The acceleration function}

Average Rate
Change in distance
Change in time

Instantaneous Rate of fat point x


f '(x)      {The derivative} Example Using Average Rate and Instantaneous Rate:

Situation: You are a prosecuter, which happens to have a degree in Calculus. You must
prove that a truck exceeded the speed limit of 55 miles per hour at some point during a 4
minute period. The truck passed two stationary patrol cars, which were 5 miles apart,
equipped with radar. As the truck passed the first patrol car, its speed was clocked at 55 miles
per hour. Four minutes later, when the truck passes the second car, its speed is clocked at 50
miles per hour. A pretty smart driver seeing the second car, but not smarter than Calculus.

Solution:

First you're going to reconstruct the crime scene in mathematical terms.


Let t = 0 be the time in hours when the truck passes the first patrol car. The time when the
truck passes the second is t = 4/60 = 1/15 hr.

Second you find the average velocity. 


Average velocity = s(1/15) - s(0) =  5  = 75 mph.
                                   (1/15) - 0       1/15
Since you find that the average velocity is above 55 miles per hour, you will then prove that
at some point the truck was traveling at the average velocity. To do this you remember the
Mean Value Theorem. Which proves that the truck was going at 75 mph at some point.

Related Rates of Change

Some problems in calculus require finding the rate of change or two or more variables that

are related to a common variable, namely time. To solve these types of problems, the

appropriate rate of change is determined by implicit differentiation with respect to time. Note

that a given rate of change is positive if the dependent variable increases with respect to time

and negative if the dependent variable decreases with respect to time. The sign of the rate of

change of the solution variable with respect to time will also indicate whether the variable is

increasing or decreasing with respect to time.

Read more: http://www.cliffsnotes.com/study_guide/Related-Rates-of-Change.topicArticleId-39909,articleId-
39897.html#ixzz14Jlet2mm

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