Section 18.1: What is Calculus?

18.1 Outline

A.  The Limit: Zeno's Paradox
      1. Achilles and the tortoise
      2. limit of a sequence
B. The derivative: the tangent problem
      1. secant line
      2. tangent line
      3. differential calculus
C. The integral: the area problem
      1. approximating the area of a circle
      2. area under a curve
      3. integral calculus
D.  Mathematical modeling
      1. modeling process
      2. developing a budget

18.1 Essential Ideas

1. 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.

2. 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.

3. 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.