Book cover for Calculus Early Transcendentals 2

Calculus Early Transcendentals 2

James Stewart

ISBN #9781285741550

8th Edition

6,468 Questions

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62,208 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Vector functions extend the concept of functions to three-dimensional vectors, allowing us to represent curves and paths in space. By analyzing the component functions, we can determine domains, compute limits, and establish continuity. Space curves, defined via parametric equations, give rise to familiar shapes such as lines, helices, and twisted cubics, and their analysis is critical in fields ranging from physics to computer graphics.

Learning Objectives

1

Define a vector function and its component functions, and determine its domain.

2

Explain how to compute limits and test for continuity of vector functions component?wise.

3

Describe space curves and convert between vector equations and parametric equations.

4

Recognize and analyze special curves like lines, helices, and twisted cubics in three-dimensional space.

5

Utilize computer tools to graph complex space curves and interpret their projections.

Key Concepts

CONCEPT

DEFINITION

Vector Function

A function whose domain is a set of real numbers and whose range consists of vectors, typically in three dimensions. It is commonly written as r(t) = <f(t), g(t), h(t)>.

Component Functions

The individual real–valued functions f(t), g(t), and h(t) that make up the vector function r(t), representing its x, y, and z components respectively.

Domain

The set of all t-values for which the vector function (and, therefore, each of its component functions) is defined.

Limit of a Vector Function

Defined by taking the limit of each component function; that is, lim(t→a) r(t) = <lim(t→a) f(t), lim(t→a) g(t), lim(t→a) h(t)>.

Continuity

A vector function is continuous at a point if each of its component functions is continuous at that point.

Space Curve

The set of points in space given by the coordinates <f(t), g(t), h(t)>, often described by parametric equations, and traced out by the tip of the vector function as the parameter varies.

Parametric Equations

Representation of a curve by expressing the coordinates x, y, and z as functions of a parameter t, which can be derived from a vector function.

Helix

A space curve defined, for example, by r(t) = <cos t, sin t, t>. Its projection onto the xy–plane is a circle while the z–component increases steadily, resulting in a spiral.

Example Problems

Example 1

$1-2$ Find the domain of the vector function. $$ \mathbf{r}(t)=\left\langle\ln (t+1), \frac{t}{\sqrt{9-t^{2}}}, 2^{t}\right\rangle $$

Example 2

Find the domain of the vector function. $$ \mathbf{r}(t)=\cos t \mathbf{i}+\ln t \mathbf{j}+\frac{1}{t-2} \mathbf{k} $$

Example 3

$3-6$ Find the limit. $$ \lim _{t \rightarrow 0}\left(e^{-3 t} \mathbf{i}+\frac{t^{2}}{\sin ^{2} t} \mathbf{j}+\cos 2 t \mathbf{k}\right) $$

Example 4

Find the limit. $$ \lim _{t \rightarrow 1}\left(\frac{t^{2}-t}{t-1} \mathbf{i}+\sqrt{t+8} \mathbf{j}+\frac{\sin \pi t}{\ln t} \mathbf{k}\right) $$

Example 5

Find the limit. $$ \lim _{t \rightarrow \infty}\left\langle\frac{1+t^{2}}{1-t^{2}}, \tan ^{-1} t, \frac{1-e^{-2 t}}{t}\right\rangle $$
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Step-by-Step Explanations

QUESTION

Find the domain of the vector function r(t) = <tÂł, ln(3t + 2), sin t>.

STEP-BY-STEP ANSWER:

Step 1: Identify each component’s domain. For t³ and sin t, they are defined for all real numbers.
Step 2: For the second component ln(3t + 2), set the argument > 0: 3t + 2 > 0.
Step 3: Solve the inequality: t > -2/3.
Step 4: The domain of r(t) is the intersection of the domains of all components, which here is t > -2/3.
Final Answer:

Determining the Domain of a Vector Function

QUESTION

Find lim(t → 0) r(t) where r(t) = <(sin t)/t, t·e^(2t), (sin t)/t>.

STEP-BY-STEP ANSWER:

Step 1: Write the limit of the vector function as the vector of the limits of its components.
Step 2: Calculate lim(t → 0) (sin t)/t, which is 1.
Step 3: Calculate lim(t → 0) t·e^(2t); as t → 0, this limit is 0.
Step 4: Compute the limit for the third component (sin t)/t which is again 1.
Step 5: Combine the results to get the final vector limit: <1, 0, 1>.
Final Answer:

Computing the Limit of a Vector Function

QUESTION

Describe the space curve defined by r(t) = <cos t, sin t, t>.

STEP-BY-STEP ANSWER:

Step 1: Identify the component functions: x = cos t, y = sin t, and z = t.
Step 2: Note that x² + y² = cos²t + sin²t = 1, so the projection on the xy–plane is a circle.
Step 3: Recognize that z = t increases linearly, which causes the circular path to lift continuously.
Step 4: Conclude that the curve is a helix that spirals upward as t increases.
Final Answer:

Describing a Space Curve from Its Vector Equation

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Common Mistakes

  • Assuming that the properties of scalar limits automatically apply without checking the domain restrictions of each component function.
  • Forgetting to take the limit component–wise, which can lead to incorrect vector limits.
  • Overlooking the importance of parameter domains when converting between vector equations and parametric equations.
  • Misinterpreting the projection of a space curve onto a plane, leading to a misunderstanding of the curve’s true three-dimensional nature.