#### Answer

(a) (i) $\$3828.84$
(ii) $\$3840.25$
(iii) $\$3850.08$
(iv) $\$3851.61$
(v) $\$3852.01$
(vi) $\$3852.08$
(b) $\frac{dA}{dt} = 0.05~A$
$A(0) = 3000$

#### Work Step by Step

(a) $A(t) = A(0)~(1+\frac{r}{n})^{nt}$
(i) $A(5) = (3000)(1+\frac{0.05}{1})^{5} = \$3828.84$
(ii) $A(5) = (3000)(1+\frac{0.05}{2})^{10} = \$3840.25$
(iii) $A(5) = (3000) (1+\frac{0.05}{12})^{60} = \$3850.08$
(iv) $A(5) = (3000) (1+\frac{0.05}{52})^{260} = \$3851.61$
(v) $A(5) = (3000) (1+\frac{0.05}{365})^{1825} = \$3852.01$
(vi) $A(5) = (3000)e^{(0.05)(5)} = \$3852.08$
(b) The rate of change of the investment value depends on the investment value $A$:
$\frac{dA}{dt} = 0.05~A$
Since the original investment is $\$3000$, the initial condition is $A(0) = 3000$