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

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1 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 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) 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 (x 2 + x 3 ) 2 is o(x 3 ). (b) R 4 e x = e x (1 + x + x 2 + x 3 + x 4 ) (c) Let f(x) and g(x) be functions whose Taylor series exist. Then for any n we have T n (f(x)g(x)) = (T n f(x))(t n g(x)). (d) The equation y = 1 is a solution to the initial value problem dy dx = y and y(0) = 1. (e) Is dy dx = x(x 2)(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 = 2y + 3x and y(1.0) = 0. Answer: (b) Find T1 2 (arctan x). You may find it helpful to recall that d 1 dx (arctan x) =. 1+x 2 You do not need to simplify any values of arctan in your answer. Answer: (c) Consider dy dx = x2 y(y 1)(sin y). When separating the variables to solve this, you introduce lost solutions. List all lost solutions for this differential equation. Answer:

5 3. (a) Let g(x) = cos(5x 3 ). Compute g (2016) (0). Answer: (b) Let F (x) be an antiderivative of e 3x2 such that F (0) = 1. Compute T 3 F (x). Answer:

6 4. Find a solution to each initial value problem. (a) dy dx ex y 2 = e x and y(0) = 0. (b) Solution Satisfying Initial Condition: y = (1 + x 2 ) dy dx + 2xy = 3(1 + x2 ) and y(1) = 5 2. 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. Start with a full 50 quart vat containing a 8% (by volume) solution of vinegar, at 10:00am. A solution of 5% vinegar flows in at a rate of 2 quarts per minute. The solution is kept thoroughly mixed and drawn off at a rate of 3 quarts per minute. We are interested in a function describing the total amount of vinegar in the vat at a given time. Variables (3pts): Differential equation (5pts) Initial condition (2pts):

8 6. We have a vat containing a mixture of acid and water. Let: t stand for time in minutes from 12:00pm A(t) denote the total amount of acid in the vat at time t V (t) denote the total volume of liquid in the vat at time t. Assume that da dt = 3 2A 20 t and A(0) = 0 and V (t) = t. What is the concentration (%) of acid in the vat after 10 minutes?

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

10 8. Let f(x) be a function satisfying the differential equation f (x) + sin(2x 2 ) f(x) = 0 and also satisfying the initial conditions f(0) = 2 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.

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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 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