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.
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.
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
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 0.0.. 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 ( 4 3 + 9t ) 7 7 (40 3 3 3 ) 7 (80 0 3 3). (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
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 4 4 + + (y + y y + y. ) The length of the curve is L 3 3 ( ) + y + 3 y3 6 y y 7 6 6 8 6 + 4 8 6 + 4 3 + 4 3 4. 5