Calculus began with the powerful idea of studying change by zooming in so closely that curved motion looks almost straight. Early mathematicians used infinitesimals, imagined as quantities smaller than any ordinary measurable amount, to describe tiny changes in position, time, area, or volume. This made it possible to find slopes of curves and areas under curves in a way that matched physical intuition.
The central picture is a secant line becoming a tangent line as two points on a curve move closer together.
Key Facts
- Average rate of change: (f(x + h) - f(x)) / h
- Derivative as a limit: f'(x) = lim h->0 [f(x + h) - f(x)] / h
- A secant line intersects a curve at two points, while a tangent line represents the limiting slope at one point.
- Infinitesimal notation: dy/dx represents the ratio of an extremely small change in y to an extremely small change in x.
- Limit notation describes a value approached by a function: lim x->a f(x) = L
- Definite integral as a limit of sums: integral from a to b f(x) dx = lim n->infinity sum f(x_i*) delta x
Vocabulary
- Infinitesimal
- An infinitesimal is an idealized quantity that is smaller than any ordinary positive number but is used to represent an extremely tiny change.
- Limit
- A limit is the value that a function or expression approaches as the input approaches a chosen number.
- Secant line
- A secant line is a line that passes through two points on a curve and gives an average rate of change.
- Tangent line
- A tangent line is the line that best matches the direction of a curve at a single point.
- Derivative
- A derivative measures the instantaneous rate of change of a function at a point.
Common Mistakes to Avoid
- Treating h as exactly zero in (f(x + h) - f(x)) / h is wrong because direct substitution often creates division by zero before the limit is taken.
- Confusing a secant slope with a tangent slope is wrong because the secant slope is an average over an interval, while the tangent slope is the limiting value at one point.
- Thinking a limit must equal the function value is wrong because a function can approach one value near a point while being undefined or assigned a different value at the point.
- Canceling terms without algebraic justification is wrong because limit problems often require factoring, expanding, or rationalizing before simplifying safely.
Practice Questions
- 1 For f(x) = x^2, compute the average rate of change from x = 2 to x = 2.1, then compare it with the derivative f'(2).
- 2 Use the limit definition f'(x) = lim h->0 [f(x + h) - f(x)] / h to find the derivative of f(x) = 3x^2 at x = 1.
- 3 Explain why the slope of a secant line can approach the slope of a tangent line even though the two points on the curve never need to become literally the same point during the limiting process.