hw01S10 - JAEHYUNPARK Math124,sectionD,Fall2016...

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Unformatted text preview: JAE HYUN PARK Math 124, section D, Fall 2016 Instructor: Dmitriy Drusvyatskiy WebAssign hw01S10.1 (Homework) Current Score : 32 / 33 Due : Friday, September 30 2016 11:59 PM PDT The due date for this assignment is past. Your work can be viewed below, but no changes can be made. Important! Before you view the answer key, decide whether or not you plan to request an extension. Your Instructor may not grant you an extension if you have viewed the answer key. Automatic extensions are not granted if you have viewed the answer key. View Key 1. 4/4 points | Previous Answers Consider the circle of radius 10 centered at the origin. Provide answers accurate to two decimal places. (a) The equation of the tangent line to the circle through the point (­6,8) has equation y= 0.75 x + 12.5 . (b) Suppose that L is a tangent line to this circle which is parallel to the line y=5x+7 and has a negative y intercept. Then the point of tangency of L with this circle is ( 9.81 , ­1.96 ). 2. 8/8 points | Previous Answers Draw the unit circle and plot the point P=(8,2). Observe there are TWO lines tangent to the circle passing through the point P. Answer the questions below with 3 decimal places of accuracy. (a) The line L1 is tangent to the unit circle at the point ( ­.123 , .992 ). (b) The tangent line L1 has equation: y= .124 x + 1.008 . (c) The line L2 is tangent to the unit circle at the point ( .358 (d) The tangent line L2 has equation: y= .384 x + ­1.071 . , ­.934 ). 3. 4/5 points | Previous AnswersSCalcET7 10.1.045. Suppose that the position of one particle at time t is given by x1 = 4 sin(t), y1 = 2 cos(t), 0 ≤ t ≤ 2π and the position of a second particle is given by x2 = −4 + cos(t), y2 = 1 + sin(t), 0 ≤ t ≤ 2π. (a) Graph the paths of both particles. How many points of intersection are there? 2 points of intersection (b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? Yes No If so, find the collision points. (Enter you answers as a comma­separated list of ordered pairs. If an answer does not exist, enter DNE.) $$(−4, 0) (c) If the x­coordinate of the second particle is given by x2 = 4 + cos(t) instead, is there still a collision? Yes No Enhanced Feedback Please try again, keeping in mind that a collision happens when x1 = x2 and y1 = y2 for the same t. Find the solutions of this system of equations. 4. 3/3 points | Previous Answers The parametric equations x = x1 + (x2 − x1)t, y = y1 + (y2 − y1)t where 0 ≤ t ≤ 1 describe the line segment that joins the points P1(x1, y1) and P2(x2, y2). Draw the triangle with vertices A(1, 1), B(3, 4), C(1, 6). Find the parametrization, including endpoints, and sketch to check. (Enter your answers as a comma­separated list of equations. Let x and y be in terms of t.) $$x=1+2t,y=1+3t A to B $$x=3−2t,y=4+2t B to C $$x=1,y=1+5t A to C 5. 8/8 points | Previous Answers An ant is moving around the unit circle in the plane so that its location is given by the parametric equations (cos(π t), sin(π t)). Assume the distance units in the plane are "feet" and the time units are "seconds". In particular, the ant is initially at the point A=(1,0). A spider is located at the point S=(7,0) on the x­axis. The spider plans to move along the tangential line pictured at a constant rate. Assume the spider starts moving at the same time as the ant. Finally, assume that the spider catches the ant at the tangency point P the second time the ant reaches P. (a) The coordinates of the tangency point P=( $$.143 , $$.990 ). (b) The FIRST time the ant reaches P is $$0.454 seconds. (c) The SECOND time the ant reaches P is $$2.454 seconds. (d) The parametric equations for the motion of the spider are: x(t)= $$−2.794 t + $$7 ; y(t)= $$0.403 t + $$0 . 6. 5/5 points | Previous Answers The graph of the quadratic function y = 2x2 – 4x + 1 is pictured below, along with the point P=(­1,7) on the parabola and the tangent line through P. A line that is tangent to a parabola does not intersect the parabola at any other point. We can use this fact to find the equation of the tangent line. (a) If m is the slope of the tangent line, then using the slope/point formula, the equation of the tangent line will be: y = m(x­ $$−1 ) + $$7 (b) The values of x for which the point (x,y) lies on both the line and the parabola satisfy the quadratic equation: 2x2 + bx + c = 0 where b= $$−(m+4) and c= $$−(m+6) (b and c should depend on m). (c) For most values of m, the quadratic equation in part (b) has two solutions or no solutions. The value of m for which the quadratic equation has exactly one solution is the slope of the tangent line. This value is m = $$−8 . ...
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