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## Lengths of parametrized curves

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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 . The curve is differentiable if it is differentiable at every parameter value. The curve is smooth if and are continuous and not simultaneously zero.**Parametric Formula for**Recall from Section 3.6: Finding the second derivative:**Parametric Formula for**Find as a function of if , Step 1: Express in terms of : Step 2: Differentiate with respect to :**Parametric Formula for**Find as a function of if , Step 3: Divide by :**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…)**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…**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…**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:**Guided Practice**Find the length of the given curve.**Guided Practice**Find the length of the given curve.**Guided Practice**Find the length of the given curve.**Guided Practice**Find the length of the given curve.