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.
a. $\frac{da}{dt}=120e^{0.06t}$
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 $ more than the future value of a 6 - year investment of $2000

Explanation:

Step1: Differentiate $A(t)$

The derivative of $y = e^{ax}$ is $y'=ae^{ax}$. Given $A(t)=2000e^{0.06t}$, using the chain - rule, $\frac{dA}{dt}=2000\times0.06e^{0.06t}=120e^{0.06t}$.

Step2: Evaluate $A^{\prime}(6)$

Substitute $t = 6$ into $\frac{dA}{dt}$. So $A^{\prime}(6)=120e^{0.06\times6}=120e^{0.36}\approx120\times1.433329 = 172$. The derivative $A^{\prime}(t)$ represents the rate of change of the balance with respect to time $t$. So $A^{\prime}(6)$ means 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 $\$2000$.

Step3: Interpretation of $A^{\prime}(6)$

The derivative $A^{\prime}(t)$ at $t = 6$ gives the instantaneous rate of change of the investment balance at $t = 6$. In the context of continuous - compounding investment, it shows how much the balance will increase for a one - unit (one - year) increase in time starting from $t = 6$.

Answer:

a. $\frac{dA}{dt}=120e^{0.06t}$
b. $A^{\prime}(6)\approx172$ dollars per year. The future value of a 7 - year investment of $\$2000$ will be $\$172$ more than the future value of a 6 - year investment of $\$2000$. So the answer is D.
c. The problem statement for part c is incomplete as it doesn't mention what the approximation of $\$172$ is being compared to in terms of the "actual change" (e.g., a different method of approximation or a discrete - change value). Without further information, we cannot solve part c.