Problem 44 Compute the derivatives of the v... [FREE SOLUTION]

91影视

Calculus Volume 3

Gilbert Strang

$Math Studyset 91影视 Explanations$ Math

3 Edition

Chapter 3: Problem 44

Compute the derivatives of the vector-valued functions. $\mathbf{r}(t)=e^{t} \mathbf{i}+2 e^{t} \mathbf{j}+\mathbf{k}$

Short Answer

Expert verified

$\mathbf{r}'(t) = e^t \mathbf{i} + 2e^t \mathbf{j} $

Step by step solution

Identify the Components of the Vector-valued Function

The given vector-valued function is $ \mathbf{r}(t) = e^{t} \mathbf{i} + 2e^{t} \mathbf{j} + \mathbf{k} $. This can be broken down into its component functions: $ x(t) = e^t $, $ y(t) = 2e^t $, and $ z(t) = 1 $.

Compute the Derivative of the First Component Function

Compute the derivative of the first component function $ x(t) = e^t $. Using the derivative rule for exponential functions, the derivative is $ \frac{dx}{dt} = e^t $.

Compute the Derivative of the Second Component Function

Compute the derivative of the second component function $ y(t) = 2e^t $. Since this function is a constant multiple of an exponential function, its derivative is $ \frac{dy}{dt} = 2e^t $.

Compute the Derivative of the Third Component Function

The third component function $ z(t) = 1 $ is a constant. The derivative of a constant function is $ \frac{dz}{dt} = 0 $.

Combine the Derivatives into the Derivative of the Vector-valued Function

Combine the computed derivatives into the vector form. Thus, the derivative of $ \mathbf{r}(t) $ is $ \mathbf{r}'(t) = e^t \mathbf{i} + 2e^t \mathbf{j} + 0 \mathbf{k} $ or simply $ \mathbf{r}'(t) = e^t \mathbf{i} + 2e^t \mathbf{j} $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differentiation

Differentiation is a fundamental concept in calculus used to determine how a function changes as its input changes. For vector-valued functions, differentiation involves finding the rate of change of each component separately. This allows us to understand the behavior of the vector function as a whole.
To differentiate a vector-valued function like $ \mathbf{r}(t) = e^{t} \mathbf{i} + 2 e^{t} \mathbf{j} + \mathbf{k} $, follow these steps:

Break the function into its component functions: $ x(t) = e^t $, $ y(t) = 2e^t $, and $ z(t) = 1 $.
Compute the derivative of each component: $ \frac{dx}{dt} = e^t $, $ \frac{dy}{dt} = 2e^t $, and $ \frac{dz}{dt} = 0 $.
Combine these derivatives to form the derivative of the entire vector-valued function: $ \mathbf{r}'(t) = e^t \mathbf{i} + 2e^t \mathbf{j} $.

This results in a new vector function representing the rate of change of $ \mathbf{r} $ with respect to $ t $. Differentiation helps us analyze and visualize how vector quantities evolve over time.

Exponential Functions

Exponential functions are a key part of both pure and applied mathematics. They are characterized by a constant base raised to a variable exponent. In our example, $ e^t $ is the exponential function in each component of $ \mathbf{r}(t) = e^{t} \mathbf{i} + 2 e^{t} \mathbf{j} + \mathbf{k} $.
Exponential functions have a unique property: their derivatives are proportional to the functions themselves.
This means:

The derivative of $ e^t $ is $ e^t $.
The derivative of $ 2e^t $ remains $ 2e^t $, because of the constant multiple rule.

This makes differentiation straightforward and predictable for exponential functions, making them highly valuable in modeling growth and decay processes in natural sciences, finance, and engineering.

Vector Calculus

Vector calculus extends calculus to vector-valued functions, encompassing operations like differentiation and integration to solve real-world problems involving curves, surfaces, and fields. Here, the focus is on differentiating vector-valued functions like $ \mathbf{r}(t) = e^{t} \mathbf{i} + 2 e^{t} \mathbf{j} + \mathbf{k} $.
Vector calculus provides tools to:

Find the rate of change in any direction by calculating derivatives of vector functions.
Understand vector fields, such as those in electromagnetism and fluid dynamics.

The derivative gives a vector that describes how each component changes, allowing for thorough analysis of motion and forces in physical systems. Understanding vector calculus helps us model and predict phenomena in multidimensional spaces, offering insights into complex patterns and behaviors.

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91影视

Compute the derivatives of the vector-valued functions. \(\mathbf{r}(t)=e^{t} \mathbf{i}+2 e^{t} \mathbf{j}+\mathbf{k}\)

Short Answer

Step by step solution

Identify the Components of the Vector-valued Function

Compute the Derivative of the First Component Function

Compute the Derivative of the Second Component Function

Compute the Derivative of the Third Component Function

Combine the Derivatives into the Derivative of the Vector-valued Function

Key Concepts

Differentiation

Exponential Functions

Vector Calculus

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