Chapter 11: Q. 17 (page 871)

$Find the derivative of the vector - valued function r (t) = <t^{3}, 5 t^{3}, - 2 t^{3}>, t > 0, and show that the derivative at any point t_{0} is a scalar multiple of the derivative at any other point . What does this property say about the graph of r ? Sketch the graph of r .$

Short Answer

Expert verified

$r' (t) = \frac{d <t^{3}, 5 t^{3}, - 2 t^{3}>}{d t} r' (t) = <3 t^{2}, 15 t^{2}, - 6 t^{2}> The graph of r (t) is a portion of straight line .$

Step by step solution

Step 1. Given information is:

$r (t) = <t^{3}, 5 t^{3}, - 2 t^{3}>, t > 0$

Step 2. Calculating derivative

$r' (t) = \frac{d <t^{3}, 5 t^{3}, - 2 t^{3}>}{d t} r' (t) = <3 t^{2}, 15 t^{2}, - 6 t^{2}>$

Step 3. Graph of r

$Let t_{0} and t_{1} be any two positive real numbers, r' (t_{0}) = <3 t_{0}^{2}, 15 t_{0}^{2}, - 6 t_{0}^{2}> r' (t_{0}) = 3 t_{0}^{2} <1, 5, - 2> r' (t_{1}) = <3 t_{1}^{2}, 15 t_{1}^{2}, - 6 t_{1}^{2}> r' (t_{0}) = 3 t_{1}^{2} <1, 5, - 2> The vectors r (t_{0}) and r' (t_{1}) are multiples of the vector <1, 5, - 2>, hence, they are scalar multiples of each other . Thus, the graph of r (t) is a portion of straight line .$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

$Find the derivative of the vector - valued function r (t) = <t^{3}, 5 t^{3}, - 2 t^{3}>, t > 0, and show that the derivative at any point t_{0} is a scalar multiple of the derivative at any other point . What does this property say about the graph of r ? Sketch the graph of r .$

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Calculating derivative

Step 3. Graph of r

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Geometry

Discrete Mathematics

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

Findthederivativeofthevector-valuedfunctionr(t)=t3,5t3,−2t3,t>0,andshowthatthederivativeatanypointt0isascalarmultipleofthederivativeatanyotherpoint.Whatdoesthispropertysayaboutthegraphofr?Sketchthegraphofr.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Calculating derivative

Step 3. Graph of r

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Geometry

Discrete Mathematics

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

$Find the derivative of the vector - valued function r (t) = <t^{3}, 5 t^{3}, - 2 t^{3}>, t > 0, and show that the derivative at any point t_{0} is a scalar multiple of the derivative at any other point . What does this property say about the graph of r ? Sketch the graph of r .$