Suggested languages for you:

Americas

Europe

Q. 11

Expert-verifiedFound in: Page 859

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Let $r\left(t\right)=\left(x\left(t\right),y\left(t\right),z\left(t\right)\right),t\in [a,b],$ be a vector-valued function, where *a < b* are real numbers and the functions *x(t), y(t),* and *z(t) *are continuous. Explain why the graph of *r* is contained in some sphere centered at the origin.

The graph of *r* is contained in a sphere ${x}^{2}+{y}^{2}+{z}^{2}=3{p}^{2}$ where $r\left(t\right)=\left(x\left(t\right),y\left(t\right),z\left(t\right)\right).$

The given vector-valued function is $r\left(t\right)=\left(x\left(t\right),y\left(t\right),z\left(t\right)\right),t\in [a,b],$ where *a < b* are real numbers and the functions *x(t), y(t),* and *z(t) *are continuous.

As it is given that *x(t), y(t),* and *z(t) *are continuous functions, then by the extreme value theorem *x(t), y(t),* and *z(t) *have minimum and maximum values.

Let ${P}_{1},{P}_{2}\mathrm{and}{P}_{3}$ be the maximum values of *x(t), y(t), *and *z(t). *Now, let *P *be the highest value of these three values. Then the graph of *r(t) *on [a.b] lies within the sphere with center *(0,0,0) *and radius $\sqrt{3}p.$

94% of StudySmarter users get better grades.

Sign up for free