Chapter 13: Problem 61
The line \(y=x\) meets \(y=k e^{x}\) for \(k \leq 0\) at (A) no point (B) one point (C) two points (D) more than two points
Short Answer
Expert verified
No point of intersection.
Step by step solution
01
Understand the Intersection of Graphs
The intersection point(s) of the lines are where their equations are equal. So you need to set the two functions equal to one another to determine where they intersect.
02
Set Up the Equations
Set the equation of the first line, y = x, equal to the second function, y = ke^x, to find the point(s) of intersection.
03
Solve for x
Since y = x and y = ke^x are equal at the point(s) of intersection, we write x = ke^x and try to solve for x.
04
Analyze the Possibility of Solutions
If k is strictly less than 0, there will be no solution to this equation since as x grows, e^x also grows and is always positive, thus ke^x will always be negative while y = x is positive or zero.
05
Conclude the Number of Intersection Points
To meet means to have at least one point of intersection but with negative k, there is no such point. Therefore, the lines do not intersect at any point.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Exponential Equations
Understanding how to solve exponential equations is critical when you encounter graphs where exponential functions are involved. Exponential equations typically take the form of \( a^x = b \), where \( a \) is the base, \( x \) is the exponent, and \( b \) is the value to which the base raised to the exponent equates.
When we attempt to solve these equations, we often look for a way to write both sides of the equation with the same base, which makes it easier to compare and solve for the exponent. However, not all exponential equations are easily transformed to have the same base. In such cases, one might use methods like taking the logarithm of both sides, which allows us to utilize the properties of logarithms to solve for the variable. This is particularly useful when graphically determining the points of intersection between an exponential function and a linear function, as seen in the example provided.
For the given problem \( x = ke^x \), with \( k \) being less than zero, there's a conceptual understanding that if \( e^x \) is always positive, multiplying it by a negative \( k \) will result in a negative outcome. Since we cannot have a positive \( x \) equal to a negative result, it means that there's no real solution to this equation, which aligns with the conclusion that no points of intersection exist on the graph when \( k \) is less than zero.
When we attempt to solve these equations, we often look for a way to write both sides of the equation with the same base, which makes it easier to compare and solve for the exponent. However, not all exponential equations are easily transformed to have the same base. In such cases, one might use methods like taking the logarithm of both sides, which allows us to utilize the properties of logarithms to solve for the variable. This is particularly useful when graphically determining the points of intersection between an exponential function and a linear function, as seen in the example provided.
For the given problem \( x = ke^x \), with \( k \) being less than zero, there's a conceptual understanding that if \( e^x \) is always positive, multiplying it by a negative \( k \) will result in a negative outcome. Since we cannot have a positive \( x \) equal to a negative result, it means that there's no real solution to this equation, which aligns with the conclusion that no points of intersection exist on the graph when \( k \) is less than zero.
Graph Analysis
Graph analysis is a powerful visual tool for understanding the behavior of various functions and their interactions. In algebra and calculus, graph analysis allows us to quickly identify points of intersection, asymptotes, extrema, and intervals of increase or decrease.
Analyzing graphs requires knowledge of how different functions behave. For instance, a linear function such as \( y = x \) is represented by a straight line with a uniform gradient, while an exponential function like \( y = ke^x \) will show a rapid increase or decrease depending on the value of \( k \) and will never touch the x-axis, as \( e^x \) never equals zero.
In the context of the given exercise, analyzing the graph means looking at the possible points where the straight line \( y = x \) can intersect with the curve of \( y = ke^x \) for negative values of \( k \) through visual representation. The analysis leads to the realization that since the exponential curve is always below the x-axis for negative \( k \) and the line \( y = x \) is always above or on the x-axis, their paths never cross, which means there are no intersection points.
Analyzing graphs requires knowledge of how different functions behave. For instance, a linear function such as \( y = x \) is represented by a straight line with a uniform gradient, while an exponential function like \( y = ke^x \) will show a rapid increase or decrease depending on the value of \( k \) and will never touch the x-axis, as \( e^x \) never equals zero.
In the context of the given exercise, analyzing the graph means looking at the possible points where the straight line \( y = x \) can intersect with the curve of \( y = ke^x \) for negative values of \( k \) through visual representation. The analysis leads to the realization that since the exponential curve is always below the x-axis for negative \( k \) and the line \( y = x \) is always above or on the x-axis, their paths never cross, which means there are no intersection points.
Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. The general form of an exponential function is \( f(x) = ab^x \) where \( a \) is the initial value, \( b \) is the base, and \( x \) is the exponent. These functions are known for their unique property of increasing or decreasing very rapidly.
One of the most important features of exponential functions is that the base, if greater than 1, results in a graph that increases rapidly. Conversely, if the base is between 0 and 1, the graph will decrease rapidly. In our example, the function \( y = ke^x \) indicates an exponential function with a base of \( e \) – the natural exponential constant approximately equal to 2.71828.
When \( k \) is positive, \( y = ke^x \) will tend to shoot upwards as \( x \) increases. Contrastingly, if \( k \) is negative, the function still grows in magnitude, but it is reflected across the x-axis, resulting in a graph that tends downward towards negative infinity, never meeting the line \( y = x \) when \( k \) is less than zero. This asymptotic nature towards the x-axis is why for negative \( k \) values, the exponential function graphed will never intersect with the linear function in the context of our exercise.
One of the most important features of exponential functions is that the base, if greater than 1, results in a graph that increases rapidly. Conversely, if the base is between 0 and 1, the graph will decrease rapidly. In our example, the function \( y = ke^x \) indicates an exponential function with a base of \( e \) – the natural exponential constant approximately equal to 2.71828.
When \( k \) is positive, \( y = ke^x \) will tend to shoot upwards as \( x \) increases. Contrastingly, if \( k \) is negative, the function still grows in magnitude, but it is reflected across the x-axis, resulting in a graph that tends downward towards negative infinity, never meeting the line \( y = x \) when \( k \) is less than zero. This asymptotic nature towards the x-axis is why for negative \( k \) values, the exponential function graphed will never intersect with the linear function in the context of our exercise.
