Math Problem Statement
Solution
To find the slope of the tangent line to the polar curve at , we need to calculate in polar coordinates.
For polar curves, the slope of the tangent line (i.e., ) is given by:
Step 1: Compute
Given:
Differentiate with respect to :
Step 2: Substitute and at
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Calculate at :
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Calculate at :
Step 3: Substitute values into formula
Now, using:
Substitute , , and :
Substitute these into the formula:
Simplify each term:
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The numerator becomes:
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The denominator becomes:
Thus:
Final Answer
The slope of the tangent line to the polar curve at is:
Would you like further explanation on any part of this solution?
Follow-up Questions
- How would the formula change if we wanted to find the slope for a different polar curve?
- What is the significance of the slope of the tangent line in the context of polar coordinates?
- How is in polar coordinates related to rectangular coordinates?
- Can we use this method for finding slopes on curves defined parametrically?
- How does symmetry affect the slope of the tangent line in polar curves?
Tip
When solving for slopes in polar coordinates, remember to convert and based on the specific angle given, as they often simplify the computation.
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Math Problem Analysis
Mathematical Concepts
Polar Coordinates
Differentiation
Slope of Tangent Line
Formulas
Slope of tangent line in polar coordinates: dy/dx = (dr/dθ * sin θ + r * cos θ) / (dr/dθ * cos θ - r * sin θ)
Theorems
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Suitable Grade Level
College Calculus
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