The three main ideas of calculus are limit, derivative, and integral

The Limit: Zeno’s Paradox

Achilles and the tortoise

limit of a sequence

The derivative: the tangent problem

secant line

tangent line

differential calculus

The integral: the area problem

approximating the area of a circle

area under a curve

integral calculus

Mathematical modeling

modeling process

developing a budget

18.1 Essential Ideas

Limit: The limit is a mathematical tool for studying the tendency of a function as its variable approaches some value. Calculus is based on the concept of limit. We shall introduce the limit of a function informally in the next section.

Derivative: The derivative is defined as a certain type of limit, and it is used initially to compute rates of change and slopes of tangent lines to curves. The study of derivatives is called differential calculus. Derivatives can be used in sketching graphs and in finding the extreme (largest and smallest) values of functions.

Integral: The integral is found by taking a special limit of a sum of terms, and the study of this process is called integral calculus. Area, volume, arc length, work, and hydrostatic force are a few of the many quantities that can be expressed as integrals.