Calculus (short explanation)
The calculus pages are highly experimental. Volunteer high
school students are needed to read through the set of pages
and comment as to which portions could benefit from better
explanations.
Differentiation determines the slope of a function.
- Differentiation converts position to velocity.
- Differentiation converts velocity to acceleration.
Integration determines the area under a function.
- Integration converts acceleration to velocity.
- Integration converts velocity to position.
Two types of problems will be used to illustrate calculus:
- At midnight (00:00:00), a car with an odometer reading of 000000 starts traveling on a highway.
- A spacecraft accelerates and we can monitor the force
applied by the engines as a function of time.
If the function is f(x), we can write the derivative as f '(x).
We haven't discussed how to differentiate or integrate yet. However, to explain the notation we will use an example: the derivative of x2 with respect to x is 2x. If
we integrate 2x with respect to x, we get x2 + C.
The C is a constant that we will talk about later. The
notation used is shown below. The first line shows the differentation of x2 with respect to x, and the second line shows the integration of 2x with respect to x.
Differentiation and integration are inverse operations. If we
differentiate x2 with respect to x, we get 2x. If
we integrate 2x with respect to x, we get x2.
The symbol, curved somewhat like an S, located at the beginning
of the second equation, is called an integrand.
The 'dx' is called a differential. Notice there is a differential in both equations. The 'dx' signifies "differientiate with respect to x."
If we had a f(x,y) = x2y3, there would be two possible differentials.
df(x,y)
-------- = 2xy3
dx
df(x,y)
-------- = 3x2y2
dy
In conclusion, we still haven't discussed how to differentiate, but we have defined the terms differential and integrand which will be needed later. We have said that in an equation with several variables, we need the differential to determine which variable to differentiate (or integrate). Also, we have explained that differentiation and integration help us convert between distance traveled, velocity, and acceleration.
In next section we will talk more about what calculus does, and in the
following section we will see examples and how to use calculus for polynomial equations.
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Last Revised 01/25/98.
Copyright ©1998 by William L. Dechent. All rights reserved.