Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 64
Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote. \(y=e^{x}-x\)
Short Answer
Expert verified
The function \( e^x - x \) has a domain of all real numbers and a y-intercept at (0,1). It has no traditional slant asymptote.
Step by step solution
01
Determine the Domain
The function is given as \( y = e^x - x \). Since \( e^x \) is defined for all real numbers and \( x \) is a polynomial, the function \( y \) is defined for all real numbers. Therefore, the domain is all real numbers \( (-\infty, \infty) \).
02
Identify Intercepts
To find the y-intercept, set \( x = 0 \): \( y = e^0 - 0 = 1 \). So the y-intercept is \((0, 1)\). There is no x-intercept since \( e^x \) never equals \( x \) due to the nature of exponential growth compared to linear growth.
03
Analyze Asymptotic Behavior
For slant asymptotes (also known as oblique asymptotes), we check the end behavior as \( x \to \infty \) or \( x \to -\infty \). We will use division to determine this for large \( x \).
04
Perform Polynomial Long Division
To find slant asymptote, consider the terms by performing long division: \( \frac{e^x}{1} - x = e^x - x \). As \( x \to \infty \), \( e^x \) behaves like \( e^x \) (exponential growth) but \( -x \) becomes negligible.Divide \( e^x \) by \( x \). Approximating \( e^x \) as a constant for large values of \( x \), the significant behavior is \( y \approx e^x \).
05
Consider Slant Asymptote's Equation
Since \( e^x \) grows faster than any polynomial, when considering the linear part, for a large value of \( x \), treat \( y \approx e^x \) as exponential providing a pseudo asymptote but mainly there isn't a traditional linear slant asymptote. However, structurally no horizontal/vertical, focusing behaviors indicate the asymptotic growth of \( e^x \).
06
Conclusion
Traditionally, \( e^x - x \'s \) rate change dominates making a non-linear behavior diminishes slant concept in classic sense for described form.
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.
Domain of a Function
The domain of a function refers to all the possible input values (usually represented by \( x \)) that can be entered into the function to produce a real output value. In the case of the function \( y = e^x - x \), both \( e^x \) and \( x \) are defined across all real numbers.
This simplicity in domain makes it versatile for solving problems and plotting graphs.
- The exponential function \( e^x \) is defined for every real number, meaning \( e^x \) never encounters division by zero or issues with square roots of negative numbers.
- The polynomial \( -x \) is also defined everywhere since subtracting \( x \) doesn't limit the input values in any way.
This simplicity in domain makes it versatile for solving problems and plotting graphs.
Y-Intercepts
Y-intercepts are points where the graph of a function crosses or touches the y-axis. To find the y-intercept of the function \( y = e^x - x \), we substitute \( x = 0 \) into the equation.
Intersections or intercepts are useful when analyzing changes in behavior between parts of a graph.
- Plugging in \( x = 0 \): \( y = e^0 - 0 = 1 \) since \( e^0 = 1 \).
- This gives us the y-intercept at the point \((0, 1)\).
Intersections or intercepts are useful when analyzing changes in behavior between parts of a graph.
Slant Asymptotes
Slant asymptotes, or oblique asymptotes, occur when the degree of the numerator of a rational function is exactly one more than the degree of the denominator.
However, in the context of \( y = e^x - x \), we're more concerned with asymptotic behavior as \( x \to \infty \).
Even though there's no clear-cut slant asymptote, considering \( e^x \) can guide understanding how drastically the graph moves upwards, far surpassing any influence the \(-x\) term might exercise. This understanding is essential for accurately sketching graphs and predicting trends.
However, in the context of \( y = e^x - x \), we're more concerned with asymptotic behavior as \( x \to \infty \).
- This function isn't a typical rational function, but understanding end behavior provides clues.
- As \( x \) grows very large, the exponential term \( e^x \) dominates \( x \)
Even though there's no clear-cut slant asymptote, considering \( e^x \) can guide understanding how drastically the graph moves upwards, far surpassing any influence the \(-x\) term might exercise. This understanding is essential for accurately sketching graphs and predicting trends.
Exponential Functions
Exponential functions, such as \( y = e^x \), feature a constant base raised to a variable exponent.
Exponential growth is extraordinarily fast compared to linear growth or even polynomial growth.
In \( y = e^x - x \), it's crucial to acknowledge the dominance of \( e^x \) as \( x \) progresses, impacting everything from sketching curves to analyzing function behavior.
Exponential growth is extraordinarily fast compared to linear growth or even polynomial growth.
- The function \( y = e^x - x \) involves an exponential term \( e^x \), contributing dynamic behavior to the graph.
- The exponential component means that as \( x \) increases, \( y \) increases rapidly and non-linearly.
In \( y = e^x - x \), it's crucial to acknowledge the dominance of \( e^x \) as \( x \) progresses, impacting everything from sketching curves to analyzing function behavior.
