Final Exam Review Part I: Unit IV Material

Final Exam Review Part I: Unit IV Material Math114 Department of Mathematics, University of Kentucky April 26, 2017 Math114 Lecture 37 1/ 11

Outline 1 Conic Sections Math114 Lecture 37 2/ 11

Outline 1 Conic Sections 2 Math114 Lecture 37 2/ 11

Outline 1 Conic Sections 2 3 Math114 Lecture 37 2/ 11

Outline 1 Conic Sections 2 3 4 Math114 Lecture 37 2/ 11

Outline 1 Conic Sections 2 3 4 Math114 Lecture 37 3/ 11

Ellipse The ellipse shown has (0, b) semi-major axis a ( a, 0) a P(x, y) (a, 0) semi-minor axis b foci at (±c, 0) where ( c, 0) (c, 0) c 2 = a 2 b 2 (0, b) The sum of distances to the two foci is 2a The equation of the ellipse is x 2 a 2 + y 2 b 2 = 1 Math114 Lecture 37 4/ 11

Ellipse If the semi-major axis lies on the y-axis, the picture changes as shown (0, a) (0, c) P(x, y) a The ellipse shown has semi-major axis a semi-minor axis b foci at (0, ±c) where ( b, 0) (b, 0) c 2 = a 2 b 2 (0, c) (0, a) The sum of distances to the two foci is 2a The equation of the ellipse is x 2 b 2 + y 2 a 2 = 1 Math114 Lecture 37 5/ 11

Parabolas (0, p) y = p Parabola with focus (0, p) and directrix y = p x 2 = 4py x = p (p, 0) Parabola with focus (p, 0) and directrix x = p y 2 = 4px Math114 Lecture 37 6/ 11

Hyperbolas y = b a x y = a b x y y = b a x y y = b a x x x Hyperbolas come in two flavors, shown at left: x 2 a 2 y 2 b 2 = 1 (foci (±c, 0), vertices (±a, 0), asymptotes y = ±(b/a)x) and y 2 a 2 x2 b 2 = 1 (foci (0, ±c), vertices (0, ±a), asymptotes y = ±(a/b)x) Math114 Lecture 37 7/ 11

Examples Identify the following conic sections. Find their vertices and foci, and sketch a graph of the conic section. 1 x 2 2x + 2y 2 8y + 7 = 0. 2 y 2 2 = x 2 + 2x Math114 Lecture 37 8/ 11

Models & Separation of Variables Key Models dp dt = kp - Exponential Growth Model dp dt = kp ( 1 P M ) - Logistic Growth Model (k = intrinsic growth rate, M = Carrying capacity) dt dt = k(t (t) T s ) - Newton s Law of Cooling (T s = surrounding temp.) dq ft = (rate in) - (rate out) - Mixture Problems Math114 Lecture 37 9/ 11

Models & Separation of Variables Key Models dp dt = kp - Exponential Growth Model dp dt = kp ( 1 P M ) - Logistic Growth Model (k = intrinsic growth rate, M = Carrying capacity) dt dt = k(t (t) T s ) - Newton s Law of Cooling (T s = surrounding temp.) dq ft = (rate in) - (rate out) - Mixture Problems Example: A 200 o F cup of tea is left in a 65 o F room. At time t = 0, the tea is cooling at 5 o F per minute. (a) Set up an IVP for the temperature of the tea at time t. (b) Determine the temperature of the tea at time t = 27 minutes. (c) How long foes it take for the tea to reach 100 o F? Math114 Lecture 37 9/ 11

Direction (Slope) Fields dy dx = f (x, y) - f (x 0, y 0 ) is the slope of a solution curve passing through the point (x 0, y 0 ). A. B. Match the direction field to the corresponding equation below: 1 y = y/2 2 y = 2x xy Math114 Lecture 37 10/ 11

Consider the first order differential equation y = f (x, y). Define y n = y n 1 + hf (x n 1, y n 1 ) where h = x n x n 1 for n = 1, 2,... The solution evaluated at x n, y(x n ) y n provided h is sufficiently small. Geogebra Worksheet - Math114 Lecture 37 11/ 11

Consider the first order differential equation y = f (x, y). Define y n = y n 1 + hf (x n 1, y n 1 ) where h = x n x n 1 for n = 1, 2,... The solution evaluated at x n, y(x n ) y n provided h is sufficiently small. Geogebra Worksheet - Example Use with a step size of 0.1 to estimate y(0.3), where y(x) is the soltion to the initial-value problem y = y + xy, y(0)=1. Math114 Lecture 37 11/ 11

Consider the first order differential equation y = f (x, y). Define y n = y n 1 + hf (x n 1, y n 1 ) where h = x n x n 1 for n = 1, 2,... The solution evaluated at x n, y(x n ) y n provided h is sufficiently small. Geogebra Worksheet - Example Use with a step size of 0.1 to estimate y(0.3), where y(x) is the soltion to the initial-value problem y = y + xy, y(0)=1. Using separation of variables, compare your answer to the exact solution. Math114 Lecture 37 11/ 11