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Chapter 3: Worksheet 2  Jack K. Cohen Colorado School of Mines

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The Derivative as the Rate of Change Function
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Suggested Problems
Section 3.1: 22, 24, 26

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;SPMlt;LI;SPMgt;Find the rate of change of the area of a circle with respect to its radius.

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;SPMlt;LI;SPMgt;A stone dropped into a pond causes an expanding circular ripple.   What is the rate of change of the area of a circle with respect to its radius when the radius is 3.5 meters?  Give the answer correct to two decimal places.

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;SPMlt;LI;SPMgt;(3.1.31) Find the rate of change of the area of a circle with respect to its circumference.

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;SPMlt;LI;SPMgt;A stone dropped into a pond makes an expanding circular ripple.  What is the rate of change of the area of a circle with respect to its circumference when the circumference is 3.5 meters?

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;SPMlt;LI;SPMgt;(3.1.32) A stone dropped into a pond causes a circular ripple that travels out from the point of impact at 5 ft/sec.  At what rate (in ft;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;/sec) is the area within the circle increasing when ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; = 10.  Hint: Find a formula for ;SPMlt;I;SPMgt;r;SPMlt;/I;SPMgt; in terms of ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;.

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;SPMlt;LI;SPMgt; (Generalization of Example 6) 

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;SPMlt;LI;SPMgt;Find the maximum height attained by a ball that is thrown straight up from the ground with initial velocity ;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;.  Assume a gravitational constant ;SPMlt;I;SPMgt;g;SPMlt;/I;SPMgt;.
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;SPMlt;LI;SPMgt;Find the velocity with which it hits the ground upon its return.
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;SPMlt;LI;SPMgt;One benefit of using parameters instead of specific numbers is the ability to deduce general laws.  Illustrate this by stating a general connection between the initial and final velocities.
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;SPMlt;LI;SPMgt;Another benefit of using parameters is the ability to check that answers have the correct dimensions.  For example, if somehow, we got the answer ;SPMlt;!-- MATH
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;SPMlt;LI;SPMgt;The treatment in the last problem assumed that the force of gravity was a constant.  Actually it falls off like 1/;SPMlt;I;SPMgt;r;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt; where ;SPMlt;I;SPMgt;r;SPMlt;/I;SPMgt; is the distance of the body (ball) from the center of the Earth (this is called the ``inverse square law'') .    Taking this variable force law into account gives the equation ;SPMlt;P;SPMgt;;SPMlt;!-- MATH
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where ;SPMlt;I;SPMgt;R;SPMlt;/I;SPMgt; is the mean radius of the Earth and ;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt; = ;SPMlt;I;SPMgt;r;SPMlt;/I;SPMgt; - ;SPMlt;I;SPMgt;R;SPMlt;/I;SPMgt; is the distance measured upward from the Earth's surface.

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;SPMlt;LI;SPMgt;Is the final velocity still the same as the initial velocity when the inverse square law is taken into account?
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;SPMlt;LI;SPMgt;What is the maximum height attained when the inverse square law is taken into account?
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;SPMlt;LI;SPMgt;Using the numbers: ;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt; = 96 ft/sec, ;SPMlt;I;SPMgt;g;SPMlt;/I;SPMgt; = 32 ft/sec;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;, ;SPMlt;I;SPMgt;R;SPMlt;/I;SPMgt; = 3960 miles, discuss whether it is worthwhile to consider the more complicated inverse square law model for this problem.  Remark: Air resistence is another effect we might consider.  Mathematical modeling is always a question of making things ``as simple as possible, but no simpler.''

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;SPMlt;LI;SPMgt;(3.1.34) A water bucket containing 10 liters of water develops a leak at time ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; = 0, and the volume ;SPMlt;I;SPMgt;V;SPMlt;/I;SPMgt; of water in the bucket ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; seconds later is given by ;SPMlt;P;SPMgt;;SPMlt;!-- MATH
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until the bucket is empty at time ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; = 100.

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;SPMlt;LI;SPMgt;At what rate is water leaking from the bucket after exactly one minute has passed?
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;SPMlt;LI;SPMgt;When is the instantaneous rate of change of ;SPMlt;I;SPMgt;V;SPMlt;/I;SPMgt; equal to the average change of ;SPMlt;I;SPMgt;V;SPMlt;/I;SPMgt; from ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; = 0 to ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; = 100?
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;SPMlt;LI;SPMgt;Does the given formula for ;SPMlt;I;SPMgt;V;SPMlt;/I;SPMgt; make sense when ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; = 200?

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;SPMlt;LI;SPMgt;(3.1.36) The following data describe the growth of the population ;SPMlt;I;SPMgt;P;SPMlt;/I;SPMgt; (in thousands) of a certain city during the 1970s.  Use the graphical method of Example 4 to estimates its rate of growth in 1975.

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;SPMlt;TD ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;1972;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;1974;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;1976;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;1978;SPMlt;/TD;SPMgt;
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;SPMlt;TD ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;324;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;358;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;395;SPMlt;/TD;SPMgt;
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