Definition of Gradient
What is the gradient of a curve?
- At a given point the gradient of a curve is defined as the gradient of the tangent to the curve at that point
- A tangent to a curve is a line that just touches the curve at one point but doesn't cut the curve at that point
- A tangent may cut the curve somewhere else on the curve
- It is only possible to draw one tangent to a curve at any given point
- Note that unlike the gradient of a straight line, the gradient of a curve is constantly changing
What is a derivative?
- Calculus is about rates of change
- the way a car’s position on a road changes is its speed (velocity)
- the way the car’s speed changes is its acceleration
- The gradient (rate of change) of a (non-linear) function varies with
- The derivative of a function is a function that relates the gradient to the value of
- The derivative is also called the gradient function
Exam Tip
- If a question asks for the "rate of change of ..." then it is asking for the "gradient"