Answered: Find the domain of the vector…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Find the domain of the vector functions, ?(?)�(�), listed below.
using interval notation. ) ?(?)=⟨ ?^-3t, 1/√t^2-4, t^1/3⟩.

i got this (-inf,-2)-(2,inf) but im still missing more intervals or sets in your union which i dont know how to do.

Expert Solution

Step 1: Introduction

To find the domain of the vector function $r (t) = ⟨ t^{-3}, t ^{2} - 4 1, t^{1/3} ⟩$ , we need to consider the values of $Error converting from MathML to accessible text.$ for which the components of the vector are defined.

The first component is $t^{-3}$ . This component is defined for all real values of $Error converting from MathML to accessible text.$ except $Error converting from MathML to accessible text.$ since division by zero is undefined. So, the domain for the first component is $Error converting from MathML to accessible text.$ .
The second component is. Here, we have a square root, so the expression under the square root must be non-negative. So, $t^{2} - 4$ must be greater than or equal to zero, which means $t^{2} \geq 4$ . Solving this inequality, we get $Error converting from MathML to accessible text.$ or $Error converting from MathML to accessible text.$ . However, we also need to exclude the points where the denominator becomes zero, which is when $t^{2} - 4 = 0$ or $Error converting from MathML to accessible text.$ . So, the domain for the second component is $Error converting from MathML to accessible text.$ .
The third component is $^{1/3}$ . This component is defined for all real values of $Error converting from MathML to accessible text.$ because taking the cube root of any real number is valid.

Now, to find the overall domain of the vector function, we need to consider the intersection of the domains for each component. So, we take the intersection of the domains from the first and second components:

notation.