Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane. The slope of tangent lines to curves. The arc-length of a curve. The arc-length function and differential. Review: Parametric curves on the plane A curve on the plane is given in parametric form iff it is given b the set of points ( (t), (t) ), where the parameter t I R. Eample Describe the curve (t) = cosh(t), (t) = sinh(t), for t, ). Solution: (t)] 2 (t)] 2 = (t) cosh 2 (t) sinh 2 (t) = 1. (t) This is a portion of a hperbola with asmptotes = ±, starting at (1, ). 1
Review: Parametric curves on the plane A ccloid with parameter a > is the curve given b (t) = a(t sin(t)), (t) = a(1 cos(t)), t R. Remar: From the equation of the ccloid we see that (t) at = a sin(t), (t) a = a cos(t). Therefore, (t) at] 2 + (t) a] 2 = a 2. Remars: This is not the equation of a circle. The point ((t), (t)) belongs to a moving circle. The ccloid plaed an important role in designing precise pendulum clocs, needed for navigation in the 17th centur. Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane. The slope of tangent lines to curves. The arc-length of a curve. The arc-length function and differential.
The slope of tangent lines to curves A curve defined b the parametric function values ( (t), (t) ), for t I R, is differentiable iff each function and is differentiable on the interval I. Theorem Assume that the curve defined b the graph of the function = f (), for (a, b), can be described b the parametric function values ( (t), (t) ), for t I R. If this parametric curve is differentiable and (t) for t I, then holds df d = (d/dt) (d/dt). Proof: Epress (t) = f ((t)), then d dt = df d d dt df d = (d/dt) (d/dt). The slope of tangent lines to curves Remar: The formula df d = (d/dt) provides an alternative wa (d/dt) to find the slope of the line tangent to the graph of the function f. = f ( ) f( ) = f()
The slope of tangent lines to curves Eample Find the slope of the tangent lines to a circle radius r at (, ). Solution: The equation of the circle is 2 + 2 = r 2. One possible set of parametric equations are: (t) = r cos(nt), (t) = r sin(nt), n 1. The derivatives of the parametric functions are (t) = nr sin(nt), (t) = nr cos(nt). The slope of the tangent lines to the circle at = cos(nt ) is ( ) = (t ) (t ) = nr cos(nt ) nr sin(nt ) 1 ( ) = tan(nt ). Remar: In the first quadrant holds ( ) = 1 ( ) 2. Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane. The slope of tangent lines to curves. The arc-length of a curve. The arc-length function and differential.
The arc-length of a curve The length or arc length of a curve in the plane or in space is the limit of the polgonal line length, as the polgonal line approimates the original curve. = f() L +1 +1 Theorem The arc-length of a continuousl differentiable curve ( (t), () ), for t a, b] is the number b L = (t) ] 2 + (t) ] 2 dt. a The arc-length of a curve Idea of the Proof: The curve length is the limit of the polgonal line length, as the polgonal line approimates the original curve. = f() L +1 +1 L N = N 1 n= ( ) 2 + ( ) 2 {a = t, t 1,, t N 1, t N = b}, L N N 1 n= L N N L = (t )] 2 + (t )] 2 t, b a (t) ] 2 + (t) ] 2 dt.
The arc-length of a curve Eample Find the length of the curve ( r cos(t), r sin(t) ), for r > and t π/4, 3π/4]. (Quarter of a circle.) Solution: Compute the derivatives ( r sin(t), r cos(t) ). The length of the curve is given b the formula L = 3π/4 π/4 r sin(t) ] 2 + r cos(t) ] 2 dt L = 3π/4 π/4 r 2( sin(t) ] 2 ] 2 ) 3π/4 + cos(t) dt = r dt. π/4 Hence, L = π 2 r. (The length of quarter circle of radius r.) The arc-length of a curve Eample Find the length of the spiral ( t cos(t), t sin(t) ), for t, t ]. Solution: The derivative of the parametric curve is ( (t), (t) ) = ( t sin(t) + cos(t) ], t cos(t) + sin(t) ]), ( ) 2 + ( ) 2 = t 2 sin 2 (t) + cos 2 (t) 2t sin(t) cos(t) ] + t 2 cos 2 (t) + sin 2 (t) + 2t sin(t) cos(t) ] We obtain ( ) 2 + ( ) 2 = t 2 + 1. The curve length is given b L(t ) = t t 1 + t 2 dt = 1 + t 2 2 + 1 2 ln( t + 1 + t )] 2 t We conclude that L(t ) = t 2. 1 + t 2 + 1 2 ln( ) t + 1 + t 2.
Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane. The slope of tangent lines to curves. The arc-length of a curve. The arc-length function and differential. The arc-length function and differential Remar: The previous eample suggests to introduce the length function of a curve. The arc-length function of a continuousl differentiable curve given b ( (t), (t) ) for t t, t 1 ] is given b L(t) = t t (τ) ] 2 + (τ) ] 2 dτ. Remars: (a) The value L(t) of the length function is the length along the curve ( (t), (t) ) from t to t. (b) If the curve is the position of a moving particle as function of time, then the value L(t) is the distance traveled b the particle from the time t to t.
The arc-length function and differential Remar: The arc-length differential is the differential of the arc-length function, that is, This is a useful notation. dl = (t) ] 2 + (t) ] 2 dt. Eample Find the length of (t) = (2t + 1) 3/2 /3, (t) = t + t 2 for t, 1]. Solution: We first compute the length differential, 1 dl = 3 3 ] 2 2 (2t + ] 2 1)1/2 2 + 1 + 2t = (2t + 1) + 1 + 4t + 4t 2 L = 1 (4t 2 + 6t + 2) dt = ( 4t 3 ) 3 + 3 1 t2 + 2t = 19 3.