Problem 68 - Brief Applied Calculus

Question

Suppose that the temperature outside your house \(x\) hours after midnight is given by a function whose derivative is negative for \(0

Step-by-Step Solution

Answer

The temperature at 6 a.m. is the lowest in the first half of the day.

Step 1: Understand the Sign of the Derivative

The derivative of the temperature function is negative for \(0 < x < 6\), which means that the temperature is decreasing during this time period. Then it becomes positive for \(6 < x < 12\), indicating that the temperature starts increasing after 6 a.m.

Step 2: Analyze Temperature at 6 a.m.

At 6 a.m., which corresponds to \(x = 6\), the derivative changes from negative to positive. This typically suggests a minimum point in the function, meaning the temperature reaches its lowest point at 6 a.m. when considering the first half of the day.

Step 3: Compare Temperatures Before and After 6 a.m.

Since the temperature is decreasing up until 6 a.m. and increasing afterward, the temperature at 6 a.m. is lower than the temperatures before and after this time. Therefore, in the period from midnight to midday, 6 a.m. represents the lowest temperature.

Key Concepts

Derivatives

The concept of a derivative is essential in calculus, acting as a tool to measure how a function changes. Simply put, the derivative tells us the rate at which a function's value is changing at any given point. Think of it as the slope of a tangent to a curve at a particular point. For a function \( f(x) \), its derivative \( f'(x) \) provides this information.
If \( f'(x) > 0 \), the function is increasing at an interval. Conversely, if \( f'(x) < 0 \), the function is decreasing. Therefore, in our exercise, when the derivative of the temperature function is negative from midnight to 6 a.m., the temperature is falling. After 6 a.m., the positive derivative indicates the temperature is rising. Such interpretations of derivatives are pivotal in understanding how functions behave over intervals.

Decreasing and Increasing Functions

Knowing when a function is decreasing or increasing can tell us a lot about its behavior. A function decreases over an interval when its derivative is negative, which means the function's value is getting smaller as the input increases. Similarly, a function is increasing when its derivative is positive. In the context of our exercise, the temperature function shows decreasing behavior for \( 0After 6 a.m., with a positive derivative \( 6

Critical Points

A critical point occurs where the derivative of a function is zero or undefined, and it often indicates a significant feature like a local minima or maxima. In our case, at exactly 6 a.m. (\(x = 6\)), not only does the derivative switch signs but it marks a critical point. This is critical because it represents the lowest temperature of the morning as the derivative changes from negative to positive.

• When the derivative moves from negative to positive, it's often indicative of a local minimum.
• If the derivative shifts from positive to negative, it can point to a local maximum.

This idea of critical points helps us predict significant changes in the behavior of the function, offering insights into peaks, troughs, and turning points that are vital in graphs of functions. These points are fundamental in optimizing real-world scenarios, designing systems, or even understanding natural processes.

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