MATH 251 Midterm: MATH 251 PSU s251Exam 1(sp04)

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turquoisegnu128

15 Feb 2019

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1: both f (t, y) = needed is that y0 3t2 y 3t2 and. 1 (y 3t2)2 have to be continuous. So the equation is an exact equation: exy2. + x4 + 2y = 5: a) the roots of the characteristic equation r2 4r 5 = 0 are r = 1, 5. Therefore, the functions y1 = e t and y2 = e5t form a pair of fundamental solutions for this second order linear di erential equation: general solution is y(t) = c1y1 + c2y2 = c1e t + c2e5t, y(t) = 7. 6e5t: a) (solved as a separable equation. ) y = ln(sin t + t + e3) (explicit solution). ey = sin t + t + e3 (implicit solution), or, (solved as a rst order linear equation. ) y(t) = By abel"s theorem, w (y1, y2) = ce r p(t) dt = cer 4.

Related Questions

1) Show that the follwing limit does not exist

lim_(x,y)-(0,0)(xy+x²)/(x^²+y²)

2) Fin all the second orderpartial derivatives of

f(x,y,z)=x²tan^-1(y+z)

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z=x²-4y² atthe piont (1,2,-15).

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ceruleancockroach23

1. (45 marks) Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answer. Unless otherwise stated, it is not necessary to simplify your answers in this question. a) Find the equation of the tangent plane to the graph of the function z = f(x,y) = sell at (0,1). Answer b) Evaluate (T, 0) if f(x,y) = sin(ry). Answer c) Graph of a function z = f(x,y) is given by: Which of the following level curves correspondes to this function. Answer d) If f(x,y) = e(3x – 2y), find lim - Math)-f(x,y) e) Find % if f(x, y) = x2 - y3 and 2 = r sin(t), y = r cos(t). Answer f) Evaluate the directional derivative of f(x, y) = !at (1, -1) in the direction of u =(2, 1). Answer 8) Does Simono converge? h) Sketch the direction field of the differential equation = 1-2y. i) Consider the differential equation ay = 15 - 31, y(0) = 0, and use Euler's method with step size 0.1 to estimate y(0.1). Answer j) Knowing that (-3,-3) is a critical point of f(x,y) = 2:3 + y2 – 4.cy -3.0, use the second derivative test to find out whether it is a local minimum, local maximum, or a saddle point. Answer k) Solve the initial value problem, dy dt 642 2y + cos(u) y(1) = . Answer 1) Is ū=(2,2, -1) perpendicular to v = (5,-4, 2)? Answer m) Find the area between the graph of y = 1 + sin(x) and y=1-sin(x) from 0 to 7. Answer n) Determine the lentgh of the curve y = 8/2,0 SIS 3. Evaluate and simplify your answer. Answer o) Express the volume of the solid obtained by rotating the bounded region that lies between curves y = (x-2)2 and y=1,150 $4 about the vertical line 2 = 5. Do not evaluate this integral. Answer

MATH 251 Midterm: MATH 251 PSU s251Exam 1(sp04)

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Single Variable Calculus: Early Transcendentals

CALCULUS:EARLY TRANSCENDENTALS

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