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# Derivatives And Differentiation Equations Research Assignment (Math Problem Sample)

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Derivatives And Differentiation Equations

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3.1 Tangents and the Derivative at a Point
30. Ball’s changing volume: What is the rate of change of the volume of a ball (V=43πr3) with respect to the radius when the radius is r=2?
Solution
V=43πr3
Step 1; Now, differentiating w.r.t, R
dVdR=4π3 (3R2)
=4πR2
Step 2; Now just plug in 2 to find the rate of change of volume with respect radius when r=2inch
dVdR4π( 22)
= 16π
Therefore the rate of change of volume is 16π
3.4 The Derivative as a Rate of Change
24. Marginal Revenue: Suppose that the revenue from selling x washing machines is rx=20,0001-1x dollars.
a. Find the marginal revenue when 100 machines are produced.
Solution
Step 1; Marginal revenue =Rate of change of revenue function
R'(x) =20000ddx (1 x )
= 20000 x2
Step 2; Marginal revenue when 100 machines are produced is:
R'(100) = 20000 1002
Step 3; marginal revenue = \$2 dollars
b. Use the function r'(x) to estimate the increase in revenue that will result from increasing production from 100 machines to 101 machines a week.
Solution
The marginal revenue function estimates the increase in revenue...
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