Book cover for University Calculus: Early Transcendentals

University Calculus: Early Transcendentals

Joel Hass, Christopher Heil, Przemyslaw Bogacki

ISBN #9780134995540

4th Edition

7,111 Questions

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266,911 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces vector-valued functions and their calculus, emphasizing that both magnitude and direction can change. By differentiating or integrating these functions, we obtain vectors that describe motion, such as velocity and acceleration. These tools are fundamental in analyzing trajectories and motion in both two-dimensional planes and three-dimensional space.

Learning Objectives

1

Explain the concept of vector-valued functions and distinguish them from scalar functions.

2

Describe how derivatives and integrals apply to vector-valued functions, including their geometric interpretation.

3

Apply vector calculus to analyze motion in space, including calculating velocity and acceleration vectors.

4

Interpret the role of magnitude and direction changes in the context of moving objects.

Key Concepts

CONCEPT

DEFINITION

Vector-Valued Function

A function that takes real numbers as inputs and returns vectors as outputs, representing quantities with both magnitude and direction.

Derivative (of a vector function)

The rate at which a vector-valued function changes. Its derivative is also a vector that captures changes in both magnitude and direction, often interpreted as the velocity of a moving object.

Integral (of a vector function)

The accumulation of a vector function over an interval, resulting in a vector that represents the net change in the function, akin to computing displacement from velocity.

Velocity Vector

The derivative of the position vector with respect to time, indicating the speed and direction of an object's motion.

Acceleration Vector

The derivative of the velocity vector with respect to time, describing how the velocity changes with time, in terms of both speed and direction.

Example Problems

Example 1

Find the given limits. $$\lim _{t \rightarrow \pi}\left[\left(\sin \frac{t}{2}\right) \mathrm{i}+\left(\cos \frac{2}{3} t\right) \mathbf{j}+\left(\tan \frac{5}{4} t\right) \mathbf{k}\right]$$

Example 2

Find the given limits. $$\lim _{t \rightarrow-1}\left[t^{3} \mathbf{i}+\left(\sin \frac{\pi}{2} t\right) \mathbf{j}+(\ln (t+2)) \mathbf{k}\right]$$

Example 3

Find the given limits. $$\lim _{t \rightarrow 1}\left[\left(\frac{t^{2}-1}{\ln t}\right) \mathbf{i}-\left(\frac{\sqrt{t}-1}{1-t}\right) \mathbf{j}+\left(\tan ^{-1} t\right) \mathbf{k}\right]$$

Example 4

Find the given limits. $$\lim _{t \rightarrow 0}\left[\left(\frac{\sin t}{t}\right) \mathrm{i}+\left(\frac{\tan ^{2} t}{\sin 2 t}\right) \mathbf{j}-\left(\frac{t^{3}-8}{t+2}\right) \mathbf{k}\right]$$

Example 5

Is the position of a particle in the $x y$ -plane at time $t .$ Find an equation in $x$ and $y$ whose graph is the path of the particle. Then find the particle's velocity and acceleration vectors at the given value of $t$. $$\mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}, \quad t=1$$
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Step-by-Step Explanations

QUESTION

How do you differentiate a vector-valued function r(t) = <f(t), g(t), h(t)> with respect to t?

STEP-BY-STEP ANSWER:

Step 1: Identify that r(t) is composed of three separate component functions: f(t), g(t), and h(t).
Step 2: Differentiate each component function with respect to t independently.
Step 3: Assemble the derivatives into a new vector: r'(t) = <f'(t), g'(t), h'(t)>.
Step 4: Interpret r'(t) as the velocity vector if r(t) represents the position of an object in space.
Final Answer: The derivative of r(t) is r'(t) = <f'(t), g'(t), h'(t)>.

Differentiation of a Vector-Valued Function

QUESTION

How do you integrate a vector-valued function r(t) = <f(t), g(t), h(t)> over an interval [a, b]?

STEP-BY-STEP ANSWER:

Step 1: Recognize that integration applies to each component separately.
Step 2: Compute the integral of each component: ∫[a to b] f(t) dt, ∫[a to b] g(t) dt, and ∫[a to b] h(t) dt.
Step 3: Combine these integrals to form the resulting vector: ∫[a to b] r(t) dt = <∫[a to b] f(t) dt, ∫[a to b] g(t) dt, ∫[a to b] h(t) dt>.
Step 4: Interpret the resulting vector as the net displacement if r(t) represents a velocity vector.
Final Answer: The integral of r(t) over [a, b] is <∫[a to b] f(t) dt, ∫[a to b] g(t) dt, ∫[a to b] h(t) dt>.

Integration of a Vector-Valued Function

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Common Mistakes

  • Confusing vector-valued functions with scalar functions, leading to omission of directional components.
  • Failing to differentiate or integrate each component individually.
  • Interpreting the derivative of a vector function as a scalar rather than as a vector with both magnitude and direction.
  • Neglecting to consider the constant of integration in vector integrals, which may represent initial conditions in motion problems.