Note I had written a different question at first for which the instruc tions

# Note i had written a different question at first for

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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. Z π 2 / 4 0 1 x e 2 sin x cos x dx Z π/ 3 π/ 6 x cos x 2 dx 3. Find the area of the region bounded by the y = x and y = x 3 . 4. Compute the following indefinite integrals.
100 CHAPTER 6. QUIZZES AND MIDTERMS Z (ln x ) 2 dx Z t 2 e t dt 5. Does the following integral converge or diverge? If it converges, com- pute its value. Z e e 1 x ln x (ln ln x ) 2 dx. 6. (Extra Credit) Consider the logistic equation dx dt = rx (1 - x ) . Find an explicit formula for x in terms of t by remembering how we went from dy/dx = ky to y = y (0) e kx . 6.3.2 Midterm 1 1. Suppose a medication has a half-life of 24 hours. A particular patient takes 50mg every day at the same time for several months. Approx- imately how much of the drug will be in the patient’s system right before taking the daily dose? Use (you don’t have to derive it) an infinite series formula. 2. Compute the following definite integrals. Simplifying is optional. Z e 2 1 3 x (ln x ) 4 dx Z 3 π/ 3 0 x 2 tan( x 3 ) dx 3. Find the area of the region bounded by x = - π/ 6, x = π/ 6, y = sec x and y = cos x + 1. 4. Compute the following indefinite integrals.
6.3. MIDTERMS 101 Z arctan x dx Z 1 x + x 2 dx 5. Does the following integral converge or diverge? If it converges, com- pute its value. Z 0 e - x sin x dx. 6. (Extra Credit) Show that k =1 1 k 2 is less than 2 by comparing it to a series or integral that you know how to compute exactly. Fun fact, it’s exactly π 2 / 6. 6.3.3 Practice Midterm 2 1. Find the second-degree Taylor approximation to the function f ( x ) = sec x centered at x = π/ 4, a.k.a. with a = π/ 4. 2. (a) Estimate the integral Z 2 0 sin( x ) dx using the Trapezoid Method with n = 5. (b) Give an upper bound on the error of this approximation. (c) Compute the integral exactly. What does your estimate say about the value of cos 2? 3. Solve the following initial value problems. (a) dN dt = N (1 - N ) , N (0) = 1 / 2 (b) ds dt = 1 t + 1 , s (0) = 1
102 CHAPTER 6. QUIZZES AND MIDTERMS 4. Graphically analyze the autonomous differential equation dL dt = 3( L + 1) 3 ( L - 1) 2 (3 - L ) in order to find the equilibria and classify their stability. Sketch the four solutions with L (0) = - 2 , 0 , 2 , 4. 5. Use the standard Ansatz to solve the following second-order initial value problems. (a) y 00 - y 0 - 2 y = 0 , y (0) = 0 , y 0 (0) = 2 (b) y 00 + y 0 + 1 4 y = 0 , y (0) = 1 , y 0 (0) = 1 (c) y 00 - 2 y 0 + 2 y = 0 , y (0) = 1 , y 0 (0) = - 1 6. (Extra Credit) (a) Write down a differential equation with two stable equilibria, one unstable equilibrium, and two saddle points. (b) Show that the solution to the differential equation dy dx = y, y (0) = 0 is not unique. In other words, find two distinct solutions with the same initial condition. Remark: The fact that the derivative of y blows up at 0 is what allows for this strange phenomenon. 6.3.4 Midterm 2 1. Find the third-degree Taylor approximation to the function f ( x ) = e x centered at x = 1, a.k.a. with a = 1.