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Will the function be changing more rapidly at . or ? Why?
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Ask by Washington Reese. in the United States
Jan 25,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The function changes more rapidly at than at because the derivative at is larger in magnitude.

Solution

To determine whether the function is changing more rapidly at or , we’ll analyze its rate of change by computing its derivative and evaluating it at these points.

Step 1: Simplify the Function

First, simplify the denominator:

Step 2: Compute the Derivative

We’ll use the quotient rule for differentiation, which states:
Let and . Then:
Applying the quotient rule:
Simplify the numerator:
Subtracting the second part from the first:
Factor out common terms:
So, the derivative simplifies to:

Step 3: Evaluate the Derivative at and

At :
At :

Step 4: Compare the Rates of Change

  • At : The derivative is -6.75.
  • At : The derivative is approximately 8.296.
Since the magnitude of the derivative at (approximately 8.296) is greater than that at (6.75), the function is changing more rapidly at .

Conclusion

The function is changing more rapidly at compared to because the absolute value of the derivative at is greater than at .

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Extra Insights

To determine how rapidly the function is changing at and , we need to analyze the derivative . The rate of change of the function is indicated by the absolute value of the derivative.
First, let’s find the first derivative using the quotient rule. After simplifying, we can evaluate and to discover which point corresponds to a greater rate of change. Generally, the function’s growth rate is often correlated with the magnitude of its slope; hence, by comparing and , we can conclude which point exhibits a steeper inclination.
Next, for a detailed exploration, you might consider graphing the function or utilizing numerical computation tools to visualize the behavior of around these points—it’s like watching a thrilling race, with colorful curves showing their speed!
The derivative evaluations will reveal the secret of which point truly zips along faster. Enjoy the mathematics ride!

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