8. [-/0.12 Points] DETAILS SCALCET9 10.2.AE.003. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Example Video Example .() Find the area under one arch of the cycloid x = r(0 - sin(0)) y = r(1 - cos(0)) 2 ar x Solution One arch of the cycloid is given by 0 s 0 S 2x. Using the Substitution Rule with y = r(1 - cos(0)) and dx = r(1 - cos(0)) do, we have 1) do - 12 / ( 1 - cos( 0) ) 2 do - 12 (1 - 2 cos(0 ) + cos ? (0) ) do = 12 [1 - 2 cos(0) + 4(1 + cos(20)| de Need Help? Read It 9. [-/0.12 Points] DETAILS SCALCET9 10.2.039. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Use the parametric equations of an ellipse, x = a cos(0), y = b sin(0), 0 s 0 s 2x, to find the area that it encloses. Need Help? Read it Watch It 10. [-/0.12 Points] DETAILS SCALCET9 10.2.038.MI. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Find the area enclosed by the given parametric curve and the y-axis. x = +2 - 2t, y = Vt 1.5 1.0 0.5 - 1.0 -0.5 0.5 1.0 1.5 Need Help? Read it Master It4. [40.12 Points] DETAILS SCALCET9 10.XP.2.040. 0/100 Submissions Used Find an equation of the tangent line to the curve at the given point by both eliminating the parameter and without eliminating the parameter. x=3+ln(t), y=t2+1. (3.2) Need new 5. [-10.12 Points] DETAILS SCALCET9 10.2.017. 011 00 Submissions Used MY NOTES ASK YOUR TEACHER For which values of t is the curve concave upward? (Enter your answer using Interval notation.) Z Need Help? _' 6. [40.12 Points] DETAILS SCALCET9 10.2.021. 01100 Submissions Used MY NOTES ASK YOUR TEACHER Find the points on the curve where the tangent Is horizontal or vertical. You may want to use a graph from a calculator or computer to check your work. (If an answer does not exist, enter DNE.) x=r33t, y=t29 horizontal tangent (x, y) = U ) vertical tangent (smaller x-value) (x, y) = U ) veltital tangent (larger x-value) (x, y) = (i ) Need Help? ii 7. [40.12 Points] DETAILS SCALCET9 10.2.033. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER At what point(s) on the curve x = 6t2 + 2, y = t3 6 does the tangent line have slope %? .n=(E) Need Help? MY NOTES ASK YOUR TEACHER Example Video Example () (a) Find the tangent line to the cycloid x = r(0 - sin(0)), y = r(1 - cos(0)) at the point where 0 = _ (b) At what points is the tangent horizontal? When is it vertical? Solution (a) The slope of the tangent line is dy dy de dx dx r(1 - cos(0)) When 0 = _, we have x = (* - sin(#)) = ( * =1(1 - cos(#)) = and sin dy = dx 1 - cos() Therefore the slope of the tangent is 2 1 - V3 and its equation is y - 1( 1 - 2 2 ) - 2(1 - V3 ) ( * - (8 -=). The tangent is sketched in the figure. (b) The tangent is horizontal when = , which occurs when sin(0) = and 1 - cos(0) # dx that is, 0 = (2n - 1)x, n an integer. The corresponding point on the cycloid is ((2n - 1)ar, 2r). When 0 = 2nx, both- dy de de are 0. It appears from the graph that there are vertical tangents at these points. We can verify this by using I'Hospital's Rule as follows. lim ay = lim sin(8) cos(0) lim 0- 2na + dx -+ 2na+ 1 - cos(0) 0 - 2nx+ sin(0) A similar computation shows that dy - -co as 0 - 2nx , so indeed there are vertical tangents when 0 = 2nx, that is, when (let n be an arbitrary integer). Need Help? Read It 2. [-/0.12 Points] DETAILS SCALCET9 10.XP.2.003. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Find dy dx X = - 4+t' y = V4+t dy dy Need Help? Read It 3. [-/0.12 Points] DETAILS SCALCET9 10.XP.2.039. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Find an equation of the tangent line to the curve at the point corresponding to the given value of the parameter. x = t cos(t), y = t sin(t); t = x Need Help? Read It