매개변수 곡선의 미적분

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

The rectangular coordinates of the point $\left(4,\frac{5\pi }{6}\right)$ are $\left(2\sqrt{3},-2\right).$

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

The equations $x=\cosh (3t),$ $y=2\,\sinh (3t)$ represent a hyperbola.

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

Given $x=f(t)$ and $y=g(t),$ if $\frac{dx}{dy}=\frac{dy}{dx},$ then $f(t)=g(t)+\text{C,}$ where C is a constant.

T/F: Given parametric equations $x=f(t)$ and $y=g(t)$, $\frac{dy}{dx} = f\,'(t)/g'(t)$, as long as $g'(t) \neq 0$.

T/F: Given parametric equations $x=f(t)$ and $y=g(t)$, to find $\frac{d^2y}{dx^2}$, one simply computes $\frac{d}{dt}\Big(\frac{dy}{dx}\Big).$

T/F: If $\frac{dy}{dx}=0$ at $t=t_0$, then the normal line to the curve at $t=t_0$ is a vertical line.