§2.2 The sine and cosine functions (continued)

Summary

An investment has value $V(t) = 24 \cdot 1.07^t + 6\sin(t)$ thousand dollars and satisfies $V''(2) \approx -5.33$. At $t = 2$ this says the rate of change of the portfolio is itself

Compare $A(t) = 24 \cdot 1.07^t$ with $V(t) = 24 \cdot 1.07^t + 6\sin(t)$. What does adding the sine term do?

With $V(t) = 24 \cdot 1.07^t + 6\sin(t)$ in thousands of dollars, compute $V'(2)$ rounded to three decimals.

For $f(x) = 3\cos(x) - 2\sin(x) + 6$, the tangent line approximation at $a = \pi$ is

For $f(x) = 3\cos(x) - 2\sin(x) + 6$, compute $f'(\frac{\pi}{4})$ rounded to three decimals.

For the same $f(x) = 3\cos(x) - 2\sin(x) + 6$, at $x = \frac{\pi}{2}$ the function is

For the same $f$, the second derivative satisfies $f''(\frac{3\pi}{2}) = -3\cos(\frac{3\pi}{2}) + 2\sin(\frac{3\pi}{2}) = -2$. Relative to the curve, the tangent line at $a = \frac{3\pi}{2}$ sits

Using small values of $h$, the two limits $\lim_{h \to 0}\frac{\sin(h)}{h}$ and $\lim_{h \to 0}\frac{\cos(h)-1}{h}$ evaluate to

Expanding $\sin(x+h) = \sin(x)\cos(h) + \cos(x)\sin(h)$ inside the difference quotient of $f(x) = \sin(x)$ splits the derivative into $\sin(x) \cdot \lim_{h \to 0}\frac{\cos(h)-1}{h} + \cos(x) \cdot \lim_{h \to 0}\frac{\sin(h)}{h}$.

The formula $\frac{d}{dx}[\sin(x)] = \cos(x)$ holds for every real number $x$.