Prmetric Equtions, Tngent Lines, & Arc Length SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chpter 10.1 of the recommended textbook (or the equivlent chpter in your lterntive textbook/online resource) nd your lecture notes. EXPECTED SKILLS: Be ble to sketch prmetric curve by eliminting the prmeter, nd indicte the orienttion of the curve. Given curve nd n orienttion, know how to find prmetric equtions tht generte the curve. Without eliminting the prmeter, be ble to find dx nd d y t given point on dx prmetric curve. Be ble to find the rc length of smooth curve in the plne described prmetriclly. PRACTICE PROBLEMS: For problems 1-5, sketch the curve by eliminting the prmeter. Indicte the direction of incresing t. x = t + 1. y = t 4 0 t y = x 17 from (, 4) to (9, 5) 1
x = cos t. y = sin t π t π x 4 + y 9 = 1 from (, 0) to (, 0). x = t 5 y = t 0 t 9 y = x + 5 from ( 5, 0) to (4, ) 4. x = sec t y = tn t 0 t < π y = x 1 for x 1
x = sin t 5. y = cos (t) π t π y = 1 x from ( 1, 1) to (1, 1) For problems 6-10, find prmetric equtions for the given curve. there re mny correct nswers; only one is provided.) (For ech, 6. A horizontl line which intersects the y-xis t y = nd is oriented rightwrd from ( 1, ) to (1, ). x = t y = 1 t 1 7. A circle or rdius 4 centered t the origin, oriented clockwise. x = 4 sin t y = 4 cos t 8. A circle of rdius 5 centered t (1, ), oriented counter-clockwise. x = 5 cos t + 1 y = 5 sin t ; Detiled Solution: Here 9. The portion of y = x from ( 1, 1) to (, 8), oriented upwrd. x = t y = t 1 t
10. The ellipse x 4 + y 16 x = cos t y = 4 sin t = 1, oriented counter-clockwise. For problems 11-1, find dx nd d y t the given point without eliminting the dx prmeter. x = sin (t) 11. The curve y = cos (t) t t = π 0 < t < π d y dx = 0; t=π dx = 1 t=π 9 x = t 1. The curve y = t t t = 1 t 0 dx = t=1 ; d y dx = t=1 4 x = tn t 1. The curve y = sec t 0 t π dx = 4, d y dx = t t = π 4 ; Detiled Solution: Here 16 x = t 14. Consider the curve described prmetriclly by y = t + 1 t 0 () Compute dx without eliminting the prmeter. t=64 dx = 1 t=64 4
(b) Eliminte the prmeter nd verify your nswer for prt () using techniques from differentil clculus. The curve is equivlent to y = x / + 1, x 0. And, t = 64 corresponds to x = 8. Thus, dx = t=64 dx = 1 x=8 (c) Compute n eqution of the line which is tngent to the curve t the point corresponding to t = 64. y 5 = 1 (x 8) 15. Consider the curve described prmetriclly by x = cos t y = 4 sin t () Compute dx without eliminting the prmeter. dx = (b) Eliminte the prmeter nd verify your nswer for prt () using techniques from differentil clculus. The curve is equivlent to the ellipse x 4 + y 16 = 1. And, t = π 4 corresponds to the point (x, y) = (, ). Thus, you cn use implicit differentition nd dx = dx = (x,y)=(, ) (c) Compute n eqution of the line which is tngent to the curve t the point corresponding to t = π 4. y ( = x ) (d) At which vlue(s) of t will the tngent line to the curve be horizontl? t = π nd t = π For problems 16-18, compute the length of the given prmetric curve. x = t 16. The curve described by y = t/ 0 t 4 5
+ 10 5 17. The curve described by x = e t y = et/ ln t ln + 16 ; Detiled Solution: Here x = 1 t 18. The curve described by y = 1 t 7 0 t 19. Compute the lengths of the following two curves: x = cos t C 1 (t) = y = sin t x = cos (t) C (t) = y = sin (t) Explin why the lengths re not equl even though both curves coincide with the unit circle. The length of C 1 (t) is π nd the length of C (t) = 6π. Notice tht C (t) is the just curve C 1 (t) trversed three times. 0. This problem describes how you cn find the re between prmetriclly defined curve nd the x-xis. The Min Ide: Recll tht if y = f(x) 0, then the re between the curve nd the x-xis on the intervl [, b] is f(x)dx = y dx. Now, suppose tht the sme curve is described prmetriclly by x = x(t), y = y(t) for t 0 t t 1 nd tht the curve is trversed exctly once on this intervl. Then, A = y dx = t1 t 0 y(t)x (t) dt. 6
x = sin t Consider the curve y = cos (t) π 4 t π 4 () Compute the re between the grph of the given curve nd the x-xis by evluting A = A = π/4 π/4 t1 t 0 y(t)x (t) dt. cos (t) cos t dt = (b) After eliminting the prmeter to express the curve s n explicitly defined function (y = f(x)), clculte the re by evluting A = A = / / ( 1 x ) dx = f(x) dx. 7