Math Problem Statement
Evaluate the difference quotient for the given function. Simplify your answer.
f(x) = 4 + 3x − x2,
f(4 + h) − f(4)
h
Solution
The difference quotient is given by:
Given the function , we'll first find and .
Step 1: Calculate
Substitute into the function :
First, simplify :
Next, expand :
So, the expression for becomes:
Simplify the expression:
Step 2: Calculate
Substitute into the function:
Simplify:
Step 3: Compute the Difference Quotient
Now, substitute and into the difference quotient formula:
Simplify the expression:
Thus, the simplified difference quotient is:
Do you have any more questions or need further clarification on any steps?
Here are 5 related questions:
- What is the difference quotient for ?
- How would the difference quotient change if were a linear function?
- What is the geometric interpretation of the difference quotient?
- How does the difference quotient relate to the derivative of a function?
- Evaluate the difference quotient for at a different point.
Tip: The difference quotient is a foundational concept in calculus, leading directly to the definition of the derivative. Practice simplifying these expressions to build your skills!
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Math Problem Analysis
Mathematical Concepts
Difference Quotient
Polynomial Functions
Function Evaluation
Formulas
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Theorems
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Suitable Grade Level
Grades 11-12