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Note: I had written a different question at first for which the instruc-tions were misleading.2. Compute the following definite integrals. Simplifying is optional.•Zπ2/401√xe2 sin√xcos√x dx•Z√π/3√π/6xcosx2dx3. Find the area of the region bounded by they=√xandy=x3.4. Compute the following indefinite integrals.
100CHAPTER 6.QUIZZES AND MIDTERMS•Z(lnx)2dx•Zt2etdt5. Does the following integral converge or diverge? If it converges, com-pute its value.Z∞ee1xlnx(ln lnx)2dx.6. (Extra Credit) Consider the logistic equationdxdt=rx(1-x).Find an explicit formula forxin terms oftby remembering how wewent fromdy/dx=kytoy=y(0)ekx.6.3.2Midterm 11. Suppose a medication has a half-life of 24 hours. A particular patienttakes 50mg every day at the same time for several months. Approx-imately how much of the drug will be in the patient’s system rightbeforetaking the daily dose?Use (you don’t have to derive it) aninfinite series formula.2. Compute the following definite integrals. Simplifying is optional.•Ze213x(lnx)4dx•Z3√π/30x2tan(x3)dx3. Find the area of the region bounded byx=-π/6,x=π/6,y= secxandy= cosx+ 1.4. Compute the following indefinite integrals.
6.3.MIDTERMS101•Zarctanx dx•Z1x+x2dx5. Does the following integral converge or diverge? If it converges, com-pute its value.Z∞0e-xsinx dx.6. (Extra Credit) Show that∑∞k=11k2is less than 2 by comparing it to aseries or integral that you know how to compute exactly. Fun fact, it’sexactlyπ2/22.214.171.124Practice Midterm 21. Find the second-degree Taylor approximation to the functionf(x) =secxcentered atx=π/4, a.k.a. witha=π/4.2. (a) Estimate the integralZ20sin(x)dxusing the Trapezoid Method withn= 5.(b) Give an upper bound on the error of this approximation.(c) Compute the integral exactly. What does your estimate say aboutthe value of cos 2?3. Solve the following initial value problems.(a)dNdt=N(1-N),N(0) = 1/2(b)dsdt=1√t+ 1,s(0) = 1
102CHAPTER 6.QUIZZES AND MIDTERMS4. Graphically analyze the autonomous differential equationdLdt= 3(L+ 1)3(L-1)2(3-L)in order to find the equilibria and classify their stability.Sketch the four solutions withL(0) =-2,0,2,4.5. Use the standard Ansatz to solve the following second-order initial valueproblems.(a)y00-y0-2y= 0,y(0) = 0,y0(0) = 2(b)y00+y0+14y= 0,y(0) = 1,y0(0) = 1(c)y00-2y0+ 2y= 0,y(0) = 1,y0(0) =-16. (Extra Credit) (a) Write down a differential equation with two stableequilibria, one unstable equilibrium, and two saddle points.(b) Show that the solution to the differential equationdydx=√y,y(0) = 0is not unique. In other words, find two distinct solutions with the sameinitial condition.Remark:The fact that the derivative of√yblows up at 0 is whatallows for this strange phenomenon.6.3.4Midterm 21. Find the third-degree Taylor approximation to the functionf(x) =excentered atx= 1, a.k.a. witha= 1.