삼각함수의 미분

(a) January 1, 2012 corresponds to the instant $t = 2$, so we compute $V'(2)$. First, we observe by the sum and constant multiple rules that $V'(t) = 24 \cdot 1.07^t \cdot \ln(1.07) + 6 \cos(t)$. Thus, $V'(2) = 24 \cdot 1.07^2 \cdot \ln(1.07) + 6 \cos(2) \approx -0.63778$. Since $V$ is measured in thousands of dollars and $t$ in years, we know that the portfolio's value is decreasing at a rate of $637.78$ dollars per year on January 1, 2012. (b) Since $V'(t) = 24 \cdot 1.07^t \cdot \ln(1.07) + 6 \cos(t)$, it follows that $V''(t) = 24 \cdot 1.07^t \cdot (\ln(1.07))^2 - 6 \sin(t)$, and thus $V''(2) = 24 \cdot 1.07^2 \cdot \ln(1.07)^2 - 6 \sin(2) \approx -5.33$, with units “thousands of dollars per year per year”. This means that around the time that $t = 2$, the portfolio's value is decreasing at while the value function is concave down. More specifically, we know that $V'$ is decreasing at the moment $t = 2$ and expect the derivative's value to decrease by about $5.33$ thousand dollars per year over the course of the next year. In other words, the portfolio's value is not only decreasing, but we expect it to decrease more rapidly as the year goes on. (c) In the figure below, we see plots of both $V$ and $A$ along with their corresponding derivatives. We see that the function $A$ is a purely exponential function that always increases at an increasing rate. By adding the term $6\sin(t)$ to $A$ to create the function $V$, we add volatility to the value of the portfolio: the oscillating nature of the sine function not only creates variance in the stock's value, but also in its rate of change. We also observe that as time progresses and the value of $24 \cdot (1.07)^t$ increases, the effect of $6 \sin(t)$ lessens.