Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 3: Problem 48
From the local linearizations of \(e^{x}\) and \(\sin x\) near \(x=\) 0, write down the local linearization of the function \(e^{x} \sin x .\) From this result, write down the derivative of \(e^{x} \sin x\) at \(x=0 .\) Using this technique, write down the derivative of \(e^{x} \sin x /(1+x)\) at \(x=0\)
Short Answer
Expert verified
The derivative of both functions at \( x=0 \) is 1.
Step by step solution
01
Review Local Linearization Formula
To find the local linearization of a function \( f(x) \) at \( x = a \), we need to express it in terms of its value and its derivative at \( a \). This is given by the formula \( f(x) \approx f(a) + f'(a)(x-a) \). Here, we need to find local linearizations at \( x = 0 \).
02
Local Linearizations of \(e^x\) and \(\sin x\) Near \(x=0\)
The local linearization of \( e^x \) at \( x=0 \) is: \( e^0 + e^0(x-0) = 1 + x \), since \( e^0 = 1 \) and \( (e^x)' = e^x \). The local linearization of \( \sin x \) at \( x=0 \) is: \( \sin 0 + \cos 0(x-0) = x \), since \( \sin 0 = 0 \) and \( (\sin x)' = \cos x \).
03
Local Linearization of \(e^x \sin x\) Near \(x=0\)
Using the product of the approximate linearizations: \( e^x \approx 1 + x \) and \( \sin x \approx x \), the local linearization of \( e^x \sin x \) near \( x=0 \) is the product of these linearizations: \( (1+x)(x) = x + x^2 \). Thus, it simplifies to \( x \) near \( x=0 \).
04
Derivative of \(e^x \sin x\) at \(x=0\)
The local linear approximation of \( e^x \sin x \) is \( f(x) \approx x \). Thus, the derivative \( f'(x) \) at \( x=0 \) is the coefficient of \( x \), which is 1.
05
Consider Function \(\frac{e^x \sin x}{1+x}\)
We write the function as \( g(x) = \frac{e^x \sin x}{1+x} \) and use the local linearization. The numerator is \( e^x \sin x \approx x \) from previous steps, and the denominator \( 1+x \approx 1 \). Thus, \( g(x) \approx \frac{x}{1} = x \).
06
Derivative of \(\frac{e^x \sin x}{1+x}\) at \(x=0\)
From the local linear approximation \( g(x) \approx x \), the derivative \( g'(x) \) at \( x=0 \) is again the coefficient of \( x \), which is 1.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Local Linearization
Local linearization is a technique used to approximate the behavior of a function near a specific point, usually denoted as \( x = a \). This approximation is achieved by considering both the value of the function and the slope of its tangent at that point.
The formula used is \( f(x) \approx f(a) + f'(a)(x-a) \), which essentially creates a linear approximation.
This linear representation is very useful in calculus as it provides a simple way to understand and predict how a function behaves, especially when dealing with complex or nonlinear functions.
For instance, we can take a difficult function like an exponential or sine function and simplify its closest behavior to a straight line when near a chosen point, like \( x = 0 \). This technique is essential for solving problems involving derivatives and further applications like integrals or differential equations.
The formula used is \( f(x) \approx f(a) + f'(a)(x-a) \), which essentially creates a linear approximation.
This linear representation is very useful in calculus as it provides a simple way to understand and predict how a function behaves, especially when dealing with complex or nonlinear functions.
For instance, we can take a difficult function like an exponential or sine function and simplify its closest behavior to a straight line when near a chosen point, like \( x = 0 \). This technique is essential for solving problems involving derivatives and further applications like integrals or differential equations.
Derivative
The derivative of a function is a key concept in calculus, representing the rate at which a function changes at any given point.
It鈥檚 like a function鈥檚 "speedometer," showing how fast or slow its value is changing as you move along the x-axis.
For any function, the derivative can tell you things like:
In the step by step solution above, the derivative helped us find the slope of the tangent line to the function \( e^x \sin x \) at \( x=0 \).
Here, it was found to be \( 1 \) by looking at the coefficient of \( x \) in the linearization, reflecting how rapidly the function \( e^x \sin x \) changes at that point.
It鈥檚 like a function鈥檚 "speedometer," showing how fast or slow its value is changing as you move along the x-axis.
For any function, the derivative can tell you things like:
- How steeply the function is rising or falling.
- Where the function reaches its maximum or minimum values.
- Where the function changes direction.
In the step by step solution above, the derivative helped us find the slope of the tangent line to the function \( e^x \sin x \) at \( x=0 \).
Here, it was found to be \( 1 \) by looking at the coefficient of \( x \) in the linearization, reflecting how rapidly the function \( e^x \sin x \) changes at that point.
Exponential Function
Exponential functions are a fundamental part of calculus, characterized by a constant rate of growth or decay.
The most common exponential function is \( e^x \), where \( e \) is a mathematical constant approximately equal to 2.71828. This function is unique because its derivative is the same as the function itself: \( \frac{d}{dx} e^x = e^x \).
This makes it especially easy to work with when using local linearization or finding derivatives.
We saw its use in linearizing \( e^x \) near \( x = 0 \), where we found the approximation \( 1 + x \).
This approximation captures how \( e^x \) increases slightly from \( 1 \) as x grows near zero. This property is pivotal for solving calculus problems that involve exponential growth patterns, such as population growth, radioactive decay, or compound interest.
The most common exponential function is \( e^x \), where \( e \) is a mathematical constant approximately equal to 2.71828. This function is unique because its derivative is the same as the function itself: \( \frac{d}{dx} e^x = e^x \).
This makes it especially easy to work with when using local linearization or finding derivatives.
We saw its use in linearizing \( e^x \) near \( x = 0 \), where we found the approximation \( 1 + x \).
This approximation captures how \( e^x \) increases slightly from \( 1 \) as x grows near zero. This property is pivotal for solving calculus problems that involve exponential growth patterns, such as population growth, radioactive decay, or compound interest.
Sine Function
The sine function, denoted \( \sin(x) \), is a classical trigonometric function involved extensively in calculus and geometry.
It represents the y-coordinate of a point on the unit circle as an angle \( x \) in radians sweeps around the circle.
Its behavior is wavelike, oscillating between -1 and 1, and is crucial for modeling periodic phenomena such as sound waves or tides.
The derivative of \( \sin(x) \) is \( \cos(x) \), showing that the rate of change of \( \sin(x) \) is dependent on \( \cos(x) \).
In this exercise, we linearized \( \sin x \) near \( x = 0 \) to get an approximation of \( x \), because \( \sin 0 = 0 \) and the slope \( \cos 0 = 1 \). This proximity allows us to simplify calculations and focus on small changes, similar to real-world applications like alternating current which relies on oscillatory wave functions.
It represents the y-coordinate of a point on the unit circle as an angle \( x \) in radians sweeps around the circle.
Its behavior is wavelike, oscillating between -1 and 1, and is crucial for modeling periodic phenomena such as sound waves or tides.
The derivative of \( \sin(x) \) is \( \cos(x) \), showing that the rate of change of \( \sin(x) \) is dependent on \( \cos(x) \).
In this exercise, we linearized \( \sin x \) near \( x = 0 \) to get an approximation of \( x \), because \( \sin 0 = 0 \) and the slope \( \cos 0 = 1 \). This proximity allows us to simplify calculations and focus on small changes, similar to real-world applications like alternating current which relies on oscillatory wave functions.
