IAPC - What is Calculus? Lesson

What is Calculus?

Calculus is the mathematics of change and motion. It provides methods for solving two large classes of problems: finding the rate at which a variable quantity is changing, which belongs to the domain of differential calculus, and finding a function when its rate of change is known, which belongs to the domain of integral calculus. In calculus functions are the major tool and the concept of a limit is the foundation upon which calculus is built. The basic idea of a limit distinguishes calculus from all other areas of mathematics. It is through calculus that we can calculate the orbits of satellites and life insurance premiums, predict population sizes, measure blood flow through the heart, and model problems in many areas related to our daily lives.

Precalculus versus Calculus

Precalculus (sometimes called analysis) addresses areas of mathematics such as algebra, geometry, and trigonometry. Many topics contained in precalculus are revisited and applied in calculus. For example, in precalculus analyzing the steepness of a line involves finding its slope between two points lying on the line, often referred to as average rate of change. In calculus you can analyze the slope of a curve at a single point, which is referred to as instantaneous rate of change. You will no longer be restricted to finding areas of geometric figures, as you were in precalculus. Calculus provides the mechanism for finding areas under curves defined in rectangular, parametric, and even polar forms.

The table below provides examples of possibilities for investigating concepts based on precalculus or calculus techniques.

Precalculus	Calculus
Slope of a line	Slope of a curve
Area of a rectangle	Area under a general curve
Motion of an object traveling at a constant velocity	Velocity of an accelerating object
Tangent line to a circle	Tangent line to a general graph
Length of a segment	Length of a curve

Examine the problem situations listed below and choose whether they could be solved using precalculus or calculus techniques.

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