Question

A tangent question Verify by graphing that the graphs of $y=e^x$ and $y=x$ have no points of intersection, whereas the graphs of $y=e^{x/3}$ and $y=x$ have two points of intersection. Approximate the value of $a>0$ such that the graphs of $y=e^{x/a}$ and $y=x$ have exactly one point of intersection.

Short answer

Answer: The approximate value of "a" for which the graphs of $y=e^{x/a}$ and $y=x$ intersect exactly once is 2.3.

Step-by-step solution

Graphing $y=e^x$ and $y=x$

First, graph the exponential function $y=e^x$ and the linear function $y=x$. You may use any graphing tool to do this, such as Desmos or a graphing calculator. Upon observing the graphs, we will see that there are no points of intersection between the two functions.

Graphing $y=e^{x/3}$ and $y=x$

Next, graph the exponential function $y=e^{x/3}$ and the linear function $y=x$. You will notice that the graphs have two points of intersection, indicating that there is a solution for which $e^{x/3}=x$ at these two points. 2. Determine the number of intersection points

Approximating the value of $a$

Our goal is to find the positive value of "a" for which $y=e^{x/a}$ and $y=x$ intersect exactly once. This means that we want one point where $e^{x/a}=x$. We can find the approximate value of $a$ by experimenting with various positive values in the equation $y=e^{x/a}$ and comparing the graphs with $y=x$. By doing this, we will observe that when $a\approx 2.3$, the graphs of $y=e^{x/a}$ and $y=x$ intersect exactly once. So, our approximation for the value of $a$ is 2.3.

Key concepts

Tangent to Exponential Functions

Understanding the behavior of tangents to exponential functions is crucial for comprehending their characteristics at a given point. A tangent line to a curve at a certain point is a straight line that touches the curve exactly at that point and has the same slope as the curve at that point. For an exponential function like y=e^{x}, to find the equation of a tangent at a particular point, we calculate the derivative of the function, which gives us the slope.
In the case of the exponential function, the derivative is unique because it is the same as the function itself; that is, the derivative of e^{x} is e^{x}. This property signifies that the slope of the tangent line at any point on the graph of y=e^{x} is equal to the value of the function at that point. For instance, to find the tangent line at x=1, we plug in the value into the function, which would be e, and this would be the slope of the tangent line at the point (1, e).

Intersection Points of Functions

The intersection points of functions are the points where their graphs meet. When graphing functions, these points are found at the coordinates where the functions share the same x and y values. Intersection points are significant because they are the solutions to the equation set by equating both functions.
For example, equating y=e^{x/3} and y=x leads to the equation e^{x/3}=x. The graph of these functions shows where the curves intersect, hence, solving this equation graphically. Students often find this approach accessible because it provides a visual perspective of the solution, which is especially helpful when dealing with transcendental functions that are not easily solved analytically.
When graphing, we observe that while y=e^{x} and y=x do not intersect, the function y=e^{x/3} and y=x intersect at two distinct points, revealing the two solutions for x where both function values are equal.

Solving Exponential Equations Graphically

Solving exponential equations graphically is an intuitive method that can help students better understand the behavior of exponential functions and their intersections with other functions. This graphical approach is particularly useful when dealing with complex equations that are difficult to solve algebraically.
In our example, we are trying to find the value of a in y=e^{x/a} such that it intersects y=x at exactly one point. This intersection is the solution to the equation e^{x/a}=x. By plotting both functions and adjusting the value of a, students can visually determine the point of intersection. In this case, experimenting with the values of a reveals that when a is approximately 2.3, the graphs intersect at a single point, giving us the visual solution to our problem.
It's important for students to understand that while graphical solutions provide a good estimate, they are not always precise due to the limitations of scaling and drawing accuracy on paper or screen. Nevertheless, graphically solving helps in conceptualizing how various values of parameters affect the position and number of intersection points between functions.