Overview
This lecture introduces the concept of the derivative, its geometric and physical interpretations, how to compute derivatives using limits, and key properties related to differentiability and continuity.
The Derivative: Definition & Meaning
- The derivative represents the slope of the curve at a specific point.
- It is also interpreted as the instantaneous rate of change or velocity at a point.
- The derivative is denoted as f'(x) or "f prime of x".
- The formal definition: f'(x) = limₕ→0 [f(x + h) - f(x)] / h.
Calculating Derivatives
- To find the derivative, compute f(x + h), subtract f(x), and divide by h, then take the limit as h approaches zero.
- For polynomials and basic functions, expand and simplify algebraically before applying the limit.
- The process eliminates all terms without h, leaving a function you can evaluate at h = 0.
- For a straight line y = mx + b, the derivative (slope) is always m.
Example Problems
- Example 1: f(x) = 2x² - 3 ⇒ f'(x) = 4x.
- The slope at x = 2: f'(2) = 8. The tangent line equation at (2, 5): y - 5 = 8(x - 2).
- Example 2: f(x) = 2x³ - x ⇒ f'(x) = 6x² - 1.
- Example 3: f(x) = √x ⇒ f'(x) = 1/(2√x). Tangent line at x = 4, y = 2: y - 2 = (1/4)(x - 4).
Applications: Instantaneous Velocity
- Instantaneous velocity is the derivative of the position function with respect to time.
- For s(t) = 1250 - 16t², velocity v(t) = s'(t) = -32t.
- To find when the object hits the ground: solve 1250 - 16t² = 0.
- Speed at impact: plug t into v(t).
Differentiability and Continuity
- A function is differentiable at a point if its derivative exists there.
- The derivative may not exist at sharp points (cusps) or vertical tangents.
- Differentiability implies continuity, but continuity does not guarantee differentiability.
- Non-differentiable points can occur at corners or where the slope is undefined.
Derivative Notation
- f'(x) or "f prime of x" denotes the derivative of f.
- d/dx [f(x)] or dy/dx are alternative notations.
- To evaluate at a specific point: f'(a) or dy/dx |{x=a}.
Key Terms & Definitions
- Derivative — The slope of a curve at a point; the instantaneous rate of change.
- Difference Quotient — [f(x+h) - f(x)] / h, used in the definition of the derivative.
- Tangent Line — A straight line that touches a curve at one point and has the same slope as the curve there.
- Differentiability — The property of a function being differentiable (having a derivative) at a point.
- Continuity — A function is continuous if its graph has no jumps, holes, or breaks.
Action Items / Next Steps
- Practice computing derivatives using the limit definition.
- Complete homework problems involving finding equations of tangent lines.
- Review concepts of continuity and differentiability for upcoming assessments.