Math Problem Statement
Solution
The problem provided involves analyzing the difference quotient for the function . We need to simplify the expression
and then express it in the form .
Step-by-Step Solution:
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Substitute into the difference quotient:
Since , we can compute and as follows:
So, the difference quotient becomes:
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Simplify the numerator:
Find a common denominator for :
So now we have:
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Cancel from the numerator and denominator:
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Final Form:
Now we have simplified the expression to:
Answer:
The quotient can be simplified to:
Would you like further details on this solution?
Related Questions:
- What is the difference quotient, and why is it used in calculus?
- How would the solution change if the function were with a different constant ?
- What is the significance of finding the limit as in the difference quotient?
- How does this result relate to finding the derivative of ?
- What other techniques could be used to simplify complex rational expressions?
Tip:
In difference quotient problems, always simplify the numerator carefully by finding a common denominator to avoid errors in handling complex fractions.
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Math Problem Analysis
Mathematical Concepts
Difference Quotient
Function Evaluation
Algebraic Simplification
Formulas
Difference Quotient Formula: (f(x + h) - f(x)) / h
Theorems
Limit Definition of Derivative (conceptually related)
Suitable Grade Level
Grades 11-12