Calculus Question

Question

4.MOBILE INSTANT MESSAGING ACCOUNTS—Mobile instant messaging (IM) is a small portion of total IM usage, but it is growing sharply. The function

𝑃(𝑡)=.257𝑡^2+0.57𝑡+3.9 (0≤𝑡≤4)

gives the mobile IM accounts in year tas a percentage of total enterprise IM accounts from 2006 (t = 0) through 2010 (t=4).

a.What percentage of total enterprise IM accounts were the mobile accounts in 2008?

b.How fast was this percentage changing in 2008?

Source: The Radical Group

Answer 1

a. To find out what percentage of the total enterprise IM accounts were mobile accounts in 2008 we first note that 2008 corresponds to t=2 since 2006 corresponds to t=0. Then we simply evaluate P(2) = 0.257*2^2+0.57*2+3.9 = 6.068, which tells us that 6.068 percent of the total enterprise IM accounts were mobile accounts in 2008.

b. To figure out how fast the percentage of total enterprise IM accounts that were mobile accounts was changing in 2008, we first note that we need to find the derivative of P(t) in order to find the rate of change of P(t). We also note that 2008 corresponds to t=2. We now calculate P'(t) = 0.257*(2t)+0.57*1 + 0 = 0.514t+0.57 and from this we can evaluate P'(2) = 0.514*2+0.57 = 1.598, which tells us that the percentage of total enterprise IM accounts that were mobile accounts was increasing by 1.598 percentage points in 2008.