 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Natural sciences course of the University of Cambridge

Year of the natural sciences course of the University of Cambridge

Year 1 of the natural sciences course of the University of Cambridge

Mathematics course of the year 1 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge

2021

Paper 2

The graph is the upper branch of the <hyperbola> $d^2-(t+1)^2=1/2$. It is symmetric about $t=-1$, positive everywhere, and has its lowest point at $(-1,1/\sqrt2)$. Its <derivatives> are
$$
d'(t)=\frac{t+1}{d(t)},\qquad d''(t)=\frac1{2d(t)^3}>0,
$$
so it is strictly convex. Since
$$
d(t)-|t+1|=\frac{1/2}{\sqrt{(t+1)^2+1/2}+|t+1|}\longrightarrow0,
$$
its two oblique <asymptotes> are
$$
\boxed{d=t+1\quad(t\to+\infty),\qquad d=-t-1\quad(t\to-\infty).}
$$
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2-q11-distance.png]
{title=Distance from a moving point to a skew line}
{description=Original plot of $d(t)=\sqrt{(t+1)^2+1/2}$, its minimum and its two asymptotes.}

Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 11 / d / ii / Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 11 / d / ii / Solution

 Ancestors (14)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 11 / d / ii / Solution

 Ancestors (14)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)