Understanding the Gradient of a Curve
The gradient of a curve at a specific point is defined as the limit of the gradient of a chord as the length of the chord approaches zero.
To visualize this, consider a curve represented by the function y = f(x). A chord is formed between two points on the curve, (x, f(x)) and (x + h, f(x + h)), where h is the distance between the two points. The gradient of the chord is given by:
Gradient of chord = (f(x + h) - f(x)) / h
As h approaches zero, the chord becomes increasingly smaller and approaches the tangent at the point (x, f(x)). Thus, we define the gradient of the curve at point x as:
Gradient at x = lim (h → 0) [(f(x + h) - f(x)) / h]
This limit represents the slope of the tangent line to the curve at that point, providing the instantaneous rate of change of the function at x.
Understanding this concept is crucial for solving problems related to motion, optimization, and various applications in calculus.
Key points to remember
- Gradient is the slope of a curve at a point.
- Defined as the limit of the gradient of a chord.
- Chord connects two points on the curve.
- As chord length approaches zero, it becomes a tangent.
- Instantaneous rate of change is given by the limit.
Worked example
Define the gradient of the curve y = x^2 at the point (2, 4).
Model Answer:
- Identify f(x) = x^2.
- Gradient of chord = (f(2 + h) - f(2)) / h = ((2 + h)^2 - 4) / h.
- Simplifying gives (4 + 4h + h^2 - 4) / h = (4h + h^2) / h = 4 + h.
- Taking the limit as h → 0, gradient = 4.