Problem 31 - Calculus with Concepts in Calculus

Question

Determine at which points the graphs of the given pair of functions intersect. $$ f(x)=e^{3 x} \text { and } g(x)=3 e^{x} $$

Step-by-Step Solution

Answer

The graphs intersect at \( \left( \frac{\ln(3)}{2}, 3^{3/2} \right) \).

Step 1: Set the Equations Equal

To find the points where the graphs intersect, set the two functions equal to each other: \[ e^{3x} = 3e^x \] This equation will help us find the values of \( x \) where the graphs intersect.

Step 2: Simplify the Equation

Divide both sides of the equation \( e^{3x} = 3e^x \) by \( e^x \): \[ e^{3x}/e^x = 3e^x/e^x \] This simplifies to \[ e^{2x} = 3 \].

Step 3: Solve for x

Take the natural logarithm of both sides of the equation \( e^{2x} = 3 \):\[ \ln(e^{2x}) = \ln(3) \]Since \( \ln(e^{2x}) = 2x \ln(e) = 2x \), we have:\[ 2x = \ln(3) \]Divide both sides by 2 to solve for \( x \):\[ x = \frac{\ln(3)}{2} \].

Step 4: Find the Corresponding y-Value

Plug \( x = \frac{\ln(3)}{2} \) back into one of the original functions to find \( y \). We will use \( f(x) = e^{3x} \):\[ y = e^{3(\frac{\ln(3)}{2})} = e^{\frac{3\ln(3)}{2}} = (e^{\ln(3)})^{3/2} = 3^{3/2} \].Thus, the \( y \)-coordinate of the intersection point is \( 3^{3/2} \).

Step 5: Conclusion

The two functions intersect at the point where \( x = \frac{\ln(3)}{2} \) and \( y = 3^{3/2} \), thus the point of intersection is \[ \left( \frac{\ln(3)}{2}, 3^{3/2} \right) \].

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In our exercise, both functions, \( f(x) = e^{3x} \) and \( g(x) = 3e^x \), are exponential. Here's what makes exponential functions unique:

• The constant \( e \) is approximately 2.718 and is also known as Euler's number. It's a critical constant in mathematics, especially in calculus.
• Exponential functions grow at rates proportional to their current value, which means they can model rapid increases or decreases.
• In our specific functions, \( e^{3x} \) means the base \( e \) is raised to a power that is three times \( x \). For \( 3e^x \), the expression implies a multiplication of the exponential part \( e^x \) with 3.

Exponential functions are used broadly in real-world applications such as population growth, compound interest, and radioactive decay.

Solving Equations

Solving equations, especially those with exponential expressions, involves finding the value(s) of the unknown variable which satisfy the equation. Here, we aimed to solve \( e^{3x} = 3e^x \) to identify the intersection points of the graphs.

• The initial step in solving involved setting the two functions equal to find where they intersect. This produced \( e^{3x} = 3e^x \).
• To simplify, both sides of the equation were divided by \( e^x \), using the property of exponents that \( \frac{e^{3x}}{e^x} = e^{3x-x} = e^{2x} \).

The result, \( e^{2x} = 3 \), is a simpler equation that allows us to proceed with logarithmic solving.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \). In solving the equation \( e^{2x} = 3 \), the natural logarithm helps to "unlock" the exponent:

• Taking \( \ln \) of both sides gives \( \ln(e^{2x}) = \ln(3) \).
• Using the logarithmic identity \( \ln(e^a) = a \), we simplify \( \ln(e^{2x}) \) to \( 2x \).
• This results in \( 2x = \ln(3) \), a much simpler linear equation.

The natural logarithm effectively converts the exponential equation into a linear form, making \( x \) easier to discover. Finally, dividing by 2 provides \( x = \frac{\ln(3)}{2} \).

Coordinate Geometry

Coordinate geometry involves the study of geometric figures through a coordinate system. In our exercise, the task was to determine where two functions intersect on the Cartesian plane.

• The solution process required finding the \( x \)-coordinate and the corresponding \( y \)-coordinate of the intersection.
• After finding \( x = \frac{\ln(3)}{2} \), we substituted back into one of the original functions, \( f(x) = e^{3x} \), to evaluate \( y \).
• The substitution calculated \( y = 3^{3/2} \), which provided us with the \( y \)-coordinate.

This intersection point \( \left( \frac{\ln(3)}{2}, 3^{3/2} \right) \) represents where both exponential functions have the same value, a crucial aspect in coordinate geometry for understanding the relationship between functions and their graphical representations.