Math Problem Statement
Solution
To address this problem thoroughly, let’s work step by step:
The function is given as: We are tasked with evaluating the difference quotient: for several small values of and interpreting the result.
Step 1: Find
Substitute into :
Step 2: Find
The function is . To find , substitute into the equation: Expand using the binomial theorem: Thus: Simplify the terms: Combine like terms:
Step 3: Difference Quotient
The difference quotient is: Since , this simplifies to: Simplify by dividing through by (assuming ):
Step 4: Evaluate for Small
We now calculate the expression for the given values of :
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:
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:
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:
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:
Step 5: Derivative Interpretation
The derivative of at is the limit of the difference quotient as : From the simplified formula , taking the limit as :
This matches the expected integer value of .
Final Answer:
The derivative of at is .
Do you need further clarifications or additional examples?
Related Questions:
- How is the difference quotient derived for a given function?
- What is the formal definition of the derivative using limits?
- How can you compute derivatives of polynomials directly?
- What role does the binomial theorem play in simplifying expressions like ?
- How does the difference quotient approximate the derivative for small ?
Tip: The derivative represents the slope of the tangent line to the curve at a given point. Approximations become more accurate as .
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Math Problem Analysis
Mathematical Concepts
Calculus
Difference Quotient
Derivatives
Formulas
f'(x) = lim(h→0) (f(x+h) - f(x))/h
Binomial Expansion for (2+h)^3
Theorems
Definition of the Derivative using Limits
Suitable Grade Level
Grades 11-12