Question 1 Suppose that r(s) is an arclength parametrized curve such that, at s = 1,7 = 2.9270, T = (0.2822, 0.9593, −0.0093), dN/ds = (−0.1038, −1.3308, 2.9262), then the value of cos(k) + sin(B · i) + sin(B · j) + sin(B · k) at s = 1 is . O 0.8985 -2.8465 1.1498 -1.8025 O 0.1268 1.0095 -0.2394 O-1.2183 1 pts
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- General Sol. of y' tanxsin2y= sin2x+sin2y isA curve is given parametrically by x = 3 + 9e tan acos 0, where a is a constant in the interval (0, π/2) and is a real-valued parameter. y = 4 + 9e tan asin 0. In this example, we can consider the whole curve by allowing the intrinsic angle to take any real value. Find an expression for tan in terms of 0 and a. Now, find an expression for the arclength s in terms of and a, choosing a sign convention such that s is positive and setting for convenience s→→0 as →∞0₁ Finally, combine the previous two results to find a relation between s and. From the form of this relation (and/or the original parametric equations) what shape must the curve have? (Choose one) OA circle OA spiral OA cycloid ONone of the other options OA straight lineAs the wheel of radius r cm in the figure rotates, the rod of length L attached to point P drives a piston back and forth in a straight line. Let x be the distance from the origin to point at the end of the rod as shown. (a) Use the Pythagorean Theorem to show that L² = (x − r cos 0)² + ² sin² 0. (b) Differentiate the equation in part (a) with respect to t to show that 0=2(x-r cos 0) (d+rsin 0df)+2r² sin cos de. dt (c) Calculate the speed of the piston when , assuming that r = 10 cm, L = 30 cm, and the wheel rotates at 4 revolutions per minute. L X = Piston moves back and forth e
- Find the expression for the tangent slope dy/dx for points on the Polar curve r= 6 +12 cosθ1. The parametric equations for the astroid below are x = sin (t), y = – cos* (t) on 0 < t < 2n. a) Find the starting point on the astroid and mark it on the graph as well as the direction of travel as t increases. dy b) Show all work to find the derivative at the point where t 2 -1 dxConsider the following. sec²(0), 0 ≤ 0 < x = 3 cos(0), y = sec T 2 (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
- Question 4 Consider the curve 5x2+4xy+4y3 = 29125 (a) Show that P(-75,-5) is on the curve i.e., show that P satisfies 5x² + 4xy + 4y³ (b) Find dy dx i.e., y'. (c) Determine the value of y' at P by.... (i) using the expression in (b), and = 29125. (ii) substituting the coordinates of the point immediately after you carry out the differen- dy tiation in (b) (i.e, right before you start solving for the d) dx What do you observe? (d) A portion of the curve of 5x² + 4xy+4y³ 29125 is illustrated below. A D B E Which of the labelled points could be P(-75, -5)? Briefly justify your answer.= 1 (h) (1- cos? r) (1+ cot² r) = 1 %3DWhat is the point of intersection of the tangent lines to the curve r(t) = sin(πt)i + 8sin(πt)j + cos(πt)k at the points t = 0 and t = 0.5 ?