Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

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2,819,387 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces vector functions as mappings from real numbers to vectors in three-dimensional space, emphasizing their component functions, limits, and continuity. It highlights how these functions form space curves—geometric paths traced by moving particles—and explains how to convert between vector, parametric, and geometric representations. Moreover, practical visualization with computer graphics is discussed to aid understanding of complex curves such as helices, twisted cubics, and more.

Learning Objectives

1

Define and describe vector functions and their component functions.

2

Determine the domain of a given vector function by analyzing its component functions.

3

Apply the concept of limits and continuity to vector functions and interpret their geometric significance.

4

Translate vector equations into parametric equations to describe space curves such as lines, helices, and twisted cubics.

5

Utilize computer graphing techniques to visualize complex space curves and understand their projections.

Key Concepts

CONCEPT

DEFINITION

Vector Function

A function whose domain is a set of real numbers and whose range is a set of vectors, typically in three-dimensional space. It is usually expressed in the form r(t) = <f(t), g(t), h(t)>.

Component Functions

The individual real-valued functions f(t), g(t), and h(t) that represent the x, y, and z components of a vector function r(t).

Domain

The set of all values of the independent variable t for which the vector function is defined, determined by the domains of its component functions.

Continuity

A vector function r(t) is continuous at a point t = a if the limit as t approaches a of r(t) equals r(a); equivalently, each component function must be continuous at t = a.

Space Curve

A set of points in space defined by parametric equations x = f(t), y = g(t), z = h(t), which represents the path traced by the tip of the vector r(t) as t varies over a given interval.

Parametric Equations

Equations that express the coordinates of the points on a curve as functions of a parameter t, which provide an alternative description of curves in space.

Example Problems

Example 1

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

Example 2

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

Example 3

Find the limit. $ \lim_{t\to 0} \left(e^{-3t} i + \frac{t^2}{\sin^2 t}\ j + \cos 2t\ k \right) $

Example 4

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

Example 5

Find the limit. $ \lim_{t\to\infty} \biggr\langle\frac{1 + t^2}{1 - t^2}, \tan^{-1} t , \frac{1 - e^{-2t}}{t} \biggr\rangle $
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Step-by-Step Explanations

QUESTION

Find lim (t → 0) r(t), where r(t) = < (1/t) * t^3, t*e^(2t), sin(t)/t >.

STEP-BY-STEP ANSWER:

Step 1: Identify the component functions: f(t) = t^3/t = t^2, g(t) = t*e^(2t), and h(t) = sin(t)/t.
Step 2: Compute the limit for each component individually.
Step 3: For f(t): lim (t → 0) t^2 = 0.
Step 4: For g(t): lim (t → 0) t*e^(2t) = 0*e^0 = 0.
Step 5: For h(t): Use the standard limit: lim (t → 0) sin(t)/t = 1.
Step 6: Combine the component limits to obtain the limit of the vector function: lim (t → 0) r(t) = <0, 0, 1>.
Final Answer: <0, 0, 1>.

Limit of a Vector Function

QUESTION

Determine the vector equation for the line segment connecting P(1, 3, 22) to Q(2, 21, 3).

STEP-BY-STEP ANSWER:

Step 1: Write the general formula for a line segment: r(t) = (1-t)r0 + t r1, where 0 ≤ t ≤ 1.
Step 2: Identify r0 = <1, 3, 22> and r1 = <2, 21, 3>.
Step 3: Substitute into the formula: r(t) = <1 + t*(2-1), 3 + t*(21-3), 22 + t*(3-22)>.
Step 4: Simplify to obtain: r(t) = <1 + t, 3 + 18t, 22 - 19t> for 0 ≤ t ≤ 1.
Final Answer: r(t) = <1 + t, 3 + 18t, 22 - 19t>, 0 ≤ t ≤ 1.

Finding a Vector Equation for a Line Segment

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Common Mistakes

  • Neglecting to consider the domain restrictions imposed by individual component functions.
  • Assuming that the limit of a vector function can be computed without separately evaluating the limit of each component.
  • Confusing parametric equations with Cartesian equations, leading to errors in sketching curves.
  • Overlooking the need to analyze both the magnitude and direction of vector functions when discussing continuity and limits.