Math 244

Math 244:F7 Exam 1 Rutgers University, Dr. Molnar, July 26 2021 This is a 80-minute exam, with a five-minute grace period. You have until 3:55 to upload your solutions to Canvas. Please do not wait until the last minute. This exam is, by default, “open book”. This should be interpreted just as it would be if we were having our exam in the classroom. Specifically, you can consult physical copies of your class notes, the notes that I have typed up for online classes, and the textbook, including solutions. No resources other than what I have listed above are allowed. You may not communicate with any other person regarding this exam, directly or indirectly. No computational assistance is allowed. This includes – but is not limited to – calculators, wolframalpha, Sage, Ramanujan, and an abacus. You need to do, and show, all the work. Answers given without justification will not receive full credit. Solutions must use techniques discussed so far in this course. (This exam counts out of 80 points.) Question Points 1 16 2 6 3 14 4 6 5 12 6 14 7 20 Total: 88 Score Cover Page Print out this page, or write the honor pledge below, on a blank piece of paper, and sign. Place your photo ID (Rutgers ID preferred) on this page when photographing. Include this page as the first page of your pdf of the exam. Name: I hereby swear/affirm that I have neither given nor received any help on this exam, of any kind, to/from any source. Signature: Math 244 Midterm 1 Page 3 of 7 1. [16 points] Solve the given IVP. pcos tq ers g t u r, R dy ` psin tq y “ 1; ypπ{4q “ dt a n l o M d i v a 21 D © 20 ty i s r e v Un?i 2. Math 244 Midterm 1 Page 4 of 7 ty i s r e v Uni 2. [6 points] Is there a second-order DE with general solution y “ c1 ` c2 e4t ? If so, find one; if not, explain why not. ers g t u r, R © v a D 1 202 a n l o id M ty i s r e v Uni 3. [14 points] Find the steady-state solution of the given equation. Then rewrite your solution in the form uptq “ R cospω0 t ´ δq. y 2 ` 2y 1 ` y “ 17 cosp4tq M d i v a D 1 2 0 ©2 er g t u R , r a n ol s Math 244 Midterm 1 Page 5 of 7 ty i s r e v Uni 4. [6 points] Rewrite uptq “ sinp10tq ´ sinp4tq as a product of two trig functions. ers g t u r, R © v a D 1 202 a n l o id M ers g t u R , r a n l 5. [12 points] Solve y ` 4y ` 29y “ 0. o M d i v a D 1 2 © 20 2 1 ty i s r e v Uni Math 244 Midterm 1 Page 6 of 7 ty i s r e v Uni 6. [14 points] A banana weighing 96 pounds is attached to a scale used to weigh preposterous produce. This causes the scale to lower by six inches. King Kong grabs the banana, pulling it downward by another six inches, and then lets go, causing the scale to oscillate in a soothing manner. An external force of F ptq “ 12 cosp8tq pounds is applied to the scale for no particular reason. ers g t u r, R a n l o a) Set up – but do not d solve M– an initial value problem governing the displacement, u, of the i v a banana as a function of time. 21 D 0 2 © b) choose the graph which would most closely resemble the graph of the solution of this IVP. B. A. D. C. E. F. Math 244 Midterm 1 Page 7 of 7 ty i s r e v Uni 7. [20 points] Since a snowball is three-dimensional while its surface area is only 2-D, the rate at which one melts is proportional to the 2{3 power of its volume. This particular snowball currently has a volume of 8 ounces, and is melting at a rate of one ounce per hour. (It’s in the shade, not especially melty.) Find the volume of the snowball as a function of time, and then determine how long it will take to melt entirely. (You must set up and solve a differential equation.) ers g t u r, R © v a D 1 202 a n l o id M
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