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Chapter 11: Q. 16 (page 872)

Findthederivativeofthevector-valuedfunctionr(t)=t2i鈭�6t2j,t>0,andshowthatthederivativeatanypointt0isascalarmultipleofthederivativeatanyotherpoint.Whatdoesthispropertysayaboutthegraphofr?Usethatinformationtosketchthegraphofr.

Short Answer

Expert verified

r'(t)=2ti-12tjThegraphofr(t)isaportionofstraightline.

Step by step solution

01

Step 1. Given information is:

r(t)=t2i鈭�6t2j,t>0

02

Step 2. Calculating derivative

r'(t)=dt2i鈭�6t2jdtr'(t)=2ti-12tj

03

Step 3. Graph of r

Lett0andt1beanytwopositiverealnumbers,r'(t0)=2t0i-12t0jr'(t1)=2t1i-12t1jThevectorsr(t0)andr'(t1)aremultiplesofthevectori-6j,hence,theyarescalarmultiplesofeachother.Thus,thegraphofr(t)isaportionofstraightline.

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Most popular questions from this chapter

Let k be a scalar and r(t)be a differentiable vector function. Prove that ddt(kr(t))=kr'(t). (This is Theorem 11.11 (a).)

Let r(t)=x(t),y(t),z(t),t鈭�[a,鈭�), be a vector-valued function, where a is a real number. Explain why the graph of r may or may not be contained in some sphere centered at the origin. (Hint: Consider the functions r1(t)=cost,sint,1/tand r2(t)=cost,sint,t,bothwithdomain[1,鈭�).)

Evaluate and simplify the indicated quantities in Exercises 35鈥�41.
5(cost,sint)

Let f(t)be a differentiable scalar function and r(t)be a differentiable vector function. Prove that ddt(f(t)r(t))=f'(t)r(t)+f(t)r'(t). (This is Theorem 11.11 (b).)

Given a vector-valued function r(t) with domain 鈩�,what is the relationship between the graph of r(t) and the graph of r(kt), where k > 1 is a scalar?
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