1. The problem involves understanding differentiability and tangent lines of a function $f$ at various points. 2. Differentiability means the function has a defined derivative (slope) at that point. A vertical tangent means the slope is infinite, so $f$ is not differentiable there. 3. Sharp corners cause non-differentiability because the slope changes abruptly. 4. Horizontal tangents mean the derivative is zero but the function can still be differentiable there. 5. Statement A is true: vertical tangent at $x=-2$ means $f$ is not differentiable there. 6. Statement B is true: discontinuity at $x=0$ and $x=2.5$ means $f$ is not differentiable there. 7. Statement C is false: the problem states sharp corners at $x=1.5$ and $x=4$, but the graph description does not mention sharp corners there, so this statement is false. 8. Statement D is true: horizontal tangents at $x=-3$ and $x=-1$ do not imply non-differentiability; the function can be differentiable with zero slope. 9. For the second problem, $g(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h}$ is the definition of the derivative of $\sin x$, so $g'(x) = \cos x$. 10. Evaluating at $x=\frac{\pi}{3}$, $g'(\frac{\pi}{3}) = \cos(\frac{\pi}{3}) = \frac{1}{2}$. 11. The value $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$ is given but not needed for the derivative. Final answers: - The false statement is C. - The instantaneous rate of change of $g$ at $x=\frac{\pi}{3}$ is $\frac{1}{2}$.