Theory

Theory 1

We can use and data to compute the slope of a parametric curve in terms of .

Slope formula

Given a parametric curve , its slope satisfies:

Concavity formula

Given a parametric curve , its concavity satisfies the formula:

Extra - Derivation of slope and concavity formulas

For both derivations, it is necessary to view as a function of through the inverse parameter function. For example if is the parametrization, then is the inverse parameter function.

We will need the derivative in terms of . For this we use the formula for derivative of inverse functions:

Given all this, both formulas are simple applications of the chain rule.

For the slope:

For the concavity:

(In the second step we inserted the formula for from the slope.)

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Pure vertical, Pure horizontal movement

In view of the formula , we see:

• Pure vertical: when and yet
• Pure horizontal: when and yet

When for the same , we have a stationary point, which might subsequently progress into pure vertical, pure horizontal, or neither.

Theory 2

Arclength formula

The arclength of a parametric curve with coordinate functions and is:

This formula assumes the curve is traversed one time as increases from to .

Counts total traversal

This formula applies when the curve image is traversed one time by the moving point.

Sometimes a parametric curve traverses its image with repetitions. The arclength formula would add length from each repetition!

Extra - Derivation of arclength formula

The arclength of a parametric curve is calculated by integrating the infinitesimal arc element:

In order to integrate in the variable, as we must for parametric curves, we convert to a function of :

So we obtain and the arclength formula follows from this:

Theory 3

Distance function

The distance function returns the total distance traveled by the particle from a chosen starting time up to the (input) time :

We need the dummy variable so that the integration process does not conflict with in the upper bound.

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Speed function

The speed of a moving particle is the rate of change of distance:

This formula can be explained in either of two ways:

1. Apply the Fundamental Theorem of Calculus to the integral formula for .
2. Consider for a small change : so the rate of change of arclength is , in other words .

Theory 4

Surface area of a surface of revolution: thin bands

Suppose a parametric curve is revolved around the -axis or the -axis.

The surface area is:

The radius should be the distance to the axis:

This formulas adds the areas of thin bands, but the bands are demarcated using parametric functions instead of input values of a graphed function.

The formula assumes that the curve is traversed one time as increases from to .