Consider the space curve C given by the vector function f(t) = (t² + 1)i + (2t4 – 3t2)j +t³k where t e [-1,2]. Í(t) %3D (a) By inspection, state and give a brief reason in words, whether or not C is smooth. (b) By inspection, state and give a brief reason in words, whether or not the tangent line to C at the point (1,0,0) exists. (You may use normal print f or “the vector function" and f' or “the derivative" for f and f' respectively in typing up your brief response when referring to the function and its derivative, respectively. Give statements like, for example “The r-component of f is not continuous", or “The z-component of the deriva- tive is not zero when t = 0", etc. You are not required to type in the actual %3D algebraic expression.)

Consider the space curve C given by the vector function f(t) = (t² + 1)i + (2t4 – 3t2)j +t³k where t e [-1,2]. Í(t) %3D (a) By inspection, state and give a brief reason in words, whether or not C is smooth. (b) By inspection, state and give a brief reason in words, whether or not the tangent line to C at the point (1,0,0) exists. (You may use normal print f or “the vector function" and f' or “the derivative" for f and f' respectively in typing up your brief response when referring to the function and its derivative, respectively. Give statements like, for example “The r-component of f is not continuous", or “The z-component of the deriva- tive is not zero when t = 0", etc. You are not required to type in the actual %3D algebraic expression.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.

Question

Consider the space curve C given by the vector function f(t) = (t² + 1)i +
(2t4 – 3t2)j + t³k where t e [-1,2].
(a) By inspection, state and give a brief reason in words, whether or not C
is smooth.
(b) By inspection, state and give a brief reason in words, whether or not the
tangent line to C at the point (1,0,0) exists.
(You may use normal print f or “the vector function" and f' or “the derivative"
for f and f' respectively in typing up your brief response when referring to the
function and its derivative, respectively. Give statements like, for example
"The r-component of f is not continuous", or “The z-component of the deriva-
tive is not zero when t = 0", etc. You are not required to type in the actual
algebraic expression.)

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