$[1$ point. Circle the letter of the correct answer.$]$ Consider the formula pertaining to interest compounded

discretely$-$i$.$e$.,A=P(1+\frac{r}{n}{)}^{nt}.$ Which of the following is true?

a$.$ In this case, if we consider the future amount as a function of the interest rate $r,$ then the derivative with

respect to $r$ is

$A^{\text{'}}(r)=Pt(1+\frac{r}{n}{)}^{nt-1}$

which is the $($instantaneous$)$ rate at which the future amount $A($for a specified principal $P,$ time $t,$ and

compounding frequency $n)$ is changing with respect to the interest rate $r,$ at exactly the interest rate $r.$

Moreover, for any specific choice of $P,n,t,$ and $r$ encountered in practice $($i$.$e$.,$ where $P,n,t,$ and $r$ are

positive$),$ we know that $A^{\text{'}}(r)>0.$ In other words, we know that the future amount is increasing with

respect to the interest rate associated with the investment or account.

b$.$ In this case, if we consider the future amount as a function of the compounding frequency $n,$ and the

derivative with respect to $n$ is $[$you are allowed to accept on faith that this formula is correct!$]$

$A^{\text{'}}(n)=Pt(1+\frac{r}{n}{)}^{nt}[ln(1+\frac{r}{n})-\frac{r}{n+r}]$

which is the $($instantaneous$)$ rate at which the future amount $A($for a specified principal $P,$ time $t,$ and

interest rate $r)$ is changing with respect to the compounding frequency $n,$ at precisely the frequency $n.$

Furthermore, for any given choice of $P,n,t,$ and $r$ encountered in practice $($i$.$e$.,$ where $P,n,t,$ and $r$ are

positive$),$ we know that $A^{\text{'}}(n)>0.[$Question $14$ below is relevant.$]$

In fact, not only is $A(n)$ strictly increasing with respect to $n,$ but we also know that $A(n)P{e}^{rt}$ as $n$

$.($Therefore$,$ the formula for interest compounded continuously is the limit of the formula for interest

compounded discretely, as $n$ increases without bound; equivalently, $y=P{e}^{rt}$ is a horizontal asymptote

of $A(n).)$

c$.$ In this instance, we consider the future amount as a function of the time $t,$ and the derivative with respect

to $n$ is

$A^{\text{'}}(t)=Pn[ln(1+\frac{r}{n})](1+\frac{r}{n}{)}^{nt}$

which is the $($instantaneous$)$ rate at which the future amount $A($for a given principal $P,$ interest rate $r,$ and

compounding frequency $n)$ is changing with respect to the time $t,$ at precisely the time $t.$ Also, for any

specific choice of $P,n,t,$ and $r$ encountered in practice $($i$.$e$.,$ where $P,n,t,$ and $r$ are positive$),$ we know

that $A^{\text{'}}(t)>0,$ so that $A(t)$ is increasing with respect to time $t.$

d$.$ All of the above.

e$.$ None of the above.