Math 307 G - Spring 2018 Midterm 1 April 20, 2018 Name: Section: Student ID Number: • There are 7 pages in total. Make sure your exam contains all these questions. • There is one bonus problem. The maximal score you can get is 50. • You are allowed to use a scientific calculator (no graphing calculators and no calculators that have calculus capabilities) and one hand-written 8.5 by 11 inch page of notes. • You must show your work on all problems. The correct answer with no supporting work may result in no credit. Put a box around your FINAL ANSWER for each problem and cross out any work that you don’t want to be graded. Give exact answers wherever possible. • If you need more room, use the backs of the pages and indicate to the grader that you have done so. • Raise your hand if you have a question. • You have 50 minutes to complete the exam. Budget your time wisely. Problem 1 10 Problem 2 8 Problem 3 8 Problem 4 12 Problem 5 12 Problem 6 (bonus) 3 Total 50 GOOD LUCK! 1
Transcript
Math 307 G - Spring 2018Midterm 1
April 20, 2018
Name:
Section:
Student ID Number:
• There are 7 pages in total. Make sure your exam contains all these questions.
• There is one bonus problem. The maximal score you can get is 50.
• You are allowed to use a scientific calculator (no graphing calculators and no calculatorsthat have calculus capabilities) and one hand-written 8.5 by 11 inch page of notes.
• You must show your work on all problems. The correct answer with no supporting work may resultin no credit. Put a box around your FINAL ANSWER for each problem and cross outany work that you don’t want to be graded. Give exact answers wherever possible.
• If you need more room, use the backs of the pages and indicate to the grader that you have doneso.
• Raise your hand if you have a question.
• You have 50 minutes to complete the exam. Budget your time wisely.
Problem 1 10
Problem 2 8
Problem 3 8
Problem 4 12
Problem 5 12
Problem 6 (bonus) 3
Total 50
GOOD LUCK!
1
1. (10 pts)
(a) (5 pts) Find the implicit solution to the equation
dy
dx=
1− e−x
y + sin(y).
(b) (5 pts) Use the substitution v = 6x+ y to find the general solution of the equation
dy
dx= 6x+ y
2
2. (8 pts)
(a) (4 pts) Classify all the equilibrium solutions of the equation y′ = −(y−a)(y2− 1)(e−y − e5)2,with a constant a > 1. Justify your answer.
(b) (4 pts) Let y(t) be a solution of the equation in (a) with y(0) = 1.5. If we know limt→+∞
y(t) = 3.
Find the value of the constant a. Justify your answer.
3
3. (8 pts) Consider the following differential equation
dy
dt= t2 + y2, y(0) = 1.
Find the approximate value of y(3) using Euler’s method, with step size h = 1.
4
4. (12 pts) A tank initially contains 100 L of water in which 10 kg of salt is dissolved. Watercontaining 1 kg/L salt flows into the tank at a constant rate of k L/min (k > 0). The mixtureflows out at a rate of k L/min. Assume that the salt is uniformly distributed in the tank.
(a) (4 pts) Let y(t) be the amount of salt in the tank after t minutes. Write a differential equationfor y(t) (with unit kg) in the tank at any time t, and write the initial condition.
(b) (3 pts) Determine limt→+∞
y(t) and justify your answer.
(c) (5 pts) If we know after 20 minutes, the concentration (NOT the mass) of the salt in thewater is (1 + e2) kg/L. Find the value of k.
5
5. (12 pts) Newton’s law of cooling states that the temperature of an object changes at a rateproportional to the difference between the temperature of the object itself and the temperatureof its surroundings.
(a) (4 pts) There is a cup of ice water in a room with ambient temperature Ts, which satisfies attime t, Ts(t) = 70 + e−t sin(t). The initial temperature of the ice water is 30◦F . Assume theabsolute value of the proportionality constant K is 1. Let T (t) be the temperature of theice water after time t. Write a differential equation with an initial value for T .
(b) (5 pts) What is the temperature of the ice water at time t?
