**Finding the Domain of Vector Functions, $ \mathbf{r}(t) $**To determine the domain of a vector function, consider the domain restrictions of each component of the vector.### a) $ \mathbf{r}(t) = \left\langle \ln(1t), \sqrt{t + 3}, \frac{1}{\sqrt{16 - t}} \right\rangle $1. **$\ln(1t)$:** The argument of the logarithm must be positive: $ t > 0 $.2. **$\sqrt{t + 3}$:** The expression inside the square root must be non-negative: $ t + 3 \geq 0 $, which simplifies to $ t \geq -3 $.3. **$\frac{1}{\sqrt{16 - t}}$:** The expression inside the square root must be positive (as it's in the denominator): $ 16 - t > 0 $, which simplifies to $ t < 16 $.**Combined Domain:** The domain for part a is $ (0, 16) $.### b) $ \mathbf{r}(t) = \left\langle \sqrt{t - 2}, \sin(9t), t^2 \right\rangle $1. **$\sqrt{t - 2}$:** The expression inside the square root must be non-negative: $ t - 2 \geq 0 $, which simplifies to $ t \geq 2 $.2. **$\sin(9t)$:** The sine function is defined for all real numbers.3. **$t^2$:** The square function is defined for all real numbers.**Combined Domain:** The domain for part b is $ [2, \infty) $.### c) $ \mathbf{r}(t) = \left\langle e^{-2t}, \frac{t}{\sqrt{t^2 - 81}}, t^{1/3} \right\rangle $1. **$e^{-2t}$:** The exponential function is defined for all real numbers.2. **$\frac{t}{\sqrt{t^2 - 81}}$:** The expression inside the square root must be positive: - \( t^2 - 81

Answered: Image /qna-images/answer/3442cc80-d78b-456c-a82b-9ff3d605d576.jpg

Answered: Find the domain of the vector…

Find the domain of the vector functions, r(t), listed below. using interval notation. 1 a) r(t) = ( In(lt), vt + 3, /16 – t b) r(t) = ( Vt – 2, sin(9t), t² ) c) r(t) = (e-", t -2t t² – 81

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Determine the domain of the vector function r() = cos(4t) i + 7in(t - 5) J - 10 Evaluate if the…

Q: Find the domain of the vector function r(t)=⟨ln⁡(19t),sqrt(t+8),1/sqrt(7−t)⟩ using interval…

A: Given vector function isrt=ln19t,t+8,17-tfind the domain.

Q: . (Section 14.4) Let curve C be described by vector function(t) = (cost-1, sint, 3/2) on 0 ≤ t ≤ b.…

A: The curve C is defined by the function,on the interval,The length of the curve on the given interval…

Q: The acceleration of an object after t seconds is given by the vector-valued function a(t) . | If…

A: to find the velocity equation, take integration .............(1)

Q: Let ß be the vector-valued function 3u B(u, v) = { 32 ß: (-2, 2) × (0, 2π) → R³, 3u² + 4 COS(V) } u…

A: Since you have asked multiple questions, we will answer the questions (a) and (b) for you. If you…

Q: 6.Consider a function f: R2 R, where f(xy) = cos(xy). Let u = be a unit vector. Use the definition…

A: Here we will use Definition of Directional derivative of f(x , y) in the direction of vector <u1…

Q: Determine the interval(s) on which the vector-valued function is continuous. (Enter your answer…

A: A vector valued function of the form r(t)= f(t)i + g(t)j is continuous at the intersection of points…

Q: If the gradient of f is Vƒ = z j - yi – 2xzk and the poi f(x, y, z) = 0, find an equation for the…

Q: Find the domain of the vector functions, r(t), listed below. You may use "-INF" for -∞ and use "INF"…

A: We have to find the domain of vector function.

Q: Find the positive t when the vector r(t) = (4t, 4t, 5t2 – 6) is perpendicular to r' (t). t =

A: We use the definition of the perpendicular of two vectors to find the positive value of t.

Q: Determine where the vector function r(t) = i + (14t) j is continuous. Use symbolic notation and…

A: Please see the below picture for detailed solution.

Q: (b) part of parabola y = −2x² from (−1, −2) to (2, —–8)

A: Given Equation of parabola y=-2x2 from -1, -2 to 2, -8.

Q: sider the vector-valued function R(t) with Ả(1) = (1,2,−1) and Ř'(t) = = 1 2 (²²/7/7 - 2) - Find…

Q: For this problem, enter your vectors with angle-bracket notation: , not in ijk- notation. (a) F =…

A: The solution are next step

Q: Given the vector-valued functions u(t)= esti+e-¹ + 2tk v(t) = 2ti - 3t² + 3k d find (u(t) v(t)) when…

Q: In(t + 4) sin2t Given the vector valued function 7(t) = (e-3t, t a) Find the domain of 7(t). b) Find…

Q: Find the domain of the vector function. lim t->infinity

A: Answer

Q: 1 2. Determine the numbers at which the vector-valued function R(t) = t²i+ ln (t-1)j + 1 ) ³ + + = =…

Q: Find the directional derivative of the function at the given point in the direction of the vector v.…

A: Directional derivative of f(x,y) at any point P in the direction of vector v is, =(∇f)P.(unit vector…

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

**Finding the Domain of Vector Functions, $ \mathbf{r}(t) $**

To determine the domain of a vector function, consider the domain restrictions of each component of the vector.

### a) $ \mathbf{r}(t) = \left\langle \ln(1t), \sqrt{t + 3}, \frac{1}{\sqrt{16 - t}} \right\rangle $

1. **$\ln(1t)$:** The argument of the logarithm must be positive: $ t > 0 $.

2. **$\sqrt{t + 3}$:** The expression inside the square root must be non-negative: $ t + 3 \geq 0 $, which simplifies to $ t \geq -3 $.

3. **$\frac{1}{\sqrt{16 - t}}$:** The expression inside the square root must be positive (as it's in the denominator): $ 16 - t > 0 $, which simplifies to $ t < 16 $.

**Combined Domain:** The domain for part a is $ (0, 16) $.

### b) $ \mathbf{r}(t) = \left\langle \sqrt{t - 2}, \sin(9t), t^2 \right\rangle $

1. **$\sqrt{t - 2}$:** The expression inside the square root must be non-negative: $ t - 2 \geq 0 $, which simplifies to $ t \geq 2 $.

2. **$\sin(9t)$:** The sine function is defined for all real numbers.

3. **$t^2$:** The square function is defined for all real numbers.

**Combined Domain:** The domain for part b is $ [2, \infty) $.

### c) $ \mathbf{r}(t) = \left\langle e^{-2t}, \frac{t}{\sqrt{t^2 - 81}}, t^{1/3} \right\rangle $

1. **$e^{-2t}$:** The exponential function is defined for all real numbers.

2. **$\frac{t}{\sqrt{t^2 - 81}}$:** The expression inside the square root must be positive:
- \( t^2 - 81

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Vector-valued Function$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

For the vector valued function below, 1. Find the domain of each component. 2. Find the domain for the entire vector-valued function (as a single function). r(t) = √√16-²³₁+ i + (-4) k i+
(19) Consider the vector function 7(t) = (cos (3t), sin (3t) , 3t). Write 7" () in the form ar () T () + aN () N () by finding all four of these quantities.
Find a vector equation for the line that goes through the points (5,9,0) and (0, 11, 1). F(1) = with -o < t<∞ %D
Find the directional derivative of the function at the given point in the direction of vector v. f(x, y) = 3 + 6x√y, (3, 4), v=(4, -3) D₁f(3, 4) = 6 X
4) For the following vector functions, identify the shape of graph they would represent: a. r(t) = (2+ 3t, 5, -2-t) b. r(t) = (2, cos(3t), sin(3t)) c. r(t) = (t,5, t²)
2.2 please solve it on paper
find the derivitive, r'(t), of the vector function. r(t)=a+3tb+t^7c
Can you explain this question? The answer shown is correct, but I don't understand why we take the derivative to get the velocity.