Math Problem Statement
Solution
To solve this problem, we will use the limit definition of the derivative to find the slope of the tangent line to the curve at the point .
Limit Definition of Derivative:
The limit definition of the derivative is given by: Where .
Step-by-Step Solution:
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Function at :
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Function at :
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Substitute into the limit definition: Simplifying the numerator:
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Rationalize the numerator: Multiply both the numerator and the denominator by the conjugate : Using the difference of squares, , we get:
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Simplify:
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Take the limit as : As , , so:
Conclusion:
The slope of the tangent line to the curve at the point is .
Would you like more details on any specific part of this solution?
Here are 5 related questions:
- What is the geometric interpretation of the derivative at a point?
- How can you verify the slope of a tangent using the chain rule instead of the limit definition?
- How does the process change if the function has a higher degree or is more complex?
- Why is it necessary to rationalize the numerator when using the limit definition of the derivative?
- How does the derivative of change as you move away from ?
Tip: Rationalizing the numerator is a common technique when dealing with square roots in limits.
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Math Problem Analysis
Mathematical Concepts
Calculus
Limit Definition of Derivative
Tangent Line Slope
Formulas
f'(x) = lim(h -> 0) (f(x+h) - f(x)) / h
Theorems
Limit Theorem
Difference of Squares
Suitable Grade Level
High School (Grades 10-12) or College-level Calculus