Math 1b Calculus Series and Differential Equations Review Guide for the Final Exam Spring 2006 Thomas W Judson Harvard University May 7 2006 Final Exam Details The final exam is comprehensive and will cover the entire course integration series and sequences and differential equations The final exam will be on Tuesday May 23 at 2 15 5 15 in Geo Lecture Hall There will also be three course wide review sessions Review session on differential equations Thursday May 11 at 2 4 PM in Science C Review session on integration Monday May 15 at 2 4 PM in Science C Review session on series and sequences Wednesday May 17 at 2 4 PM in Science C We plan to videotape the review session and you should be able to access the video by clicking on Lecture Videos at the course website Studying and Reviewing You can find copies of old exams clicking on Previous Exams at the course website http www courses fas harvard edu math1b prevexams You should also try working some of the problems in the review sections of Chapters 5 8 Solutions will be posted on the exam page Be sure to take advantage of the TF office hours CA sections and the MQC in Science 309 Office hours are listed at http www courses fas harvard edu math1b exams 1 Topics for the Final Exam For a list of topics on integration and series and sequences see the review guides for the first and second midterms at http www courses fas harvard edu math1b exams We will list the topics for differential equations below To understand how differential equations can be used to model problems from the natural sciences engineering economics and the social sciences 7 1 in Stewart 31 1 in Gottlieb To understand what it means to be the solution of a differential equation 7 1 in Stewart 31 2 in Gottlieb To understand the idea of a slope field 7 1 in Stewart 31 2 in Gottlieb To understand and be able to apply the existence and uniqueness theorem weak form for differential equations 7 1 in Stewart 31 2 in Gottlieb To be able to solve differential equations of the form dy dt g t 7 1 in Stewart 31 2 in Gottlieb To be able to solve differential equations of the form dy dt ky 7 1 in Stewart 31 2 in Gottlieb To understand that autonomous equations of the form dy dt f y are time independent 7 1 in Stewart 31 2 in Gottlieb To understand and be able to find equilibrium solutions for autonomous differential equations 7 2 in Stewart 31 3 in Gottlieb To understand and be able to determine the stability of equilibrium solutions for autonomous differential equations 7 2 in Stewart 31 3 in Gottlieb To understand and be able to draw the phase line for an autonomous differential equation and be able to use this information to sketch solutions of equations of the form dy dt f y 7 2 in Stewart 31 3 in Gottlieb To understand and be able to solve first order separable differential equations 7 3 5 in Stewart 31 4 in Gottlieb To be able model situations requiring first order separable differential equations such as mixing problems 7 3 5 in Stewart 31 4 in Gottlieb To be able to solve first order linear differential equations of the form y 0 p t y q t Handouts on first order linear differential equations available at http www courses fas harvard edu math1b handouts linear first order pdf http www cours 2 To be able to model applications using first order linear differential equations Handouts on first order linear differential equations available at http www courses fas harvard edu math1b handouts linear first order pdf http www cours To understand and be able to set up systems of first order differential equations 7 6 in Stewart 31 5 in Gottlieb To be understand and be able to analyze first order systems by examining the phase plane 7 6 in Stewart 31 5 in Gottlieb To understand and to be able to apply phase plane analysis to dx dt dy dt f x y g x y 7 6 in Stewart 31 5 in Gottlieb To understand that the system dx dt dy dt f x y g x y is completely predictive If you choose a starting point in the xy plane then there is exactly one solution that starts at your chosen point 7 6 in Stewart 31 5 in Gottlieb To understand how use the equation mx00 bx0 kx 0 to model a harmonic oscillator 31 6 in Gottlieb To be able to solve equations of the form ax00 bx0 cx 0 when the roots of the characteristic equations and real and distinct 31 6 in Gottlieb To be able to solve equations of the form ax00 bx0 cx 0 when the roots of the characteristic equations and repeated and real or complex 31 6 in Gottlieb 3