Chapter 11: Q. 16 (page 872)

$Find the derivative of the vector - valued function r (t) = t^{2} i - 6 t^{2} j, t > 0, and show that the derivative at any point t_{0} is a scalar multiple of the derivative at any other point . What does this property say about the graph of r ? Use that information to sketch the graph of r .$

Short Answer

Expert verified

$r' (t) = 2 ti - 12 tj The graph of r (t) is a portion of straight line .$

Step by step solution

Step 1. Given information is:

$r (t) = t^{2} i - 6 t^{2} j, t > 0$

Step 2. Calculating derivative

$r' (t) = \frac{d <t^{2} i - 6 t^{2} j>}{d t} r' (t) = 2 ti - 12 tj$

Step 3. Graph of r

$Let t_{0} and t_{1} be any two positive real numbers, r' (t_{0}) = 2 t_{0} i - 12 t_{0} j r' (t_{1}) = 2 t_{1} i - 12 t_{1} j The vectors r (t_{0}) and r' (t_{1}) are multiples of the vector i - 6 j, hence, they are scalar multiples of each other . Thus, the graph of r (t) is a portion of straight line .$

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$Find the derivative of the vector - valued function r (t) = t^{2} i - 6 t^{2} j, t > 0, and show that the derivative at any point t_{0} is a scalar multiple of the derivative at any other point . What does this property say about the graph of r ? Use that information to sketch the graph of r .$

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)=t2i−6t2j,t>0,andshowthatthederivativeatanypointt0isascalarmultipleofthederivativeatanyotherpoint.Whatdoesthispropertysayaboutthegraphofr?Usethatinformationtosketchthegraphofr.

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

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Study anywhere. Anytime. Across all devices.

$Find the derivative of the vector - valued function r (t) = t^{2} i - 6 t^{2} j, t > 0, and show that the derivative at any point t_{0} is a scalar multiple of the derivative at any other point . What does this property say about the graph of r ? Use that information to sketch the graph of r .$