Calculus studies change, and rates of change are one of its central ideas. When a quantity depends on another variable, such as position depending on time, the graph of that relationship can show how quickly it changes. The average rate of change compares two points on a curve and is represented by the slope of a secant line.
The instantaneous rate of change describes what is happening at one exact point and is represented by the slope of a tangent line.
The bridge between these ideas is the difference quotient, which measures the slope between x and x + h. As h gets closer to 0, the two points move together and the secant line approaches the tangent line. This limiting value is the derivative, written f'(x), and it gives the instantaneous rate of change of f at x.
Derivatives are used in physics for velocity and acceleration, in biology for growth rates, and in economics for marginal change.
Key Facts
- Average rate of change from x = a to x = b is [f(b) - f(a)] / (b - a).
- The slope of a secant line gives the average rate of change over an interval.
- Instantaneous rate of change at x = a is f'(a).
- Derivative definition: f'(a) = lim h->0 [f(a + h) - f(a)] / h.
- If s(t) is position, then velocity is v(t) = s'(t).
- A positive derivative means f is increasing, and a negative derivative means f is decreasing at that point.
Vocabulary
- Rate of change
- A measure of how much one quantity changes compared with a change in another quantity.
- Secant line
- A line that passes through two points on a curve and shows average rate of change.
- Tangent line
- A line that touches a curve at one point and has the same instantaneous slope as the curve there.
- Difference quotient
- The expression [f(x + h) - f(x)] / h that gives the slope between two nearby points on a function.
- Derivative
- The limit of the difference quotient as h approaches 0, representing instantaneous rate of change.
Common Mistakes to Avoid
- Using f(b) - f(a) without dividing by b - a. This gives only the change in output, not the rate of change per unit input.
- Treating a secant slope as an instantaneous slope. A secant line measures change over an interval, while a tangent line measures change at one point.
- Substituting h = 0 directly into [f(a + h) - f(a)] / h. This usually creates division by zero, so the expression must be simplified before taking the limit.
- Ignoring units in rate problems. If position is in meters and time is in seconds, the derivative has units of meters per second, not meters.
Practice Questions
- 1 For f(x) = x^2 + 3x, find the average rate of change from x = 1 to x = 4.
- 2 Use the limit definition to find f'(2) for f(x) = x^2.
- 3 A graph has a secant line between two points with positive slope, but the tangent line at the right point has negative slope. Explain what this says about average change over the interval compared with instantaneous change at that point.