Question

The rate of change of the value of investment, S , with respect to time, t>= 0, given by dS/dt = 2000(r/100)e^(rt/100).Where r is the annual interest rate (assumed constant) and the principal of the investment is S(0) = 2 000. (i)Find an expression for S(t), that is, the value of the investment at time t. (ii) Verify that your expression for S(t) is correct by computing S'(t).

Answer 1

i. $ds\over dt$ $= 2000$ $($ $r\over 100$ $)$ $e$ $^ {rt \over 100}$

$=\intop$ $2000$ $($ $r \over100$ $)$ $.$ $100 \over r$ $e$ $^{rt \over 100}$ $+ C$

$S$ $(t) =2000 e$ $^{rt\over 100}$ $+C$

$t=0$

$S(0)=2000$

$2000=2000e$$^0$ $+C$

$C=0$

$=2000e$ $^{rt\over 100}$

ii. ${dS(t)\over dt}=$

$2000$$($ $r\over 100$ $)e$ $^{rt \over 100}$