/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Find a tangent vector at the giv... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 19

Find a tangent vector at the given value of \(t\) for the following parameterized curves. $$\mathbf{r}(t)=2 t^{4} \mathbf{i}+6 t^{3 / 2} \mathbf{j}+\frac{10}{t} \mathbf{k}, t=1$$

Short Answer

Expert verified
Answer: The tangent vector at \(t=1\) is \(\mathbf{T}(1) = 8\mathbf{i} + 9\mathbf{j} - 10\mathbf{k}\).

Step by step solution

01

Find the derivative of the parameterized curve with respect to \(t\)

To find the derivative of the given curve, we need to find the derivatives of its components with respect to \(t\). So, the derivative of the curve is given by: $$ \frac{d\mathbf{r}}{dt} = \frac{d(2 t^{4} \mathbf{i})}{dt}+\frac{d(6 t^{3 / 2} \mathbf{j})}{dt}+\frac{d(\frac{10}{t} \mathbf{k})}{dt} $$ Now let's find the derivative of each component: 1. \(\frac{d(2 t^{4} \mathbf{i})}{dt} = 8t^3\mathbf{i}\) 2. \(\frac{d(6 t^{3 / 2} \mathbf{j})}{dt} = 9t^{1/2}\mathbf{j}\) 3. \(\frac{d(\frac{10}{t} \mathbf{k})}{dt} = -\frac{10}{t^2}\mathbf{k}\) So, the derivative \(\frac{d\mathbf{r}}{dt}\) is given by: $$ \frac{d\mathbf{r}}{dt} = 8t^3\mathbf{i}+ 9t^{1/2}\mathbf{j}-\frac{10}{t^2}\mathbf{k} $$
02

Evaluate the derivative at \(t=1\) to find the tangent vector at that point

Now, we need to find the tangent vector at \(t=1\). To do this, substitute \(t=1\) in the derivative formula: $$ \frac{d\mathbf{r}}{dt}(1) = 8(1)^3\mathbf{i}+ 9(1)^{1/2}\mathbf{j}-\frac{10}{(1)^2}\mathbf{k} $$ This simplifies to: $$ \frac{d\mathbf{r}}{dt}(1) = 8\mathbf{i} + 9\mathbf{j} - 10\mathbf{k} $$ So, the tangent vector at \(t=1\) is: $$ \mathbf{T}(1) = 8\mathbf{i} + 9\mathbf{j} - 10\mathbf{k} $$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Cauchy-Schwarz Inequality The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\) (because \(|\cos \theta| \leq 1\) ). This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. Verify that the Cauchy-Schwarz Inequality holds for \(\mathbf{u}=\langle 3,-5,6\rangle\) and \(\mathbf{v}=\langle-8,3,1\rangle\)

Evaluate the following definite integrals. $$\int_{0}^{2} t e^{t}(\mathbf{i}+2 \mathbf{j}-\mathbf{k}) d t$$

An object moves along a path given by $$\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle$$ for \(0 \leq t \leq 2 \pi\) a. Show that the curve described by \(\mathbf{r}\) lies in a plane. b. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the curve described by \(\mathbf{r}\) is a circle?

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in the \(x y\) -plane and a and \(c\) are scalars. $$(a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}$$

Compute the following derivatives. $$\frac{d}{d t}\left(\left(t^{3} \mathbf{i}-2 t \mathbf{j}-2 \mathbf{k}\right) \times\left(t \mathbf{i}-t^{2} \mathbf{j}-t^{3} \mathbf{k}\right)\right)$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.