Math 250B (Spring 2010)  Quiz 1 SOLUTIONS 1. (5 points) Calculate the following integral if it converges.
0

ex dx 9 + ex
We can quickly solve the integral if we let u = 9 + ex .
0
a a
lim
ex dx = lim ln (9 + ex )0 = ln (9 + 1)  lim ln (9 + ea ) a a
Math 250B (Spring 2010)  Quiz 2 SOLUTIONS 1. (5 points) Find the value of k such that the following limit exists. x2  kx + 6 x1 x1 lim Our goal is to factor our numerator in such a way that we can cancel out the denominator. There are a few ways to acc
Math 250B (Spring 2010)  Quiz 3 SOLUTIONS 1. (5 points) Find the sum of the following series.
( 1 )t t=3
2
We can see that our series is geometric, and since the rate is less than one, we can use the 1 formula 1r to calculate the sum of the series. Fir
Math 250B (Spring 2010)  Quiz 4 SOLUTIONS 1. (5 points) Use the ratio test to decide if the series converges or diverges. Make sure to clearly explain why it has that particular behavior.
(!)2 =1
(2)!
a+1  ( + 1)2 [( + 1)!]2 (2)! 1 = lim = lim = (2 +
Math 250B (Spring 2010)  Quiz 5 SOLUTIONS 1. (5 points) Find the radius of convergence for the following power series.
2n (x  1)n n=1
n
Applying the Ratio Test, we find:
[
n
lim
2n+1 x  1n+1 n+1
][
2nx  1 n = lim = 2x  1 < 1 n x  1n n n + 1
Math 250B (Spring 2010)  Quiz 6 SOLUTIONS 1. (5 points) Find the first four nonzero terms of the Taylor polynomial for the following function near = 0. g() = cos()
First, we will need to find the first four nonzero derivatives. g(0) = 1 g () =  sin() g
Math 250B (Spring 2010)  Quiz 7 SOLUTIONS 1. (5 points) Solve the given initial value problem. y + y = 0 y(0) = 4; y (0) = 3
Since we have constant coefficients for our secondorder ODE, we assume y(t) to have the form et . We now arrive with the followi
Math 250B (Spring 2010)  Quiz 8 SOLUTIONS 1. (5 points) Find the value for the following complex number. Write your answer in the form z = a + bi. z = (1 + i)36
Our complex number lies in the first quadrant with an angle of . The value of R is 4 our com
Math 250B (Spring 2010)  Quiz 9 SOLUTIONS 1. (5 points) Find the general solution for this second order differential equation. x  5x + 3x = 0 Also, sketch a few example solution curves. We first find our characteristic equation to be 2  5 + 3 = 0 which
Math 250B (Kennedy)  Exam 3  Spring '08 SHOW YOUR WORK. Correct answers with no work will get no credit. You may not use anything on the web other than the PPlane programs. There are 5 problems for a total of 100 points. 1. (20 points) Consider the hom
Math 250B (Kennedy)  Exam 2 Solutions  Spring '08 SHOW YOUR WORK. Correct answers with no work will get no credit. You may not use anything on the web other than the pplane and RK programs. There are 5 problems for a total of 100 points. 1. (22 points)
Math 250b (Kennedy)  Quiz 1 Solutions  Spring '08 1. Calculate the following integral if it converges
0
ex dx 1 + ex
Solution: Let u = 1 + ex . Then du = ex dx. So 1 ex dx = du =  ln u =  ln(1 + ex ) x 1+e u So b ex ex dx = lim dx b 0 1
Math 250b (Kennedy)  Quiz 2  Spring '08 1. For each of the following sequences, state whether it converges or diverges. If it converges, give its limit. sn = (0.99)n converges to 0 since 0.99 < 1. 2n2  n + 7 7  n2
sn = converges to
2 1
= 2 sin(n) n
Math 250b (Kennedy)  Quiz 3  Spring '08 1. For each of the following series, determine whether it converges or diverges. You should explain your reasoning; in particular cite the test that you use.
n=1
n2 3n
Use ratio test: an+1 (n + 1)2 3n (n + 1)2 1 =
Math 250b (Kennedy)  Quiz 4 Solutions  Spring '08 1.
n=1
x2n n!
x2(n+1) n! x2 2n = (n + 1)! x n+1 This coverges to 0 as n for all x, so the series converges for all x.
Ratio test:
n=1
(x + 2)n n 3n
Ratio test: 3n x + 2 n + 1 x + 2n+1 (n + 1) = n+1 n
Math 250b (Kennedy)  Quiz 5 Solutions  Spring '08 1. Find the solution of each of the following second order differential equations with initial conditions. (a) x  3x + 2x = 0, x(0) = 1, x (0) = 0 Characteristic equation is r 2  3r + 2 = 0. This has r
Math 250b (Kennedy)  Quiz 6  Spring '08 1. Consider the first order system x y = y = 3x2  3
(a) Find a second order differential equation for x that corresponds to this system. Solution: x = y = 3x2  3 So second order equation for x is x = 3x2  3 or
Math 250b (Kennedy)  Quiz 7 Solutions  Spring '08 1. Find the general solution of x  4x + 5x = cos(2t). Solution: We start by looking for one solution with the guess: x = A sin(2t) + B cos(2t) x = 2A cos(2t)  2B sin(2t) x = 4A sin(2t)  4B cos(2t) wh
Math 250b (Kennedy)  Quiz 8  Spring '08 Each of the following systems has only one equilibrium. For each system, (a) Find the equilibrium (b) Classify it as stable node, unstable node, saddle, stable spiral, unstable spiral, periodic (center). (c) Find
Math 250b (Kennedy)  Quiz 9 Solutions  Spring '08 Variation of parameters:
z1 x1 + z2 x2 = 0, z1 x1 + z2 x2 = f (t) a2
1. (a) t2 is one solution of the homogeneous equation t2 x  2x = 0. Find another (linearly independent) solution of the homogeneous
Math 250B (Kennedy)  Exam 1 Solutions  Spring '08 SHOW YOUR WORK. Correct answers with no work will get no credit. There are 6 problems for a total of 100 points. p(p  1) 2 p(p  1)(p  2) 3 x + x + 2! 3! The error in the nth order Taylor polynomial ha
Math 250B (Spring 2010)  Quiz 10 SOLUTIONS 1. (5 points) Find the general solution to the following differential equation. y  y = 5t  3 Let our solution be of the form y(t) = yh (t) + yp (t), where yh solves the homogeneous differential equation y  y