Lecture Notes on Gradients and Tangents

Key Concepts

Gradient at a Point: Difficult to define as there are no two points to compare directly.
Tangent and Secant Lines:
- A tangent touches the curve at one point.
- A secant intersects the curve at two points.

Circle Example:
- Imagine a circle with a tangent line.
- Introduce parallel lines (secants) intersecting the circle at two points.
- As secants approach the tangent, the length of the chord (the line segment) between intersection points decreases.
- When secant becomes tangent, the chord length = 0.

Limits: Understanding behavior of secants as they approach a tangent.
- Use limits to analyze values as they approach a particular point.
- The idea is to calculate things we cannot measure directly.

Setting Up the Axes: Draw a curve representing some function of X.
Gradient of Secant:
- Select two points on the curve to find the secant's gradient.
- The gradient formula:
  
  [ \text{Gradient of Secant} = \frac{\text{rise}}{\text{run}} = \frac{f(x + h) - f(x)}{h} ]
- Where rise = difference in Y-values, run = difference in X-values (h).
Finding the Tangent:
- As h approaches 0, the secant approaches the tangent.
- Hence, we want to find the limit as h approaches 0.

Gradient Function: Changes based on the value of x; reflects the notion that gradients differ across a function.
Change Notation:
- Introduced the notation ( \Delta ) for change.
- Use ( dy ) for change in y and ( dx ) for change in x.
- Rise over run = ( \frac{dy}{dx} ) (change in y over change in x).

Gradient of tangent derived by understanding secants and limits.
Reflects the foundational concepts that underpin calculus.
They found a way to perform calculations even when direct values are not available.

Understanding gradients involves both geometric intuition (secants and tangents) and analytical tools (limits and calculus).
The transition from secant to tangent via limits is crucial to defining derivatives in calculus.