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Chapter 11: Q. 1 (page 901)

Sketching vector functions: Sketch the following vector functions.

r(f)=t,t3,t鈭�鈩�

Short Answer

Expert verified

r(f)=t,t3,t鈭�鈩�

Step by step solution

01

Step 1. Given information

r(f)=t,t3,t鈭�鈩�

02

Step 2. Sketching the vector

On X-axis, x=t and Y-axis, y=t3

Plotting the graph,

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Most popular questions from this chapter

Every description of the DNA molecule says that the strands of the helices run in opposite directions. This is meant as a statement about chemistry, not about the geometric shape of the double helix. Consider two helices

h1(t)=鉄�cos鈦�t,sin鈦�t,伪t鉄�andh2(t)=鉄�sin鈦�t,cos鈦�t,伪t鉄�

(a) Sketch these two helices in the same coordinate system, and show that they run geometrically in different directions.

(b) Explain why it is impossible for these two helices to fail to intersect, and hence why they could not form a configuration for DNA.

For each of the vector-valued functions in Exercises 22鈥�28, find the unit tangent vector.

r(t)=(cos3t,sin3t)

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

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

Explain why the graph of every vector-valued function r(t)=cost,sint,f(t)lies on the surface of the cylinder x2+y2=1for every continuous functionf.
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