(a) Domain of the given vector valued function.

Question

Want to see more full solutions like this?

Answer 1

Question

Chapter 13, Problem 1CRE

To determine

(a)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(- 1, 0) \cup (0, 1]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$\Rightarrow r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Domain of the first component $t^{- 1} = \frac{1}{t}$ is all real numbers except 0. Thus, domain of the first component is $(- \infty, 0) \cup (0, \infty)$ .

Domain of the second component ${(t + 1)}^{- 1} = \frac{1}{t + 1}$ is all real numbers except $- 1$ . Thus, domain of the second component is $(- \infty, - 1) \cup (- 1, \infty)$ .

Domain of the third component $\sin^{- 1} t$ is $[- 1, 1]$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(- 1, 0) \cup (0, 1]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

To determine

(b)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(0, 2]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Domain of the first component $\sqrt{8 - t^{3}}$ is $(- \infty, 2]$ .

Domain of the second component $\ln t$ is $(0, \infty)$ .

Domain of the third component $e^{\sqrt{t}}$ is $[0, \infty)$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(0, 2]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Students have asked these similar questions

question 8

Find the area of the surface obtained by rotating the circle x² + y² = r² about the line y = r.

question 4 a and b

Answer 2

Question

Chapter 13, Problem 1CRE

To determine

(a)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(- 1, 0) \cup (0, 1]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$\Rightarrow r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Domain of the first component $t^{- 1} = \frac{1}{t}$ is all real numbers except 0. Thus, domain of the first component is $(- \infty, 0) \cup (0, \infty)$ .

Domain of the second component ${(t + 1)}^{- 1} = \frac{1}{t + 1}$ is all real numbers except $- 1$ . Thus, domain of the second component is $(- \infty, - 1) \cup (- 1, \infty)$ .

Domain of the third component $\sin^{- 1} t$ is $[- 1, 1]$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(- 1, 0) \cup (0, 1]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

To determine

(b)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(0, 2]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Domain of the first component $\sqrt{8 - t^{3}}$ is $(- \infty, 2]$ .

Domain of the second component $\ln t$ is $(0, \infty)$ .

Domain of the third component $e^{\sqrt{t}}$ is $[0, \infty)$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(0, 2]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Question

Chapter 13, Problem 1CRE

To determine

(a)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(- 1, 0) \cup (0, 1]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$\Rightarrow r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Domain of the first component $t^{- 1} = \frac{1}{t}$ is all real numbers except 0. Thus, domain of the first component is $(- \infty, 0) \cup (0, \infty)$ .

Domain of the second component ${(t + 1)}^{- 1} = \frac{1}{t + 1}$ is all real numbers except $- 1$ . Thus, domain of the second component is $(- \infty, - 1) \cup (- 1, \infty)$ .

Domain of the third component $\sin^{- 1} t$ is $[- 1, 1]$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(- 1, 0) \cup (0, 1]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

To determine

(b)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(0, 2]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Domain of the first component $\sqrt{8 - t^{3}}$ is $(- \infty, 2]$ .

Domain of the second component $\ln t$ is $(0, \infty)$ .

Domain of the third component $e^{\sqrt{t}}$ is $[0, \infty)$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(0, 2]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Question

Chapter 13, Problem 1CRE

To determine

(a)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(- 1, 0) \cup (0, 1]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$\Rightarrow r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Domain of the first component $t^{- 1} = \frac{1}{t}$ is all real numbers except 0. Thus, domain of the first component is $(- \infty, 0) \cup (0, \infty)$ .

Domain of the second component ${(t + 1)}^{- 1} = \frac{1}{t + 1}$ is all real numbers except $- 1$ . Thus, domain of the second component is $(- \infty, - 1) \cup (- 1, \infty)$ .

Domain of the third component $\sin^{- 1} t$ is $[- 1, 1]$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(- 1, 0) \cup (0, 1]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

To determine

(b)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(0, 2]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Domain of the first component $\sqrt{8 - t^{3}}$ is $(- \infty, 2]$ .

Domain of the second component $\ln t$ is $(0, \infty)$ .

Domain of the third component $e^{\sqrt{t}}$ is $[0, \infty)$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(0, 2]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Chapter 13, Problem 1CRE

To determine

(a)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(- 1, 0) \cup (0, 1]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$\Rightarrow r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Domain of the first component $t^{- 1} = \frac{1}{t}$ is all real numbers except 0. Thus, domain of the first component is $(- \infty, 0) \cup (0, \infty)$ .

Domain of the second component ${(t + 1)}^{- 1} = \frac{1}{t + 1}$ is all real numbers except $- 1$ . Thus, domain of the second component is $(- \infty, - 1) \cup (- 1, \infty)$ .

Domain of the third component $\sin^{- 1} t$ is $[- 1, 1]$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(- 1, 0) \cup (0, 1]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

To determine

(b)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(0, 2]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Domain of the first component $\sqrt{8 - t^{3}}$ is $(- \infty, 2]$ .

Domain of the second component $\ln t$ is $(0, \infty)$ .

Domain of the third component $e^{\sqrt{t}}$ is $[0, \infty)$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(0, 2]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

Answer 6

Chapter 13, Problem 1CRE

To determine

(a)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(- 1, 0) \cup (0, 1]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$\Rightarrow r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Domain of the first component $t^{- 1} = \frac{1}{t}$ is all real numbers except 0. Thus, domain of the first component is $(- \infty, 0) \cup (0, \infty)$ .

Domain of the second component ${(t + 1)}^{- 1} = \frac{1}{t + 1}$ is all real numbers except $- 1$ . Thus, domain of the second component is $(- \infty, - 1) \cup (- 1, \infty)$ .

Domain of the third component $\sin^{- 1} t$ is $[- 1, 1]$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(- 1, 0) \cup (0, 1]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

Answer 7

Chapter 13, Problem 1CRE

To determine

(a)

Domain of the given vector valued function.

Answer 8

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(- 1, 0) \cup (0, 1]$ .

Answer 9

Expert Solution

Answer 10

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$\Rightarrow r_{1} (t) = 〈 t^{- 1}, {(t + 1)}^{- 1}, \sin^{- 1} t 〉$

Domain of the first component $t^{- 1} = \frac{1}{t}$ is all real numbers except 0. Thus, domain of the first component is $(- \infty, 0) \cup (0, \infty)$ .

Domain of the second component ${(t + 1)}^{- 1} = \frac{1}{t + 1}$ is all real numbers except $- 1$ . Thus, domain of the second component is $(- \infty, - 1) \cup (- 1, \infty)$ .

Domain of the third component $\sin^{- 1} t$ is $[- 1, 1]$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(- 1, 0) \cup (0, 1]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

Answer 13

To determine

(b)

Domain of the given vector valued function.

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(0, 2]$ .

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Domain of the first component $\sqrt{8 - t^{3}}$ is $(- \infty, 2]$ .

Domain of the second component $\ln t$ is $(0, \infty)$ .

Domain of the third component $e^{\sqrt{t}}$ is $[0, \infty)$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(0, 2]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.

Answer 14

To determine

(b)

Domain of the given vector valued function.

Answer 15

Expert Solution

Answer to Problem 1CRE

Solution:

Domain of the given vector-valued function is $(0, 2]$ .

Answer 16

Expert Solution

Answer 17

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

Given:

We have been given a vector valued function:

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Key concepts used:

Domain of a rational function is all real numbers except when denominator is zero. Domain of inverse sine function is $[- 1, 1]$

Calculation:

In order to find the domain of a vector valued function, we find the domains of all the components of the vector function and then we consider the intersection of all the domains.

$r_{2} (t) = 〈 \sqrt{8 - t^{3}}, \ln t, e^{\sqrt{t}} 〉$

Domain of the first component $\sqrt{8 - t^{3}}$ is $(- \infty, 2]$ .

Domain of the second component $\ln t$ is $(0, \infty)$ .

Domain of the third component $e^{\sqrt{t}}$ is $[0, \infty)$ .

For writing the domain of the entire vector valued function, we consider the intersection of all three domains. Thus, the domain of the given vector valued function is $(0, 2]$ .

Conclusion:

The domain of the given vector valued function has been found by first finding the domains of each of the components and then considering the intersection.