Calculus III

Math 285 Midterm 2 Practice Full Name: Net ID: UIN: Circle your class section: B1 C1 F1 • You must not communicate with other students during this test. • No written materials of any kind allowed. • No phones, calculators, iPods or electronic devices of any kind allowed for ANY reason, including checking the time (you may use a simple wristwatch). • Do not turn this page until instructed to do so. • There are many di ↵ erent versions of this exam. • Violations of academic integrity (in other words, cheating) will be taken extremely seriously. Fill in the following answers on the scantron form now: 93. A 94. A 95. D 96. C Multiple Choice Questions. Mark answers to questions 1 to 7 on your scantron form 1. (3 points) The di ↵ erential equation u 00 +6 u 0 +9 u =0 corresponds to an oscillator that is (A) undamped (B) overdamped (C) underdamped (D) F critically damped (E) forced Solution. The characteristic equation is r 2 +6 r + 9 = 0 which has the double root r = 3. Therefore the oscillator is critically damped. 2. (3 points) Which is the solution of the given boundary value problem? y 00 +4 y =0 y (0) = 0 y ( ⇡ / 4) = 1 (A) y ( x )=0 (B) y ( x ) = sin ( x ) (C) F y ( x ) = sin (2 x ) (D) y ( x ) = cos (2 x ) (E) y ( x ) = cos ( x ) Solution. The correct answer is the only one that satisﬁes the equation and the boundary conditions. 3. (3 points) For which value of ! is the following oscillator in resonance? u 00 + 16 u = sin ( ! t ) (A) for no value of ! (B) F ! =4 (C) ! =0 (D) ! = 16 (E) ! =2 Solution. The complementary solutions is c 1 cos (4 t )+ c 2 sin (4 t ) which shows that the natural frequency is 4. There is resonance if the forcing frequency, ! , is also 4. 4. (0 points) Which of the following ingredients is not used to create Polyjuice Potion? (A) ﬂuxweed (B) F powdered unicorn horn (C) leeches (D) lacewing ﬂies (E) knotgrass Solution. Powdered Bicorn horn is an ingredient of Polyjuice Potion, not powdered unicorn horn. Therefore powdered unicorn horn is the correct answer. 5. (3 points) The interval of existence of the solution of the initial value problem ( t + 3)( t + 1)( t - 4) y 0000 + 5( t + 4) y 000 - 4 y = 10 sin t, y ( - 2) = 2 ,y 0 ( - 2) = - 1 ,y 00 ( - 2) = 3 ,y 000 ( - 2) = - 5 is (A) ( - 1 , 4) (B) ( -1 , - 3) (C) F ( - 3 , - 1) (D) ( -1 , 1 ) (E) (4 , 1 ) Solution. The solution exists in one of the intervals of continuity of the coe ﬃ cient functions: ( -1 , - 3), ( - 3 , - 1), ( - 1 , 4), (4 , 1 ). Choose the one containing t = - 2. 6. (3 points) Identify the correct form of a particular solution for the following di ↵ erential equation. y 00 +2 y 0 = t + 2e t (A) At + B + C e t (B) t +e t (C) F At 2 + Bt + C e t (D) At + B e t (E) A + B e t + C e - 2 t Solution. The characteristic equation is r 2 +2 r = 0 and its roots are r = 0 and r = - 2. A complementary solution is therefore y c ( t )= c 1 + c 2 e - 2 t . By looking at the right hand side of the original equation we deduce that an appropriate guess for a particular solution is At + B + C e t . Because the second term duplicates part of y c , we need to multiply the polynomial part of the solution by t .

MATH | 285 | Calculus III | study guides | Math 285 Midterm 2 Practice Fu

Midterm_1_Practice_Solutions.pdf-Math 285 Midterm 1 Practice Full

Math 285 Final - Practice Full Name: Net ID: UIN: Circle your class section: • You must not communicate with other students during this test. • No written materials of any kind allowed. • No phones, calculators, iPods or electronic devices of any kind allowed for ANY reason, including checking the time (you may use a simple wristwatch). • Do not turn this page until instructed to do so. • There are many di ↵ erent versions of this exam. • Violations of academic integrity (in other words, cheating) will be taken extremely seriously. Fill in the following answers on the scantron form now: 93. A 94. A 95. D 96. C Multiple Choice Questions. Mark answers to questions 1 to 12 on your scantron form 1. (2 points) What is the order of the following di ↵ erential equation for y ( x )? y 4 + sin ( y ) y 0 +( y 00 ) 2 + x 5 =0 (A) Fourth order (B) None of these (C) Second order (D) First order (E) Third order 2. (5 points) The solution of the initial value problem 2 yy 0 =3 e 3 x ,y (0) = - 4 is (A) None of these choices is correct. (B) y = - p e 3 x + 15 (C) y = p e 3 x - 17 (D) y = p e 3 x + 15 (E) y = ± p e 3 x + 15 3. (3 points) Consider the solution curves in the following ﬁgure: 0.5 1.0 1.5 2.0 2.5 3.0 - 2 2 4 6 8 10 Which model equation corresponds to the above ﬁgure? (A) dp dt =3 p (2 - p )(8 - p ) (B) dp dt = - 3 p (2 - p )(8 - p ) (C) dp dt = - 3 p (2 - p )(9 - p ) (D) dp dt =3 p (1 - p )(5 - p )(9 - p ) (E) dp dt =3 p 2 (8 - p ) 4. (5 points) Consider the di ↵ erential equations I: y 00 + xy 0 + x 2 y =4 e x II: xyy 0 y 00 =1 III: yy 0 y 00 =3 IV: yy 00 =3 y 0 V: y 00 + x 2 y 0 = sin x VI: yy 0 + sin( y 00 ) = cos x Which of these, if any, is reducible to a pair of ﬁrst order di ↵ erential equations by a substitution? (A) Each of I and II (B) Each of III, IV, and V (C) Just V (D) None of them (E) Each of IV and VI 5. (2 points) Consider the following di ↵ erential equation for y ( x ): x 2 y 00 + y 0 = x 4 Which of these statements is true? (A) The equation is linear and homogeneous (B) None of these (C) The equation is linear and non-homogeneous (D) The equation is non-linear 6. (5 points) Consider the following oscillator equation for x ( t ): x 00 +4 x = 2 cos ( ! t ) First determine the value of ! so that the oscillator is in resonance and then determine which of the following is the correct form for the particular solution: (A) None of these (B) 1 3 cos (3 t ) (C) 1 2 t sin (2 t ) (D) 1 4 sin (4 t ) (E) 1 4 t sin (4 t ) 7. (5 points) Which choice below is the correct form of a particular solution of the di ↵ erential equation y 000 - y 00 +4 y 0 - 4 y =3 e t + 5 sin 4 t ? (A) Y = Ate t + B cos 4 t + C sin 4 t (B) Y = Ae t + B cos 4 t + C sin 4 t (C) Y = Ae t + Bt cos 4 t + Ct sin 4 t (D) None of these is correct. (E) Y = Ate t + Bt cos 4 t + Ct sin 4 t 8. (3 points) The amplitude/phase shift form of the function f ( x ) = cos ⇡ x - sin ⇡ x is (A) p 2 cos( ⇡ x - ⇡ 4 ) (B) 2 cos( ⇡ x + ⇡ 3 ) (C) 2 cos( ⇡ x + ⇡ 2 ) (D) p 2 cos( ⇡ x + ⇡ 4 ) (E) 2 cos( ⇡ x + ⇡ 6 )

MATH | 285 | Calculus III | study guides | Math 285 Final - Practice Full

Math 285 Spring 2017 Midterm 1: Results Name: Paez, D. UIN: 674727762 Exam code: CDCEEC NetID: dpaez2 Summary of answers: Question Library Question Correct Answer Your Answer Points 1 4/1 D D 7 2 2/2 B B 5 3 5/2 B B 8 4 1/2 C C 5 5 6/1 C B 0 6 3/1 D D 5 7 7/3 * 0 8 8/3 * 0 Total 30 Multiple Choice Questions. Mark answers to questions 1 to 6 on your scantron form 1. (7 points.) Determine the correct integrating factor for the given ODE for y ( x ) xy 0 +3 y = - x 2 (A) None of these (B) μ ( x )= x 2 (C) μ ( x )=e x 2 (D) μ ( x )= x 3 (E) μ ( x )=e 3 x Correct answer: D. Your answer: D. 7 out of 7 points received Solution. After dividing by x (so that the coeﬃcient of y 0 is 1) we can calculate the integrating factor as μ ( x )= e R 3 /x d x 2. (5 points.) The interval of existence of the solution y ( x ) of the initial value problem ( x + 3)( x - 1)( x - 2) y 00 + 5( x + 4) y 0 - 4 x 2 y = 10 sin x ; y (3 / 2) = - 2; y 0 (3 / 2) = 2 is (A) ( - 3 , 1) (B) (1 , 2) (C) (2 , ∞ ) (D) ( -∞ , ∞ ) (E) ( -∞ , - 3) Correct answer: B. Your answer: B. 5 out of 5 points received Solution. The solution exists in one of the intervals of continuity of the coeﬃcient functions: ( -∞ , - 3), ( - 3 , 1), (1 , 2), (2 , ∞ ). Choose the one containing x =3 / 2. 3. (8 points.) Given the following equation for P ( t ) d P d t =( P - 1) P 2 ( P + 3) determine where the equilibrium solutions are and their stability. (A) P = 3, unstable; P = 0, stable; P = - 1, semi-stable. (B) P = 1, unstable; P = 0, semi-stable; P = - 3, stable. (C) P = 1, unstable; P = 0, stable; P = - 3, semi-stable. (D) Not these (E) P = 3, semi-stable; P = 0, unstable; P = - 1, unstable. Correct answer: B. Your answer: B. 8 out of 8 points received Solution. The equilibrium solutions are the values of P that make the right hand side equal to 0, so P =1 , 0 , - 3. Drawing the slope ﬁeld lets us determine their stability. 4. (5 points.) Determine the order of the following ODE for y ( x ). x 3 y 000 - y 2 + x 2 ( y 00 ) 4 = x 3 (A) None of these (B) Second order (C) Third order (D) Fourth order (E) First order Correct answer: C. Your answer: C. 5 out of 5 points received Solution. The highest derivative is the third so this is a third order ODE. 5. (0 points.) The three main characters of the movie Hidden Figures each follow a career as: (A) Secretary, cook, engineer (B) Computer, secretary, Fortran programmer (C) Computer, engineer, Fortran programmer (D) Secretary, computer, driver (E) Cook, engineer, driver Correct answer: C. Your answer: B. 0 out of 0 points received Solution. It’s great movie, watch it.

MATH | 285 | Calculus III | study guides | Math 285 Spring 2017 Midterm 1

Midterm_3_Practice_Solutions.pdf-Math 285 Midterm 3 - Practice

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