Chapter 5.2, Problem 27E

(a)

To determine

The balance after $20$ years where the function $A(t)$ gives the balance in a savings account after $t$ years with interest compounded continuously and $A\text{'}\left(t\right)$ is the derivative of $A\left(t\right)$ The graphs as follows,

(b)

To determine

The balance increasing after $20$ years where the function $A(t)$ gives the balance in a savings account after $t$ years with interest compounded continuously and $A\text{'}\left(t\right)$ is the derivative of $A\left(t\right)$ The graphs are as follows,

(c)

To determine

Theinterest rate by using answers of part (a) and (b).The function $A\left(t\right)$ gives the balance in savings account after $t$ years with interest compounded continuously and $A\text{'}\left(t\right)$ shows the derivative of $A\left(t\right)$.

(d)

To determine

The balance is $\$300$ using the graphs of the function $A\left(t\right)$ which gives the balance in savings account after $t$ years with interest compounded continuously and $A\text{'}\left(t\right)$ shows the derivative of $A\left(t\right)$. The graphs are as follows,

(e)

To determine

Thebalance is increasing at the rate of $\$12$ per year using the graphs of the function $A\left(t\right)$ which gives the balance in savings account after $t$ years with interest compounded continuously and $A\text{'}\left(t\right)$ shows the derivative of $A\left(t\right)$. The graphs are as follows,

(f)

To determine

why the graphs $A(t)$ and $A\text{'}(t)$ look the same using the graphs of the function $A\left(t\right)$ which gives the balance in savings account after $t$ years with interest compounded continuously and $A\text{'}\left(t\right)$ shows the derivative of $A\left(t\right)$. The graphs are as follows,