Question
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
.
Answered by UpStudy AI and reviewed by a Professional Tutor
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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!