Recall that the compound interest formula for continuous compounding is

$A(P,r,t)=P{e}^{rt}$

where $A$ is the future value of an investment of $P$ dollars after $t$ years at an interest rate of $r.$

$($a$)$ Calculate $\frac{\text{d}\text{e}\text{l}\text{A}}{\text{d}\text{e}\text{l}\text{P}},\frac{\text{d}\text{e}\text{l}\text{A}}{\text{d}\text{e}\text{l}\text{r}},$ and $\frac{\text{d}\text{e}\text{l}\text{A}}{\text{d}\text{e}\text{l}\text{t}},$ all evaluated at $(120,0\mathrm{.}3,9).($Round your answers to two decimal places.$)$

$\frac{\text{d}\text{e}\text{l}\text{A}}{\text{d}\text{e}\text{l}\text{P}}=$

$\frac{\text{d}\text{e}\text{l}\text{A}}{\text{d}\text{e}\text{l}\text{r}}=$

$\frac{\text{d}\text{e}\text{l}\text{A}}{\text{d}\text{e}\text{l}\text{t}}=$

Interpret your answers.

For a $ $$ investment at $q,\%$ interest invested for $$ years and compounded continuously, the accumulated amount is increasing at a rate of $ per $$1$ of principal, at a rate of $ per increase of $1$ in $r,$ and at a rate of $$|||$ per year.

$($b$)$ What does the function $\frac{\text{d}\text{e}\text{l}\text{A}}{\text{d}\text{e}\text{l}\text{P}}{|}_{(120,0\mathrm{.}3,t)}||$ of $t$ tell about your investment?

$A_p(120,0\mathrm{.}3,t)$ tells you the rate at which the accumulated amount in an account bearing $\%$ interest, compounded continuously, with a principal of $ $,$ is growing per $$1---$Select$---$V in the

$$

$---$Select$---$ years after the investment. $---$Select$---$

interest rate

principal

SU

future value of investment