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Lengths of parametrized curves. Section 10.1a. Definition: Derivative at a Point. A parametrized curve , , . , has a derivative at if and. have derivatives at .

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slide2

Definition: Derivative at a Point

A parametrized curve , ,

, has a derivative at if and

have derivatives at .

The curve is differentiable if it is differentiable at every

parameter value. The curve is smooth if and

are continuous and not simultaneously zero.

slide3

Parametric Formula for

Recall from Section 3.6:

Finding the second derivative:

slide4

Parametric Formula for

Find as a function of if ,

Step 1: Express in terms of :

Step 2: Differentiate with respect to :

slide5

Parametric Formula for

Find as a function of if ,

Step 3: Divide by :

slide6

Length of a Smooth Parametrized Curve

If a smooth curve , , ,

is traversed exactly once as increases from to ,

the curve’s length is

(This formula is based on logic similar to that which we

used to find the lengths of curves in Section 7.4…)

slide7

Length of a Smooth Parametrized Curve

Find the length of the astroid:

First, graph the astroid in [–1.5, 1.5] by [–1, 1]

The curve is traced once on this interval. Because of the

curve’s symmetry, its length is four times the length of the

first quadrant portion…

slide8

Length of a Smooth Parametrized Curve

Find the length of the astroid:

Because we’re only working in the first quadrant, we won’t

need the absolute value…

slide9

Length of a Smooth Parametrized Curve

Find the length of the astroid:

Length of the first quadrant portion of the curve:

Length of the entire astroid:

Support numerically:

slide10

Guided Practice

Find the length of the given curve.

slide11

Guided Practice

Find the length of the given curve.

slide12

Guided Practice

Find the length of the given curve.

slide13

Guided Practice

Find the length of the given curve.