Question

the equation $a(t)=2000e^{0.06t}$ gives the balance after $t$ years of an initial investment of 2000 dollars which pays 6.00% compounded continuously.
a. find a formula for $\frac{da}{dt}$
b. find and interpret $a(6)$. include appropriate units.
c. compare the approximation of $172$ to the actual change. report your answer to two decimal places.

$\frac{da}{dt}=120e^{0.06t}$
b. $a(6)=172$ dollars increased in the total amount per yearly increase in the investment time period (round up to the nearest dollar)
interpret $a(6)=172$
a. the future value of a 7 - year investment of $2001 will be $ more than the future value of a 6 - year investment of $2000
b. the future value of a 6 - year investment of $2000 at 7.00% will be $ more than the future value of a 6 - year investment of $2000 at 6.00%
c. the future value of a 6 - year investment of $2001 will be $ more than the future value of a 6 - year investment of $2000
d. the future value of a 7 - year investment of $2000 will be $172 more than the future value of a 6 - year investment of $2000
c. $a(7)-a(6)=$ dollars per year (round to two decimal places as needed.)

Explanation:

Step1: Differentiate $A(t)$

We know that if $y = e^{ax}$, then $y^\prime=ae^{ax}$. Given $A(t)=2000e^{0.06t}$, by the chain - rule, $\frac{dA}{dt}=2000\times0.06e^{0.06t}=120e^{0.06t}$.

Step2: Evaluate $\frac{dA}{dt}$ at $t = 6$

Substitute $t = 6$ into $\frac{dA}{dt}$. So $A^\prime(6)=120e^{0.06\times6}=120e^{0.36}\approx172$ dollars per year. The interpretation of $A^\prime(6)$ is that the future value of a 7 - year investment of $\$2000$ will be $\$172$ more than the future value of a 6 - year investment of $\$2000$ (since the derivative represents the rate of change of the function, here the rate of change of the investment value with respect to time).

Step3: Calculate $A(7)-A(6)$

First, find $A(7)=2000e^{0.06\times7}=2000e^{0.42}$ and $A(6)=2000e^{0.06\times6}=2000e^{0.36}$. Then $A(7)-A(6)=2000(e^{0.42}-e^{0.36})=2000(e^{0.36}(e^{0.06}-1))$.
$e^{0.06}\approx1.061836547$ and $e^{0.36}\approx1.433329415$.
$A(7)-A(6)=2000\times1.433329415\times(1.061836547 - 1)\approx177.62$ dollars per year.

Answer:

a. $\frac{dA}{dt}=120e^{0.06t}$
b. $A^\prime(6)\approx172$ dollars per year; D. The future value of a 7 - year investment of $\$2000$ will be $\$172$ more than the future value of a 6 - year investment of $\$2000$
c. $177.62$