A lemur rancher needs to invest in some high-tech lemur grooming machines. She d

Question

A lemur rancher needs to invest in some high-tech lemur grooming machines. She determines that the value of the machines will depreciate at a rate f(t) = e^-at^2 (for some constant a), and the cost of keeping them in top running condition is given by another function g(t) = B ln(1 + t), where t is the time that the machines have been running. The cost of keeping the machines around (instead of replacing them with new ones) is given by C(t) = 1/t integral_0^t (f(x) + g(x)) dx Show the critical points of C(t) occur when C(t) = f(t) + g(t) by calculating the derivative of C(t) and setting it to zero.

Explanation / Answer

Let C(t) = (1/t) * (x = 0 to t) (f(x) + g(x)) dx.

Applying the product rule, along with the fundamental theorem of calculus yields
C'(t) = (-1/t^2) * (x = 0 to t) (f(x) + g(x)) dx + (1/t) * (f(t) + g(t))
       = (-1/t) * [(1/t) * (x = 0 to t) (f(x) + g(x)) dx] + (1/t) * (f(t) + g(t))
       = (-1/t) * C(t) + (1/t) * (f(t) + g(t)), by the definition of C(t)
       = [-C(t) + (f(t) + g(t))]/t.

For the critical points set C'(t) = 0.
==> -C(t) + (f(t) + g(t))]/t = 0
==> -C(t) + (f(t) + g(t)) = 0
==> C(t) = f(t) + g(t).

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.