Question Partpoint Submissions Used Assume That X And Y Are

Question Partpointsubmissions Usedassume Thatxandyare Both Differenti

Question Part points submissions used assume that x and y are both differentiable functions of t and find the required values of dy / dt and dx / dt. y = x (a) Find dy / dt, when x = 16, given that dx / dt = 4. (b) r = 25 inches. in 2 /min decreasing. Find the open intervals on which the function is increasing or decreasing. Use a graphing utility to verify your results. (Enter your answer using interval notation. If an answer does not exist, enter DNE.) increasing decreasing STEP 2: Find the critical numbers. (Enter your answers as a comma-separated list.) x = STEP 3: Test the intervals to determine the open intervals on which the function is increasing or decreasing. (Enter your answer using interval notation.) increasing decreasing

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Question Partpointsubmissions Usedassume Thatxandyare Both Differenti

The problem involves analyzing the derivatives of functions with respect to a parameter t, specifically focusing on functions x(t) and y(t) which are differentiable. The key aspect is to compute the derivatives dy/dt and dx/dt and analyze the behavior of the function y in relation to x over specific intervals. The context includes a scenario where y equals x, and also involves calculating critical points and determining intervals of increasing or decreasing behavior, often visualized using graphing utilities.

Part (a): Calculating dy/dt given dx/dt and x

Given that y = x, the relationship between their derivatives with respect to t is directly tied through the chain rule. Specifically, since y = x, then dy/dt = dx/dt. When x equals 16 and dx/dt is 4 (units per minute), the value of dy/dt is immediately derived as 4 units per minute. This relationship holds because y and x are directly proportional at the point in question, simplifying the derivative calculation.

Part (b): Analyzing the behavior of r = 25 inches

In this part, the problem involves a radius r = 25 inches decreasing at a rate of 2 inches per minute. The objective is to determine the intervals on which the function describing r is increasing or decreasing. Since the radius is decreasing, the function r(t) is decreasing over its domain. To identify where the function is increasing or decreasing, critical points, or points where the derivative equals zero, are computed.

Step 2: Find the critical numbers

Critical numbers are identified by setting the derivative of the radius function to zero. Given that r(t) is decreasing at 2 inches per minute, the derivative dr/dt is constant and negative. Since the derivative does not change sign, there are no critical points within the interval, Hence, the set of critical points is empty, which can be denoted as DNE or "does not exist."

Step 3: Determine the increasing and decreasing intervals

Because r(t) is decreasing at a constant rate, the entire function is decreasing over its domain. Therefore, the function is decreasing on the whole real line, which in interval notation is (-∞, ∞). There are no intervals where the function is increasing, given its decreasing nature across all t.

Graphical verification and conclusion

Using a graphing utility confirms the decreasing behavior of the radius over time. The illustrated graph shows a consistent downward slope, verifying that the function decreases throughout the interval. This visual evidence supports the interval analysis and the derivative sign considerations.

Summary

To summarize, the derivative dy/dt equals dx/dt when y = x. The critical points for the radius function are nonexistent due to its constant decreasing rate, and the entire domain corresponds to the function decreasing. No regions of increase are present, which aligns with the decreasing rate defined. This analysis underscores the importance of derivatives and critical points in understanding the behavior of differentiable functions over various intervals.

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