Allen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park. James Hanson Julia Lindberg Mao Li Polly Yu Qiao He

Size: px

Start display at page:

Download "Allen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park. James Hanson Julia Lindberg Mao Li Polly Yu Qiao He"

Simon Harris
5 years ago
Views:

1 MATH 222 (Lectures 2,3, and 4) Fall 2017 Midterm 2 Name: Circle your TA s name from the following list. Allen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park James Hanson Julia Lindberg Mao Li Polly Yu Qiao He Tejas Bhojraj Weitong Wang Yu Sun Zihao Zheng Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Instructions On Problems 1 and 2 only the answer will be graded. On all other problems, you must show your work and we will grade the work and your justification. Problem 1,2,3,4 are multipart problems, and they are worth 14 or 15 points each. Problem 5,6,7,8 are worth 10 or 11 points each. No calculators, books, or notes (except for those notes on your 3 inch by 5 inch notecard.) Please simplify any formula involving a trigonometric function and an inverse trigonometric function. For example, please write cos(arcsin x) = 1 x 2. Final answers should not involve functions applied to either infinity or applied to a point outside of their domain. For instance, arctan( ) and ln(0) and 5 will not be accepted as a final answer. In a question involving a limit, we will accept a final answer of as synonymous with The limit does not exist. Please simplify any binomial expression ( b k).

2 Formulas T e x = k=0 xk k! T sin x = x2k+1 k=0 ( 1)k (2k+1)! T cos x = x2k k=0 ( 1)k (2k)! T 1 1 x = k=0 xk 1 T 1+x = k=0 ( 1)k x k T (1 + x) b = ( b ) k=0 k x k where ( ) b = b(b 1)(b 2) (b k+1) T ln(1 + x) = xk k=1 ( 1)k+1 k. k k!

3 1. On this page are four True/False statements and one question on slope fields. CIRCLE the correct answers below. (a) (b) (c) (d) (e) True False True False True False True False I II (a) The function ( cos(x 2 ) 1 ) is o(x 3 ). (b) T x = 1 x6 1 x. (c) If f(x) is polynomial then T n f(x) = f(x). (d) 2x 4 x 2x 4 x for all 1 x 1. (e) Is dy dx = (x 1)(y x)(1 y) the equation of slope field I or slope field II? I. II.

4 2. On this page only the answer will be graded. (a) Use Euler s method with step size h = 0.1 to approximate y(1.1) where y(x) is the solution of dy dx = x2 + xy + 2y 2 and y(1.0) = 0. Answer: (b) Compute T π 2 cos x. Answer: (c) Consider dy. When separating the variables to solve this, you introduce lost solutions. List all lost solutions for this differential equation. dx = x(y2 1) y(x 2 1) Answer:

5 3. On this page, partial credit is available. (a) Let g(x) = ln(1 + 3x 2 ). Compute g (100) (0). Answer: (b) Let f(x) = e x5 +x 7. Compute T 9 f (x). Answer:

6 4. On this page, partial credit is available. Find a solution to each initial value problem. (a) dy 1 + 2x + 3x2 = dx 1 + 2y and y(0) = 2. (b) Solution Satisfying Initial Condition: y = dy dx 2x 1 + x 2 y = 1 + x2 and y(0) = 7. Solution Satisfying Initial Condition: y =

7 5. In the problem below: 1. Clearly define variables (including units!); 2. Set up the appropriate differential equation; and 3. Write down the appropriate initial condition. DO NOT SOLVE THE DIFFERENTIAL EQUATION. At 9:00am, a 1000 liter tank starts out with 200 liters of fluid containing 10 grams/liter of dye. Pure water is poured in at 20 liters/minute and the tank is being drained at a rate of 15 liters/min. We are interested in a function describing the amount of dye in the tank at any time t. Variables (3pts): Differential equation (5pts) Initial condition (2pts):

8 6. On this page, partial credit is available. Find a solution to the initial condition problem: da dt = (4 A2 ) and A(0) = 3. (Your final answer should be an explicit solution, i.e. a solution of the form A = f(t).)

9 7. On this page, partial credit is available. You must justify your work on this problem. Let f(x) = 4x 5 + 3e 1 2x. (a) Find a number M such that M f (3) (x) for all 2 x 2. (b) Use your answer in part (a) to find a number B such that f(x) T 2 f(x) B for all x in the range 2 x 2.

10 8. On this page, partial credit is available. Let f(x) be a function satisfying the differential equation xf 1 (x) 1 + 3x 2 f(x) = 0 and also satisfying the initial conditions f(0) = 1 and f (0) = 0. Compute T 4 f(x). Note: it is essential that you use notation correctly in your answer, as part of what we are testing is whether you understand what the notation means.

11 This page left blank for additional work.

12 This page left blank for additional work.

13 This page left blank for additional work.

Allen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park. James Hanson Julia Lindberg Mao Li Polly Yu Qiao He

MATH 222 (Lectures 2,3, and 4) Fall 2017 Practice Midterm 2.1 Name: Circle your TA s name from the following list. Allen Zhang Bobby Laudone Dima Kuzmenko Geoff Bentsen Jaeun Park James Hanson Julia Lindberg