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

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Derivatives And Differentiation Equations

source..Content:

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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