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Chapter 11: Q. 4 (page 871)

State what it means for a vector function rt=xt,yt,ztto be differentiable.

Short Answer

Expert verified

Proved that the given function is said to be differentiable

Step by step solution

01

Step 1. Given information

The given vector functionrt=xt,yt,zt

02

Step 2. The object is to define the derivative of the vector-function r(t)

Defination of the derivative of the vector function

Let rt=xt,yt,ztwhere xt,yt,ztare three real-valued function which are differentiable at every point in the interval I鈯�Rthen the derivative of

rt=xt,yt,ztis

r'(t)=drdtrt=ddtxt,yt,zt=x't,y't,z't=x'ti+y'tj+z'tk

Now the function is said to be differentiable

03

Step 3. Take an example

Let r(t)=t,t2,t4

r'(t)=drdt=ddrr(t)=ddrt,t2,t4=dtdt,ddtt2,ddt(t4)=1,2t,4t3

Thus ,r'(t)=1,2t,4t3

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Most popular questions from this chapter

The DNA molecule takes the shape of a double helix鈥攖wo helices that stay a roughly uniform distance apart.

(a) Neglecting actual dimensions, we can model one strand of DNA using the vector function h1(t)=鉄�cos鈦�(t),sin鈦�(t),伪t鉄�..

Sketch the graph of h1. What is the effect of the parameter 伪?

(b) The second strand of DNA can be constructed by shifting the first. Does the graph of h2(t)=鉄�cos鈦�(t),sin鈦�(t),伪t+尾鉄�ever intersect that of h1?

(c) The distance between two curves is the minimum distance between any two points on the curves. What is the distance between h1and h2if 伪=1and尾=蟺? (Hint: Write two points on the curves using parameters t1and t2, expand the formula for the distance between them, and then use a trigonometric identity for addition. Then let

s=t1-t2and minimize.).

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

Let t=f(蟿)be a differentiable real-valued function of 蟿, and let r(t)be a differentiable vector function with three components such that f(蟿)is in the domain of rfor every value of 蟿on some interval I. Prove that drd蟿=drdtdtd蟿. (This is Theorem 11.8.)

In Exercises 19鈥�21 sketch the graph of a vector-valued function r(t)=x(t),y(t) with the specified properties. Be sure to indicate the direction of increasing values oft.

Domain

t鈭�鈩�,r2=1,1,r0=-3,3,r-2=-5,-5limt鈫�-鈭�x(t)=鈭�,limt鈫�-鈭�y(t)=-鈭�,andlimt鈫�鈭�r(t)=0,0.

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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