Question

$$\begin{gathered} \mathrm{The}\mathrm{table}\mathrm{that}\mathrm{follows}\mathrm{describes}\mathrm{the}\mathrm{activity}\mathrm{in}\mathrm{a}\mathrm{college}\mathrm{tuition}\mathrm{savings}\mathrm{account}\mathrm{over}\mathrm{four}\mathrm{years}. \\ \mathrm{Notice}\mathrm{that}2008\mathrm{was}\mathrm{a}\mathrm{particularly}\mathrm{bad}\mathrm{year}\mathrm{for}\mathrm{investing}! \\ \mathrm{Let}I\left(t\right)\mathrm{be}\mathrm{the}\mathrm{amount}\mathrm{by}\mathrm{which}\mathrm{your}\mathrm{account}\mathrm{increased}\mathrm{or}\mathrm{decreased}\mathrm{in}\mathrm{year}t, \\ \mathrm{and}\mathrm{let}B\left(t\right)\mathrm{be}\mathrm{the}\mathrm{balance}\mathrm{of}\mathrm{your}\mathrm{account}\mathrm{at}\mathrm{the}\mathrm{end}\mathrm{of}\mathrm{year}t. \end{gathered}$$

Year	2005	2006	2007	2008
Deposited	\(600	\)1200	\(1200	\)1200
Earnings	\(10	\)183	\(317	\)-1650
Increase	\(610	\)1383	\(1517	\)-450
Balance	\(610	\)1994	\(1312	\)3061

$$\begin{gathered} \left(\mathrm{a}\right)\mathrm{Describe}\mathrm{in}\mathrm{your}\mathrm{own}\mathrm{words}\mathrm{how}B\left(t\right)\mathrm{is}\mathrm{the}\mathrm{accumulation}\mathrm{function}\mathrm{of}I\left(t\right). \\ \left(\mathrm{b}\right)\mathrm{Plot}a\mathrm{step}-\mathrm{function}\mathrm{graph}\mathrm{of}I\left(t\right),\mathrm{and}\mathrm{describe}\mathrm{how}B\left(t\right)\mathrm{relates}\mathrm{to}\mathrm{the}\mathrm{area}\mathrm{under}\mathrm{graph}. \\ \left(\mathrm{c}\right)\mathrm{What},\mathrm{if}\mathrm{anything},\mathrm{can}\mathrm{you}\mathrm{say}\mathrm{about}B\left(t\right)\mathrm{when}I\left(t\right)\mathrm{is}\mathrm{positive}?\mathrm{Negative}? \\ \mathrm{If}\mathrm{you}\mathrm{had}\mathrm{to}\mathrm{guess}\mathrm{that}\mathrm{one}\mathrm{of}\mathrm{these}\mathrm{functions}\mathrm{was}\mathrm{related}\mathrm{to}\mathrm{the}\mathrm{derivative}\mathrm{of}\mathrm{other},\mathrm{which}\mathrm{one}\mathrm{would}\mathrm{it}\mathrm{be}? \end{gathered}$$

Short answer

$$\begin{gathered} \left(\mathrm{a}\right).B\left(t\right)=I\left(1\right)+I\left(2\right)+....+I\left(t\right) \\ \left(\mathrm{b}\right).B\left(t\right)\mathrm{is}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{areas}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{through}{t}^{th}\mathrm{rectangles}. \\ \left(\mathrm{c}\right).I\mathrm{could}\mathrm{be}\mathrm{derivative}\mathrm{of}B. \end{gathered}$$

Step-by-step solution

Step 1. Given Information

Year	2005	2006	2007	2008
Deposited	$600	$1200	$1200	$1200
Earnings	$10	$183	$317	$-1650
Increase	$610	$1383	$1517	$-450
Balance	$610	$1994	$3512	$3061

$$\begin{gathered} \mathrm{Where}I\left(t\right)\mathrm{is}\mathrm{increased}\mathrm{or}\mathrm{decreased}\mathrm{amount}\mathrm{in}\mathrm{the}\mathrm{amount}\mathrm{and} \\ B\left(t\right)\mathrm{is}\mathrm{the}\mathrm{balance}\mathrm{of}\mathrm{the}\mathrm{amount}\mathrm{at}\mathrm{the}\mathrm{end}\mathrm{of}\mathrm{the}\mathrm{year}. \end{gathered}$$

Step 2. Solution (a) : How B(t) is accumulation graph of I(t).

$$\begin{gathered} \mathrm{From}\mathrm{the}\mathrm{table},\mathrm{it}\mathrm{is}\mathrm{seen}\mathrm{that}\mathrm{the}\mathrm{balance}B\left(t\right)\mathrm{is}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{increase}\mathrm{in}\mathrm{any}\mathrm{year}t. \\ \mathrm{So}, \\ B\left(t\right)=I\left(1\right)+I\left(2\right)+....+I\left(t\right) \\ \mathrm{Therefore},\mathrm{we}\mathrm{can}\mathrm{say}\mathrm{that}B\left(t\right)\mathrm{is}\mathrm{accumulation}\mathrm{graph}\mathrm{of}I\left(t\right). \end{gathered}$$

Step 3. Solution (b) : How B(t) relates to the area under graph

$B\left(t\right)$ is the sum of the areas of the first through $t^{th}$rectangle.

Step 4. Solution (c) : About B(t) when I(t) is negative or positive. 

From table, B will increase when I is positive and decrease when $I\left(t\right)$is negative.

Therefore, we can say that $I$could be derivative of $B$.