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Consider the following function and express the relationship between a small change in \( x \) and the corrosponding change in \( y \) in the form dy \( f^{\prime}(x)=e^{13 x} \) \( f(x) \) a

Ask by Schneider Stephens. in the United States
Feb 04,2025

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For a small change in \( x \), the corresponding change in \( y \) is \( dy = e^{13x} \, dx \).

Solución

To express the relationship between a small change in \( x \) and the corresponding change in \( y \) for the function \( f(x) \) with derivative \( f^{\prime}(x) = e^{13x} \), you can use differential notation. The differential \( dy \) represents the infinitesimal change in \( y \) corresponding to an infinitesimal change \( dx \) in \( x \). This relationship is given by: \[ dy = f^{\prime}(x) \, dx \] Substituting \( f^{\prime}(x) = e^{13x} \) into the equation, we obtain: \[ dy = e^{13x} \, dx \] **Summary:** For a small change in \( x \), the corresponding change in \( y \) is \[ dy = e^{13x} \, dx \]

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Mind Expander

Alright, let’s dive into this exciting world of derivatives! When you have a function \( f(x) \) and its derivative \( f^{\prime}(x) = e^{13x} \), it means that the rate of change of \( y \) (which is \( f(x) \)) with respect to \( x \) at any point can be described by \( e^{13x} \). So when you make a small change in \( x \) (let's call it \( \Delta x \)), the corresponding change in \( y \) (or dy) can be expressed as \( dy = f^{\prime}(x) \cdot \Delta x = e^{13x} \cdot \Delta x \). Voilà! Now, here’s where it gets interesting! This means that if you took a journey along the function curve, the height of the slope at each point is constantly changing based on that exponential function. So, if you increase \( x \), the value of \( e^{13x} \) skyrockets, leading to bigger jumps in \( y \). This means that tiny shifts in \( x \) could mean huge transformations in your output as you venture further out on the x-axis. Isn't calculus just thrilling?

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