/*! 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 44 Compute the derivatives of the v... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 44

Compute the derivatives of the vector-valued functions. $$\mathbf{r}(t)=e^{t} \mathbf{i}+2 e^{t} \mathbf{j}+\mathbf{k}$$

Short Answer

Expert verified
The derivative is \( e^t \mathbf{i} + 2e^t \mathbf{j} \).

Step by step solution

01

Identify the Components

The vector-valued function \( \mathbf{r}(t) \) is given by \( e^{t} \mathbf{i} + 2e^{t} \mathbf{j} + \mathbf{k} \). Identify the individual components as functions of \( t \): \( x(t) = e^t \), \( y(t) = 2e^t \), and \( z(t) = 1 \).
02

Differentiate Each Component

Differentiate each component of \( \mathbf{r}(t) \) with respect to \( t \):1. \( \frac{dx}{dt} = \frac{d}{dt}(e^t) = e^t \).2. \( \frac{dy}{dt} = \frac{d}{dt}(2e^t) = 2e^t \).3. \( \frac{dz}{dt} = \frac{d}{dt}(1) = 0 \).
03

Combine the Derivatives

Combine the derivatives of the individual components into a single vector:\[ \mathbf{r}'(t) = e^t \mathbf{i} + 2e^t \mathbf{j} + 0 \cdot \mathbf{k} = e^t \mathbf{i} + 2e^t \mathbf{j} \].

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector-Valued Functions
A vector-valued function associates a vector to each value of a given variable, often denoted as \( t \). These functions have multiple components, each of which is a scalar function of \( t \). For instance, a vector-valued function \( \mathbf{r}(t) \) might be written in the form \( x(t)\mathbf{i} + y(t)\mathbf{j} + z(t)\mathbf{k} \).
  • Components: These are like the individual parts of the vector function. Instead of just one output, the function gives a vector, with each part following its own function of \( t \).
  • Example: In the function \( \mathbf{r}(t) = e^{t} \mathbf{i} + 2e^{t} \mathbf{j} + \mathbf{k} \), the components are \( e^t \), \( 2e^t \), and \( 1 \).
Understanding vector-valued functions is important as they are used to describe motions in space, complex systems in engineering, and fields in physics.
Derivatives
The derivative of a vector-valued function is very similar to that of scalar functions, but we find the derivative of each component individually. The result is a new vector comprised of the derivatives of the original components.
  • Symbol: Just like scalar functions, the derivative of a vector \( \mathbf{r}(t) \) is denoted as \( \mathbf{r}'(t) \).
  • Example: If \( \mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j} + z(t)\mathbf{k} \), then \( \mathbf{r}'(t) = x'(t)\mathbf{i} + y'(t)\mathbf{j} + z'(t)\mathbf{k} \).
Calculating derivatives helps in understanding rates of change and motion along paths, making them essential in calculus and physics.
Differentiation
Differentiation is the process of finding a derivative, which involves calculating the rate at which a function changes. For vector-valued functions, differentiation is applied component by component.When you differentiate \( \mathbf{r}(t) = e^{t} \mathbf{i} + 2e^{t} \mathbf{j} + \mathbf{k} \), you treat each component as a separate function of \( t \):
  • For \( x(t) = e^t \): The derivative is \( e^t \).
  • For \( y(t) = 2e^t \): The derivative is \( 2e^t \).
  • For \( z(t) = 1 \): The derivative is 0, since constants have a derivative of zero.
This step-by-step differentiation shows how each part of the vector is treated separately yet combined back into a single vector derivative.
Components of Vectors
Vectors in three-dimensional space have three components, often associated with the standard basis vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \), which align with the x-, y-, and z-axes, respectively.
  • Identifying Components: Each scalar expression multiplied by the unit vector (\( \mathbf{i} \), \( \mathbf{j} \), \( \mathbf{k} \)) is a component. For example, in \( \mathbf{r}(t) = e^{t}\mathbf{i} + 2e^{t}\mathbf{j} + \mathbf{k} \), \( e^t\mathbf{i} \) is a component.
  • Analysis and Application: Understanding these components helps in breaking down complex motions into simpler parts. It allows easier manipulation and analysis of changes in systems.
The components of a vector reflect how much influence each axis has in defining the direction and magnitude in space.

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

Evaluate the following integrals. $$ \int_{1}^{4} \mathbf{u}(t) d t, \text { with } \mathbf{u}(t)=\left\langle\frac{\ln (t)}{t}, \frac{1}{\sqrt{t}}, \sin \left(\frac{t \pi}{4}\right)\right\rangle $$

Consider the curve described by the vector-valued function \(\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}\) What is \(\lim _{t \rightarrow \infty} \mathbf{r}(t) ?\)

Calculate the curvature of the circular helix \(\mathbf{r}(t)=r \sin (t) \mathbf{i}+r \cos (t) \mathbf{j}+t \mathbf{k}\)

Given the vector-valued function \(\mathbf{r}(t)=\langle\tan t, \sec t, 0\rangle\) (graph is shown here), find the following: Speed

Find the unit tangent vector \(\mathbf{T}(t)\) for the following vector-valued functions. $$\mathbf{r}(t)=\langle t+1,2 t+1,2 t+2\rangle$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Calculus

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.