Mobile instant messaging (IM) is a small portion of total IM...

Mobile instant messaging (IM) is a small portion of total IM usage, but it is growing sharply. The function $$ P(t)=0.257 t^{2}+0.57 t+3.9 \quad(0 \leq t \leq 4) $$ gives the mobile IM accounts in year $t$ as 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 $2008 ?$

Mobile instant messaging (IM) is a small portion of total IM usage, but it is growing sharply. The function
$$
P(t)=0.257 t^{2}+0.57 t+3.9 \quad(0 \leq t \leq 4)
$$
gives the mobile IM accounts in year $t$ as 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 $2008 ?$

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Key Concepts

Derivatives and Instantaneous Rates of Change

In calculus, the derivative of a function represents its instantaneous rate of change. This key concept is applied here by differentiating the percentage function to determine how quickly the mobile IM account percentage is changing at a specific time, giving insight into the trend's velocity.

Polynomial Function Modeling

A polynomial function, especially a quadratic function, can be used to model real-world phenomena with curvature or acceleration in growth. This concept is central in understanding and analyzing the proportional growth of mobile IM accounts as compared to total enterprise IM accounts over time.

Function Evaluation

This concept involves substituting a specific value into a function to calculate its output. In contexts like this, it is used to determine the percentage of mobile IM accounts at a particular time by plugging the corresponding time value into the provided function.

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