Math Problem Statement
Find the tangent line and normal line
Solution
To find the tangent line and normal line to the function at , we need to follow these steps:
-
Find and at :
- Evaluate at .
- Compute the derivative and evaluate it at for the slope of the tangent line.
-
Equation of the Tangent Line:
- Use the point-slope form of the line equation: , where .
-
Equation of the Normal Line:
- The slope of the normal line is the negative reciprocal of the slope of the tangent line.
Step 1: Evaluate at
Given:
At :
Thus,
Step 2: Compute
Use the chain rule: Let , then .
The derivative is:
Now, find :
Thus,
Evaluate at :
Step 3: Equation of the Tangent Line
Using the point and slope : So, the tangent line equation is:
Step 4: Equation of the Normal Line
The slope of the normal line is the negative reciprocal of , which is .
Thus, the normal line equation is: So,
Summary
- Tangent line equation:
- Normal line equation:
Would you like further details or have any questions?
Further Exploration
- How do we derive the equation of a tangent line in parametric form?
- What is the significance of the negative reciprocal in finding the normal line?
- How can the derivative be interpreted geometrically?
- What happens to the tangent and normal lines if we choose a different point ?
- Can we generalize this approach to three-dimensional functions?
Tip
Always check if the function simplifies before differentiating to avoid complex calculations.
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Math Problem Analysis
Mathematical Concepts
Calculus
Trigonometric Functions
Derivative
Tangent Line
Normal Line
Formulas
Point-slope form of the line equation: \( y - y_1 = m(x - x_1) \)
Chain rule: \( (f(g(t)))' = f'(g(t)) \cdot g'(t) \)
Theorems
Chain Rule for Differentiation
Suitable Grade Level
Grades 11-12