Book cover for Calculus Early Transcendental Functions

Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

ISBN #9781285774770

6th Edition

8,973 Questions

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64,375 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces vector–valued functions as mappings from real numbers to vectors and explains how they serve as a framework for describing curves in both the plane and space. Key concepts such as parametric equations, domain restrictions, and the component–wise approach for limits, continuity, and differentiation are emphasized. Understanding these ideas provides a solid foundation for analyzing the geometry of curves and the dynamics of particle motion.

Learning Objectives

1

Describe the definition and properties of vector?valued functions in both the plane and space.

2

Explain how to represent curves using parametric equations and convert them into rectangular forms when possible.

3

Apply the concepts of limits and continuity to vector?valued functions through component–wise evaluation.

4

Differentiate vector–valued functions to obtain tangent vectors and analyze particle motion along space curves.

Key Concepts

CONCEPT

DEFINITION

Vector–Valued Function

A function that maps real numbers to vectors. In the plane, it is often expressed as r(t) = <f(t), g(t)>, and in space, r(t) = <f(t), g(t), h(t)> where f, g, and h are real–valued functions of t.

Space Curve

The set of points in three-dimensional space described by a vector–valued function. It is given by all ordered triples (x, y, z) determined by the component functions.

Parametric Equations

Equations that express the coordinates of the points on a curve in terms of a parameter (t). They are derived from a vector–valued function by equating the vector’s components.

Limit of a Vector–Valued Function

Defined component–wise; the limit of r(t) as t approaches a is <lim f(t), lim g(t), lim h(t)> provided these limits exist.

Continuity

A vector–valued function is continuous at a point t = a if the limit of r(t) as t approaches a equals r(a), which holds if each component function is continuous at a.

Derivative of a Vector–Valued Function

Obtained by differentiating the component functions; r'(t) = <f'(t), g'(t), h'(t)>, which represents the tangent vector to the curve at t.

Example Problems

Example 1

Find the domain of the vector-valued function. $$\mathbf{r}(t)=\frac{1}{t+1} \mathbf{i}+\frac{t}{2} \mathbf{j}-3 t \mathbf{k}$$

Example 2

Find the domain of the vector-valued function. $$\mathbf{r}(t)=\sqrt{4-t^{2}} \mathbf{i}+t^{2} \mathbf{j}-6 t \mathbf{k}$$

Example 3

Find the domain of the vector-valued function. $$\mathbf{r}(t)=\ln t \mathbf{i}-e^{t} \mathbf{j}-t \mathbf{k}$$

Example 4

Find the domain of the vector-valued function. $$\mathbf{r}(t)=\sin t \mathbf{i}+4 \cos t \mathbf{j}+t \mathbf{k}$$

Example 5

Find the domain of the vector-valued function. $$\begin{aligned} &\mathbf{r}(t)=\mathbf{F}(t)+\mathbf{G}(t), \text { where }\\ &\mathbf{F}(t)=\cos t \mathbf{i}-\sin t \mathbf{j}+\sqrt{t} \mathbf{k}, \quad \mathbf{G}(t)=\cos t \mathbf{i}+\sin t \mathbf{j} \end{aligned}$$
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Step-by-Step Explanations

QUESTION

Given the vector–valued function r(t) = <4 cos(t), 4 sin(t)>, how do you sketch the corresponding curve and determine its orientation?

STEP-BY-STEP ANSWER:

Step 1: Write the parametric equations: x = 4 cos(t) and y = 4 sin(t).
Step 2: Eliminate the parameter t by using the identity cos²(t) + sin²(t) = 1, yielding (x/4)² + (y/4)² = 1, which is the equation of a circle of radius 4.
Step 3: Determine the orientation by considering the sign of the trigonometric functions as t increases (the curve is traced in a counterclockwise or clockwise direction based on the parameter’s progression).
Final Answer: The curve is a circle with a radius of 4 traced in the direction determined by the increasing value of t.

Sketching a Plane Curve

QUESTION

How do you verify the continuity of a vector–valued function r(t) = <f(t), g(t), h(t)> at a point t = a?

STEP-BY-STEP ANSWER:

Step 1: Evaluate each component function f(t), g(t), and h(t) at t = a to ensure r(a) is defined.
Step 2: Compute the limit of each component, i.e., lim (t → a) f(t), lim (t → a) g(t), and lim (t → a) h(t).
Step 3: Verify that each of these limits equals the corresponding component of r(a).
Final Answer: r(t) is continuous at t = a if and only if lim (t → a) f(t) = f(a), lim (t → a) g(t) = g(a), and lim (t → a) h(t) = h(a).

Determining Continuity

QUESTION

Given r(t) = <f(t), g(t), h(t)>, how do you determine its derivative?

STEP-BY-STEP ANSWER:

Step 1: Differentiate each component function with respect to t to obtain f'(t), g'(t), and h'(t).
Step 2: Form the derivative vector r'(t) = <f'(t), g'(t), h'(t)>, which represents the tangent vector to the curve at any t.
Final Answer: The derivative r'(t) is found by differentiating each component, yielding r'(t) = <f'(t), g'(t), h'(t)>.

Differentiation of Vector–Valued Functions

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Common Mistakes

  • Failing to evaluate the limit and continuity component–wise, which may lead to incorrect conclusions about smoothness and existence of limits.
  • Confusing the graph (or trace) of a curve with its parametrization; different parameterizations can trace the same curve in different ways.
  • Overlooking domain restrictions imposed by individual component functions.
  • Misinterpreting the orientation of the curve by not carefully considering the increasing parameter value.