1 Section 0. The Clculus of Prmetric Curves Even though curve defined prmetricly my not be function, we cn still consider concepts such s rtes of chnge. However, the concepts will need specil tretment. For exmple, let s think bout rtes of chnge. The prmeter t controls the vribles x nd y, so it mkes sense to consider nd. However, we might lso be curious bout ; in prticulr, this would be helpful if we wished to find the eqution of tngent line to the curve. Unfortuntely, determining this rte of chnge is not s strightforwrd s it ws when we were differentiting functions f(x) of x. To solve the problem, we think of y s function of both x nd t, y y(x(t)). Then using the chin rule, we see tht. Finlly, we hve, if 0. Exmple The prmetric curve shown below is defined by x sin t, y t + : Find the eqution in terms of x nd y of tngent line to the curve t t π. To write the eqution for the tngent line, we must know its slope, i.e. bove; using the eqution, which ws given requires us to evlute nd. Since y(t) t +, t; nd x(t) sin t mens tht cos t. At t π, we hve t π π nd t π ; so t π π π.
2 Section 0. Now the eqution for line pssing through (x, y ) with slope m is y y m(x x ); we hve lre determined the slope m, so we need to find point through which the line psses. This, of course, is best chosen s the point on the curve where t π, whose coordintes re x sin( π) 0 nd y ( pi) + π +. The eqution for the tngent line is The tngent line is grphed in red below. y πx + π +. In this section, we will lern how to determine the length of curve. For instnce, we might wnt to determine the length of the curve below: To think bout the mening of curve length, imgine lying piece of string on top of the curve; once we strightened the string, we could mesure it with ruler. The length would be the length of the curve.
3 Section 0. To determine the length of curve mthemticlly, we will gin employ the technique we hve seen so often in clculus: we mke n pproximtion to wht we wnt using something we know, mke the pproximtions better, then use clculus to go from n pproximtion to the exct vlue. Given the curve defined by x f(t), y g(t), our first pproximtion for the length of the curve below is dmittedly poor: We use the length of the line joining the beginning nd ending points of the curve. The length of line is the distnce between its beginning nd ending points, so L L (x x ) + (y y ) (f(t ) f(t )) + (g(t ) g(t )). If we use two lines insted of just one, then the pproximtion is probbly closer to the ctul curve length: To simplify nottion, set x i x i+ x i nd y i y i+ y i. Then L L + L ( x ) + ( y ) + ( x ) + ( y ). We get better pproximtions by using more lines: Our pproximtion of the length of the curve is now L k ( xi ) + ( y i ). i We will mke the pproximtion exct by tking limit; however, we first note tht, for smll chnges in t, f (t i ) x i t. So we rewrite x i f (t i ) t nd y i g (t i ) t, which llows us to 3
4 Section 0. write L k ( xi ) + ( y i ) i k (f (t i )) + (g (t i )) t. Tking the limit is bit more complicted thn it ppers on the surfce, nd we will not del with the detils; however the rgument cn be mde precise. We rrive t the following: Theorem If the curve C is defined by the prmetric equtions x f(t) nd y g(t) on [, b], f (t) nd g (t) re continuous on [, b] nd not both 0 t ny point in [, b], nd C is trversed exctly once s t increses from t to t b, then the length of the curve C is precisely b b ( ) ( ) L (f (t)) + (g (t)) +. i Exmple Find the length of the curve defined by the prmetric equtions x t nd y t 3 from t to t. Since t nd 3t, the formul for rc length tells us tht the length of the curve is L (t) + (3t ) 4t + 9t 4 t 4 + 9t ( t ) 7 7 ( ) 7 ( ). (using the u-substitution u 4 + 9t ) If the curve f(x) is chosen so tht f (x) is continuous on [, b], we cn consider it s prmetric curve by setting x t nd g(t) f(x), so tht nd f (x); then the originl formul b ( ) ( ) L + becomes L b ( ) b + + (f (x)). 4
5 Section 0. Alterntively, if x g(y) is defined s function of y, then the rc length of x is given by b ( ) b L + + (g (y)). Find the length of the curve x y3 6 + y Since y y, we hve from y to y 3. ( ) y4 4 + so tht ( ) + + y4 4 + y (y + y y + y. ) The length of the curve is L 3 3 ( ) + y + 3 y3 6 y y