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

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

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

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Calculating derivative

Step 3. Graph of r

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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

Decision Maths

Probability and Statistics

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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