Fix the mistake: Applications of Differentiation ① (Tangents, Monotonicity, and Extrema)
Equations of Tangent and Normal Lines
At the point (a, f(a)),
Tangent line: y − f(a) = f'(a)(x − a)
Normal line: y − f(a) = −1/f'(a)·(x − a) (a line perpendicular to the tangent)
Example ① Find the tangent and normal lines to y = x² at x=1.
Solution: f(1)=1, f'(x)=2x, and f'(1)=2.
Tangent: y − 1 = 2(x−1) → y = 2x − 1
Normal: y − 1 = −(1/2)(x−1) → y = −x/2 + 3/2
Solution: f(1)=1, f'(x)=2x, and f'(1)=2.
Tangent: y − 1 = 2(x−1) → y = 2x − 1
Normal: y − 1 = −(1/2)(x−1) → y = −x/2 + 3/2
Key Point
- Tangent slope = f'(a); normal slope = −1/f'(a)
- At a point where f'(a) = 0, the normal is the vertical line x = a, parallel to the y-axis
Practice Problem
- Find the tangent line to y = e^x at x=0
Answer and Explanation
- Answer:y = x + 1
f(0)=1 and f'(0)=1.
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