Definition: The slope of a line describes the rate of change of a vertical variable (y) with respect to a horizontal variable (x).
Formula: Slope = Change in y / Change in x (( \Delta y / \Delta x ))
Terminology: Often referred to as "rise over run".
Characteristics: For any line, the slope is constant. The calculation of slope between any two points on the line will yield the same result.
Calculus and Changing Rates
Curves: Unlike lines, curves have a changing rate of change.
Secant Line: The average rate of change between two points on a curve can be found using the slope of the secant line connecting those points.
Instantaneous Rate of Change: The slope at a specific point on a curve, known as the tangent line. This is the central concept of differential calculus.
Example: Position vs. time graph for a sprinter, where the tangent line at a point represents the sprinter's instantaneous speed.
Derivative
Definition: The derivative represents the slope of the tangent line or the instantaneous rate of change at a point.
Importance: Fundamental concept in differential calculus.
Notation:
Leibniz's Notation: ( \frac{dy}{dx} ) (super small changes in y for super small changes in x as x approaches zero).
Lagrange's Notation: ( f'(x) ) (denotes the derivative of the function f at point x).
Physics Notation: ( \dot{y} ) (less common in calculus, more in physics).
Mathematics Notation: ( y' ).
Moving Forward in Calculus
Limits: Essential for understanding and calculating derivatives.
General Equations: Ability to describe derivatives for any given point using equations.
Excitement: Encouragement to be excited about learning these tools and concepts.