Mathematics: Rate of change of revenue the rate of change of revenue (in dollars per calculator) from the sale of x calculators is r (x)=(x+1)ln(x+1). find the total revenue from the sale of the first 12 calculators. (hint: in this exercise, it simplifies matters to write an anti-derivative of x+1 as (x+1)2/2 rather than x^2/2+x).

Ithink the answer is a. cosine function....

Rate of change of revenue the rate of change of revenue

rkingmitch8074

15.01.2022

Assuming no revenue from no calculators being sold, so that R(0)=0 , we have

$R(12)=R(0)+\displaystyle\int_0^{12}R'(x)\,\mathrm dx$

$R(12)=\displaystyle\int_0^{12}(x+1)\ln(x+1)\,\mathrm dx$

Integrate by parts, taking

$u=\ln(x+1)\implies\mathrm du=\dfrac{\mathrm dx}{x+1}$

$\mathrm dv=x+1\,\mathrm dx\implies v=\dfrac{(x+1)^2}2$

Then

$R(12)=\displaystyle\frac{(x+1)^2}2\ln(x+1)\bigg|_0^{12}-\frac12\int_0^{12}x+1\,\mathrm dx$

$R(12)=\dfrac{169}2\ln13-\dfrac{(x+1)^2}4\bigg|_0^{12}$

$R(12)=\dfrac{169}2\ln13-\left(\dfrac{169}4-\dfrac14\right)\approx174.74$

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