Sure, let's go through a detailed, step-by-step solution for the given function [tex]\( e^x \cdot \sin(2x) \)[/tex].
### 1. Understanding the Function
The function in question is [tex]\( e^x \cdot \sin(2x) \)[/tex]. This is a product of the exponential function [tex]\( e^x \)[/tex] and the sine function [tex]\( \sin(2x) \)[/tex].
### 2. Breakdown of the Components
- Exponential Function: [tex]\( e^x \)[/tex] is a fundamental exponential function where the base is Euler's number [tex]\( e \approx 2.718 \)[/tex].
- Sine Function: [tex]\( \sin(2x) \)[/tex] is a trigonometric function that gives the sine of [tex]\( 2x \)[/tex]. This implies a periodic oscillation.
### 3. Combining the Functions
When we multiply [tex]\( e^x \)[/tex] by [tex]\( \sin(2x) \)[/tex], we get a function that has both the exponential growth due to [tex]\( e^x \)[/tex] and the oscillatory behavior due to [tex]\( \sin(2x) \)[/tex].
Putting these together, the detailed form of the function is simply:
[tex]\[ e^x \cdot \sin(2x) \][/tex]
This product involves:
1. The exponential growth from [tex]\( e^x \)[/tex], which increases rapidly as [tex]\( x \)[/tex] increases.
2. The oscillation from [tex]\( \sin(2x) \)[/tex], which varies between -1 and 1 over the interval [tex]\([0, \pi]\)[/tex].
### 4. Interpretation
- Growth and Decay: As [tex]\( x \)[/tex] increases, [tex]\( e^x \)[/tex] grows exponentially, which means the amplitude of the oscillations will also grow exponentially.
- Oscillation: The [tex]\( \sin(2x) \)[/tex] term causes the function to oscillate with a period of [tex]\( \pi \)[/tex] (since the sine function repeats every [tex]\( 2\pi \)[/tex]), causing the product to oscillate but with increasingly larger swings as [tex]\( x \)[/tex] increases.
### 5. Conclusion
So, when the function is [tex]\( e^x \cdot \sin(2x) \)[/tex], it captures the behavior of an exponentially growing wave, where the wave part comes from the sine function and the exponential growth part comes from the [tex]\( e^x \)[/tex].
The final form is:
[tex]\[ \boxed{e^x \cdot \sin(2x)} \][/tex]
