study guide exam 2 mth

study guide exam 2 mth - Study Guide for the 2nd Exam...

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Unformatted text preview: Study Guide for the 2nd Exam MTH241 Exam Date: 28th October 2011. In class Exam Covers Sections 14.1 − 14.8 You must know the followings 1. Domain of the function. 2. Limit and Continuity Prove or disprove the existance of limit. Do all the problems from the handout which I distributed in class 3. Partial Derivatives. Find fx , fy , fxx , fyy , fxy etc 4. Tangent plances and linearizations. Equation of the tangent plane of the curve z = f (x, y ) at (a, b) is given by fx (a, b)(x − a) + fy (a, b)(y − b) − (z − f (a, b)) = 0 Linearization of f (x, y ) = 0 around (a, b) is given by fx (a, b)(x − a) + fy (a, b)(y − b) − (z − f (a, b)) = 0. 5. Chain Rule. Must know ∂ f dx ∂ f dy ∂ x dt + ∂ y dt z = f (x, y ), x = g (s, t), y = h(s, t) ⇒ dz = ∂ f ∂ x + ∂ f ∂ y ds ∂x ∂s ∂y ∂s ⇒ dz = ∂ f ∂ x + ∂ f ∂ y dt ∂x ∂t ∂y ∂t dy Implicit Differentiation: F (x, y ) = 0, y = f (x) ⇒ dx = − Fx Fy z = f (x, y ), x = g (t), y = h(t) ⇒ dz dt = 6. Directional Derivatives, Gradiant, Maximum rate of change. Finding the equations of Tangent plane and Normal lines. 7. Maximum and Minimum: Suppose the second partial derivative of f are continuous on a disk with center (a.b) and suppose that fx (a, b) = 0 and fy (a, b) = 0 i.e (a, b) are the co-ordiantes of the critical point. Let D = D(a, b) = fxx (a, b)fyy (a, b) − [fxy (a, b)]2 a. If D > 0 and fxx (a, b) > 0 ⇒ f (a, b) is a Local Minimum Point b. If D > 0 and fxx (a, b) < 0 ⇒ f (a, b) is a Local Maximum Point c. If D < 0 then f (a, b) is not local max or min point. It is called Saddle Point If D = 0 then no information about the point (a, b) 8. a. b. c. Absolute maxima and minima in a closed bounded set. Find the values of f at the critical points of f in D. Find the extreme values of f on the boundary of D. The largest of the values from steps 1 and 2 is the absolute maximum value etc... 9. Lagrange Multipliers. Sample problems These problems are not the exact problems but by solving these you can understand the hardness of the original problems. Do the homework problems and the problems which I discussed in class. 1. Find the limit 4 it exists or show that 2 limit doesn’t exist. if 4 the 2 y + a. lim(x,y)→(0,0) x2 −y2 b. lim(x,y)→(0,0) x2x2sin 2y x+ +y 2. Determine the set of points at which the following function is continuous. f (x, y, z ) = √ y x2 −y 2 +z 2 3. Find the indicated partial derivative of the function ω = . x 2y +z Find ∂3ω ∂ x∂ y ∂ z and ∂3ω ∂ x2 ∂ y . 4. Find the equation of the tangent plane at the point (2, −2, 12) to the surface z = 3(x − 1)2 + 2(y + 3)2 + z . 5. Find the linearization L(x, y ) of the function f (x, y ) = sin(2x + 3y ) at the point (−3, 2). 6. Use chain rule to find ∂R ∂x and ∂R ∂y where R = ln(u2 + v 2 + w2 and u = x + 2y, v = 2x − y, w = 2xy . √ 7. Find the directional derivative of the function f (x, y ) = 1 + 2x y at (3, 4) along the vector v = < 4, − 3 > . 8. Find the equation of the plane and the normal line to the surface y = x2 − z 2 at the point (4, 7, 3). 9. Find the local maximum and minimum values and the saddle point (if any) of the function f (x, y ) = 2x3 + xy 2 + 5x2 + y 2 . 10. Use Lagrange multipliers to find the maximum and minimum values of the function f (x, y, z ) = 2x + 6y + 10z subject to the given condition x2 + y 2 + z 2 = 35. Rules of Exam 1. Exam is closed book and closed notes. 2. You can carry a 4￿￿ × 6￿￿ Formula card which can be used to write only important formulae and results. You’re not allowed to write any problem there. After the exam, you turn it back along with your exam packet. 3. You’re not allowed to use any kind of Calculator. 4. You’re not allowed to go outside the exam hall without the permission of the proctor. ...
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