When the annual rate of inflation averages 6% over the next 10 years, the approximate cost c of goods or services during any year in that decade is given below, where t is the time in years and p is the present cost. c(t) = p(1.06)t (a) the price of an oil change for your car is presently $21.34. estimate the price 10 years from now. (round your answer to two decimal places.) c(10) = $ 38.216 (b) find the rates of change of c with respect to t when t = 1 and t = 9. (round your coefficients to three decimal places.) at t = 1 at t = 9 (c) verify that the rate of change of c is proportional to c. what is the constant of proportionality?

(a) Putting the given information into the formula gives
  c = $21.34(1.06)^10 ≈ $38.22 . . . . . you knew that

(b) It is convenient to use the derivative function of a graphing calculator for this pupose. In the attached, we have defined Y1(x) = 1.06^x, which is to say that we have chosen p=1. Then we used the calculator to find the rate of change at the designated points.
  c'(1) ≈ 0.062p
  c'(9) ≈ 0.098p

(c) When these values are divided by c(1) and c(9) respectively, the constant of proportionality is revealed to be
  c'(t) = k·c(t) . . . . . . . where k ≈ 0.0582689

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Differentiating according to the rules of calculus, you find
  c'(t) = ln(1.06)·p·(1.06)^t = ln(1.06)·c(t)
  c'(t) ≈ 0.0582689·c(t)

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