Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

William Briggs, Lyle Cochran, Bernard Gillet

ISBN #9780134763644

3rd Edition

8,978 Questions

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Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces vector-valued functions as a powerful tool to represent dynamic motion in space. It explains the equivalence of writing parametric equations in vector form and emphasizes how calculus operations can be extended to these functions. By studying limits, continuity, and the geometric properties of space curves (e.g., tangents and curvature), students learn to model and analyze complex trajectories like spirals, roller coaster paths, and slinky curves.

Learning Objectives

1

Describe the concept of vector-valued functions and their role in representing continuous motion in space.

2

Explain how vector-valued functions are equivalent to sets of parametric equations and how to interpret their graphs.

3

Apply calculus concepts such as limits, derivatives, and integrals to vector-valued functions.

4

Analyze the orientation, curvature, and other geometric properties of space curves generated by vector functions.

Key Concepts

CONCEPT

DEFINITION

Vector-Valued Function

A function of the form r(t) = ⟨f(t), g(t), h(t)⟩, where each component is a function of a single variable t (often time), representing a curve in space.

Parametric Equations

A set of equations x = f(t), y = g(t), and z = h(t) that describe a curve in space; equivalently written in vector form as r(t) = ⟨f(t), g(t), h(t)⟩.

Tangent Vector

A vector that touches a curve at a point and has the same direction as the curve’s instantaneous motion (i.e., derivative of the vector function).

Limit of a Vector-Valued Function

Defined as lim(t→a) r(t) = L if and only if lim(t→a) ||r(t) - L|| = 0, meaning that each component of the vector function approaches the corresponding component of L.

Continuity

A vector function r(t) is continuous at t = a if lim(t→a) r(t) = r(a); this ensures that the curve it represents has no breaks or gaps.

Example Problems

Example 1

How many independent variables does the function $\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle$ have $?$

Example 2

How many dependent scalar variables does the function $\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle$ have?

Example 3

Why is $\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle$ called a vector-valued function?

Example 4

In what plane does the curve $\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{k}$ lie?

Example 5

$$\text { How do you evaluate } \lim \mathbf{r}(t), \text { where } \mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle ?$$
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Step-by-Step Explanations

QUESTION

How do you construct a vector function for the line passing through two points P and Q?

STEP-BY-STEP ANSWER:

Step 1: Identify the starting point P = ⟨x0, y0, z0⟩ and the ending point Q = ⟨x1, y1, z1⟩.
Step 2: Calculate the direction vector by subtracting P from Q, i.e., v = Q - P.
Step 3: Write the parametric equations: x = x0 + v₁t, y = y0 + v₂t, z = z0 + v₃t.
Step 4: Express the line as a vector function: r(t) = ⟨x0 + v₁t, y0 + v₂t, z0 + v₃t⟩.
Final Answer: The vector function representing the line is r(t) = ⟨x0 + (x1 - x0)t, y0 + (y1 - y0)t, z0 + (z1 - z0)t⟩.

Finding a Vector Function for a Line

QUESTION

How do you compute the limit of a vector-valued function as t approaches a point a?

STEP-BY-STEP ANSWER:

Step 1: Write the vector-valued function r(t) = ⟨f(t), g(t), h(t)⟩.
Step 2: Compute the limits of the individual components: L₁ = lim(t→a) f(t), L₂ = lim(t→a) g(t), and L₃ = lim(t→a) h(t).
Step 3: Combine these limits to get the limit of the vector function: lim(t→a) r(t) = ⟨L₁, L₂, L₃⟩.
Final Answer: The limit is ⟨lim(t→a) f(t), lim(t→a) g(t), lim(t→a) h(t)⟩.

Evaluating Limits of a Vector Function

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Common Mistakes

  • Assuming that vector functions are fundamentally different from parametric equations instead of alternate representations.
  • Ignoring the component-wise approach when evaluating limits, derivatives, or integrals.
  • Forgetting to consider the domain of all component functions when analyzing continuity or motion.
  • Misinterpreting the orientation of a curve and the direction of its tangent vectors.