Q. 55 - Calculus

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

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

Step-by-Step Solution

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 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 $B$.