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# 693 Review Sheet for MATH M0050 at Purdue

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This 4 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 80 views.

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Date Created: 02/06/15
Review 2 Math 266 Spring 2001 APPLICATIONS You should be able to set up and solve up and solve problems of the types 0 Rate of change of y is proportional to y7 y Icy o Newtonls Law of Cooling7 T kT Te7 Where Te is the temperature of the surrounding environment 0 Problems in mechanics7 F ma or mdi Sum of external forces Forces on an object that is moving vertically near the surface of the earth are gravity7 w mg7 and a force proportional to velocity7 so Trad 3 tmg lw F ket ble the f e d e t v39ty 39 F m 2 or roc ro ms orc u o ra 1 1s p 7 g 10 R2 do do Use d7 vi and nd the velocity as a function of the height above the 6 surface of the earth 0 Mixing problems Let Qt be the amount of Whatever is in solution and let Vt be the amount of solution CM m so Qt ctVt Rate of Q coming in is Concentration in Rate of solution in Rate of Q going out is Concentration at time 23 Rate of solution out dQ dt Caution if Rate of solution in 7E Rate of solution out then you must The concentration of Q at time t is then Ct Rate of Q coming in Rate of Q going out dV solve for Vt from E Rate of solution in Rate of solution out If the in rates and out rates are constant7 then Vt V0 kt Where k is the difference of the rates 2 MA 266 SPR 01 REVIEW 2 PRACTICE QUESTIONS 1 Suppose y is proportional to y y0 47 and y2 2 Find y in terms of t For what value oft does yt 3 2 A thermometer reads 360 when it is moved into a 700 room Five minutes later the thermometer reads 500 Find the thermometer reading t minutes after it is moved into the room What will it read ten minutes after it is moved into the room 3 Determine the vertical velocity of a 1284b parachutist t seconds after jumping from an airplane that is ying slowly and horizontally at an altitude of 5000 feet Assume that air resistance is eight times the speed and ignore horizontal motion 4 A rocket is launched from earth with initial vertical velocity Find a formula that relates the velocity of the rocket and its height What is the maximum height of the rocket Assume that gravity is the only force acting on the rocket and that the mass of the rocket is constant 5 A 500gal tank contains 200 gal of brine with salt concentration of 2 oz gal Pure water ows into the tank at a rate of 10 gal min7 while the mixture ows out of the tank at a rate of 5 gal min Find the salt concentration in the tank at the time the tank becomes completely lled d 6 Consider the differential equation d z yy 27 t 2 07 Hint draw a rough graph of F yy 2 to analyse where y is positive and negative a Sketch the graph of the solution of the differential equation for t 2 0 with each of the initial values y0 237 y0 07 y0 237 y 437 310 27 310 83 b What are the equilibrium solutions c Which equilibrium solutions are stable d For which intervals of y is the graph of yt increasing e For which intervals of y is the graph of yt concave upward 7 Solve the initial value problem 3 2y 07 y0 3 8 Solve the initial value problem 3 2y 17 y0 3 1 W7 210 1 What is the 9 Solve the initial value problem 3 largest open interval on which the solution is valid 3 10 Solve the initial value problem 3 yy 17 y0 2 What is the largest open interval on Which the solution is valid 11 Solve the initial value problem y ylac7 y1 47 y 1 2 12 Solve the initial value problem y y 2y7 y0 e y 0 1 13 For each of the initial value problems determine the largest interval for Which a unique solution is guaranteed a y 2y 1 y10 t t b y tan ty sec 75 y0 0 t 1 I 01 Cyt29y 7t2y 1 ltdgtltt4gty tyt 21 14 For each of the initial value problems determine all initial points to7 yo for Which a unique solution is guaranteed in some interval to h lt t lt to h a y 762 92 W0 yo b y 762 312 31050 yo C 2 7621 We yo 01 y 7613212 We yo 4 MA 266 SPR 01 REVIEW 2 PRACTICE QUESTION ANSWERS 1 y 4611105 t 39 m 083 7 ln05 2 T 70 34elt1nlt1017gtgtt5 T10 5820 3 1126 16 166 i 2R2g R 0 5 C60 032 OZgal 4 v2 Rg wmax B when U 0 6 y gt01feitherylt00rygt2 y lt01f0ltylt2 a For y0 23 yt increases asympt to line y 0 For y0 0 stays on y 0 line For y0 23 or 43 yt decreases asympt to y 0 line For y0 2 stays on y 2 line For y0 83 decreases asympt to liney2 by0 y2 Cy0 dylt0 ygt2 e0ltylt1 ygt2 7y3e 2t 8y12e2t 1 9 t 1 y t1 gt 2 10 ym 11 y 3 32 12 y 6 605 7T 7 14 a All t0y0 b All t0y0 except 00 c All t0y0 with yo 74 0 d All to yo with yo 74 0 13 atgt0 b glttlt c 3lttlt2d 4lttlt0

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