Question

Identify whether the slope field given represents an increasing or decreasing function in the region where \( x < 0 \) and \( y > 0 \) for the differential equation \( \frac{dy}{dx} = 5 - y \).

Ask by Whittaker Rodriquez. in the Philippines
Dec 08,2024

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Tutor-Verified Answer

Answer

In the region where \( x < 0 \) and \( y > 0 \), the function is increasing if \( 0 < y < 5 \), at a critical point if \( y = 5 \), and decreasing if \( y > 5 \).

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Beyond the Answer

In the region where \( x < 0 \) and \( y > 0 \) for the differential equation \( \frac{dy}{dx} = 5 - y \), we need to analyze the slope given by \( 5 - y \). When \( y > 5 \), the expression \( 5 - y \) is negative, indicating that the slope is downward, which means that the function is decreasing. Conversely, when \( y < 5 \), \( 5 - y \) is positive, showing that the slope is upward, indicating that the function is increasing. Since we are specifically looking at the region where \( y > 0 \), if \( y < 5 \), the function will be increasing, while if \( y > 5 \), it will be decreasing. Therefore, \( \frac{dy}{dx} \) can be either increasing or decreasing depending on the value of \( y \) in that region.

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