Chapter 5.5, Problem 75E

Depreciation of Equipment For assets such as machines, whose market values drop rapidly in the early years of usage, businesses often use the double declining-balance method. In practice, a business firm normally employs the double declining-balance method for depreciating such assets for a certain number of years and then switches over to the linear method (see Exercise 50. page 92). The double declining-balance formula is

$V(n)=C{\left(1-\frac{2}{N}\right)}^n$

where C is the initial value of the asset in dollars, V(n) denotes the book value of the assets at the end of n years, and N is the number of years over which the asset is depreciated.

1. a. Find V′(n).

Hint: Use logarithmic differentiation.

1. b. What is the relative rate of change of V′(n)?

Hint: Find [V′(n)]/[V(n)]. See Section 3.5.