Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.08 Textbook: Calculus by Varberg, Purcell, Rigdon, eight edition. April 9 12.5, 12.6 Rotation and Polar coordinates p539: 1-4,(6),11,(12),14,15,23 p544: 1,5,7,9,11-14,17-21,(22),(24),(28),29,(32),33,37 Main Idea: 10 miles that-a-way. Key Words: polar coordinates, limacon, cardioid, spiral Goal: learn rotation of axes and polar coordinates. ------------------------------------------------------------------ Previous assignment: Extra Credit Hand in home work assignment. The equation Sqrt[x] + Sqrt[y] = 1 is a parabola. Find the equation of the directrix and the focus. Page 530 Problem 2 The sum of the distances of P from (+/-4,0) is 14. c = 4 a = 7 b = Sqrt[49-16] = Sqrt[33] x^2/49 + y^2/33 = 1 --------------------------------------------------- Page 530 Problem 4 The difference of the distances of P from (0,+/-6) is 10. c = 6 a = 5 b = Sqrt[36-25] = Sqrt[11] -x^2/11 + y^2/25 = 1 ------------------------------------------------------- Page 530 Problem 6 Find the equation of the tangent line to the given curve at the given point. x^2/27 + y^2/9 = 1 at (3,Sqrt[6]) 2x/27 + 2y/9 y' = 0 6/27 + 2 Sqrt[6] y' = 0 -6 -Sqrt[6] y' = -------------- = ------------ 27 2 Sqrt[6] 54 y-Sqrt[6] = -Sqrt[6]/54 (x-3) -------------------------------------------------------- Page 530 Problem 31 Listeners A(-8,0), B(8,0), and C(8,10) recorded the exact times at which they heard an explosion. If B and C heard the explosion at the same time and A heard it 12 seconds later, where was the explosion. Assume that distances are in kilometers and that sound travels 1/3 kilometer per second. | C (8,10) | | | | o | o ....................|............o........ | | o | | o | -----A--------------------+----------o---------B---------- (-8,0) | (8,0) | o | | o | c = 8 a = 2 b = Sqrt[64-4] = Sqrt[60] x^2/4 - y^2/60 = 1 when y = 5 x^2/4 - 25/60 = 1 x^2/4 = 85/60 x^2 = 85/15 = 17/3 x = +/- Sqrt[17/3] Answer (Sqrt[17/3], 5) Sqrt[ (Sqrt[17/3]+8)^2 + 25 ] = 11.5219 km or 34.5657 seconds Sqrt[ (Sqrt[17/3]-8)^2 + 25 ] = 7.5219 km or 22.5657 seconds ----------------------------------------------------------------------- Page 535 Problem 34 Find the foci of the ellipse 16(x-1)^2 + 25 (y+2)^2 = 400 (x-1)^2 (y+2)^2 ------------ + ----------- = 1 400/16 400/25 Center is (1,-2) a = 5 b = 4 c = 3 Focus is (4,-2) (-2,-2) --------------------------------------------------------------- Page 535 Problem 36 Find the equation of the given conic. Hyperbola with center (2,-1), vertex at (4,-1) and focus at (5,-1) | | | | ---------------------------+-------------------------------- | (2,-1) (4,-1)F | | | | | | | | | C V (5,-1) | | a = 2 c = 3 b = Sqrt[9-4] = Sqrt[5] (x-2)^2 (y+1)^2 ------ - -------- = 1 4 5 ----------------------------------------------------- Page 535 Problem 40 Ellipse with foci at (2,0) and (2,12) and a vertex at (2,14) | (2,14) V | | (2,12) F | | | | ----------------------------+------(2, 0)F-------------------- | | c = 6 a = 8 b = Sqrt[64-36] = Sqrt[28] (x-2)^2 (y-6)^2 -------- + ------- = 1 28 64 -------------------------------------------------------- Page 535 Problem 44 Hyperbola with foci (0,0) and (0,4) that passes through (12,9) x^2 (y-2)^2 - ----- + --------- = 1 b^2 a^2 - 144/b^2 + 49/a^2 = 1 a^2 + b^2 = 4 Solve[{ -144/b^2 + 49/a^2 ==1, a^2 + b^2 ==4}] a = 1, b= Sqrt[3] -x^2/3 + (y-2)^2 = 1 ----------------------------------------------------------------- Rotation of Axes. x = u Cos[t] - v Sin[t] y = u Sin[t] + v Cos[t] Example 1. xy = 1. Rotate the axis by 45 degrees. u^2/2 - v^2/2 = 1 -------------------------------------------------- The general quadratic Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 x = u Cos[t] - v Sin[t] y = u Sin[t] + v Cos[t] The new equation is au^2 + buv + cv^2 + du + ev + f = 0 Where b = B(Cos[t]^2 - Sin[t]^2) - 2(A-C) Sin[t] Cos[t] To make b = 0 b = B Cos[2t] - (A-C) Sin[2t] If B Cos[2t] = (A-C) Sin[2t] then Ctn[2t] = (A-C)/B Example 2: What angle should be used to eliminate the xy-term in 4x^2 + 2 Sqrt[3] xy + 2 y^2 + 10 Sqrt[3] x + 10 y = 5 4-2 Ctn[2t] = ------------ = 1/Sqrt[3] 2t = 60 degrees so t = 30 degrees. 2 Sqrt[3] -------------------------------------------------------------------- Show that B^2 - 4 A C is invariant under rotation. This tells us what type of quadratic the curve is. ---------------------------------------------------------- Polar graphing. Graph theta = Pi/4 r = 3 r = theta r = Cos[t] r = Sin[t] r = 1 + Cos[t] r theta = 1