/*! 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 14 Differentiate the following func... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 14

Differentiate the following functions. $$\mathbf{r}(t)=\left\langle(t+1)^{-1}, \tan ^{-1} t, \ln (t+1)\right\rangle$$

Short Answer

Expert verified
Question: Find the derivative of the vector function \(\mathbf{r}(t) = \left\langle (t+1)^{-1}, \tan^{-1} t, \ln (t+1) \right\rangle\). Answer: The derivative of the vector function \(\mathbf{r}(t)\) is \(\mathbf{r}'(t) = \left\langle -\frac{1}{(t+1)^2}, \frac{1}{1+t^2}, \frac{1}{t+1} \right\rangle\).

Step by step solution

01

Differentiate the first component

To differentiate the first component \((t+1)^{-1}\), use the power rule: If \(f(t) = t^n\), then \(f'(t) = n t^{n-1}\). So, the derivative of \((t+1)^{-1}\) is: $$\frac{d}{dt} \left((t+1)^{-1}\right) = -1 (t+1)^{-1-1} = -\frac{1}{(t+1)^2}$$
02

Differentiate the second component

To differentiate the second component \(\tan ^{-1} t\), recall that the derivative of \(\tan^{-1} t\) is: $$\frac{d}{dt} \left(\tan ^{-1} t\right) = \frac{1}{1+t^2}$$ So, the derivative of the second component is \(\frac{1}{1+t^2}\).
03

Differentiate the third component

To differentiate the third component \(\ln (t+1)\), recall that the derivative of \(\ln x\) is \(\frac{1}{x}\). Using the chain rule, we get the derivative of \(\ln(t+1)\) as: $$\frac{d}{dt} \left(\ln(t+1)\right) = \frac{1}{t+1}$$
04

Write the derivative in the vector form

We now have the derivatives of each of the three components: - First component: \(-\frac{1}{(t+1)^2}\) - Second component: \(\frac{1}{1+t^2}\) - Third component: \(\frac{1}{t+1}\) Now, we can write the derivative of the vector function \(\mathbf{r}(t)\) as: $$\mathbf{r}'(t) = \left\langle -\frac{1}{(t+1)^2}, \frac{1}{1+t^2}, \frac{1}{t+1} \right\rangle$$ This is the final answer for the differentiation of the given function.

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

A pair of lines in \(\mathbb{R}^{3}\) are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect. determine the point(s) of intersection. $$\begin{array}{l} \mathbf{r}(t)=\langle 4+t,-2 t, 1+3 t\rangle ;\\\ \mathbf{R}(s)=\langle 1-7 s, 6+14 s, 4-21 s\rangle \end{array}$$

Consider the ellipse \(\mathbf{r}(t)=\langle a \cos t, b \sin t\rangle\) for \(0 \leq t \leq 2 \pi,\) where \(a\) and \(b\) are real numbers. Let \(\theta\) be the angle between the position vector and the \(x\) -axis. a. Show that \(\tan \theta=(b / a) \tan t\) b. Find \(\theta^{\prime}(t)\) c. Note that the area bounded by the polar curve \(r=f(\theta)\) on the interval \([0, \theta]\) is \(A(\theta)=\frac{1}{2} \int_{0}^{\theta}(f(u))^{2} d u\) Letting \(f(\theta(t))=|\mathbf{r}(\theta(t))|,\) show that \(A^{\prime}(t)=\frac{1}{2} a b\) d. Conclude that as an object moves around the ellipse, it sweeps out equal areas in equal times.

Show that the two-dimensional trajectory $$x(t)=u_{0} t+x_{0}\( and \)y(t)=-\frac{g t^{2}}{2}+v_{0} t+y_{0},\( for \)0 \leq t \leq T$$ of an object moving in a gravitational field is a segment of a parabola for some value of \(T>0 .\) Find \(T\) such that \(y(T)=0\)

Suppose an object moves on the surface of a sphere with \(|\mathbf{r}(t)|\) constant for all \(t\) Show that \(\mathbf{r}(t)\) and \(\mathbf{a}(t)=\mathbf{r}^{\prime \prime}(t)\) satisfy \(\mathbf{r}(t) \cdot \mathbf{a}(t)=-|\mathbf{v}(t)|^{2}\)

Determine whether the following statements are true using a proof or counterexample. Assume that \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are nonzero vectors in \(\mathbb{R}^{3}\). $$\mathbf{u} \times(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \cdot \mathbf{w}) \mathbf{v}-(\mathbf{u} \cdot \mathbf{v}) \mathbf{w}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.