Overview
This lecture introduces the concept of the tangent line to a curve and formally defines the derivative as the limit of the slope of secant lines, followed by several concrete examples.
Tangent Lines and Slope
- To define a tangent line to a curve at point (a, f(a)), we need to understand the concept of slope.
- The slope of a line between points (x₀, y₀) and (x₁, y₁) is (y₁ - y₀) / (x₁ - x₀).
- For circles, the tangent line is perpendicular to the radius at the point of tangency.
- The tangent line to a general curve doesn't necessarily just touch at one point and can cross the curve.
Defining the Tangent Line for General Curves
- To approximate the tangent line's slope, use two points on the curve: (a, f(a)) and (x, f(x)).
- The slope of the secant line is [f(x) - f(a)] / (x - a).
- As x approaches a, the secant line slope approaches the tangent line slope.
- This limiting process gives the formal definition of the tangent line.
The Derivative: Definition and Notation
- The slope of the tangent line at x is defined as lim_{h→0} [f(x+h) - f(x)] / h.
- The result is called the derivative of f at x, denoted f'(x).
- Other notations: y', dy/dx, or d/dx [f(x)].
- The process of finding a derivative is called differentiation.
- A function f is differentiable at x if f'(x) exists; not all functions are differentiable everywhere.
Example Calculations of Derivatives
- For f(x) = x² + 1, f'(x) = 2x using the definition.
- For f(x) = 1/x, f'(x) = -1/x².
- For f(x) = √x, f'(x) = 1/(2√x); f'(0) is undefined.
Key Terms & Definitions
- Tangent Line — a line that locally approximates a curve at a specific point.
- Secant Line — a line that passes through two points on a curve.
- Derivative — the limit of the secant line slope as the two points approach each other, giving the instantaneous rate of change.
- Differentiable — a function is differentiable at a point if its derivative exists there.
- Differentiation — the process of finding the derivative of a function.
Action Items / Next Steps
- Memorize the formal definition of the derivative.
- Practice finding derivatives using the limit definition.
- Review examples from the lecture and try similar problems.