2. Partial Differentiation

2. Partial Differentiation - 2. Partial Differentiation 2A....

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2. Partial Differentiation 2A. Functions and Partial Derivatives 2A-1 Sketch five level curves for each of the following functions. Also, for a-dl sketch the portion of the graph of the function lying in the first octant; include in your sketch the traces of the graph in the three coordinate planes, if possible. a) 1 - x - y b) Jw c) x2 + Y2 d) 1 - x2 - y2 e) x2 - y2 2A-2 Calculate the first partial derivatives of each of the following functions: x w = x3y - 3xy2 + 2y2 b) z = - c) sin(3x + 2y) ex2y Y e) z = x ln(2x + y) f) x2z - 2yz3 2A-3 Verify that f,, = fyx for each of the following: x a) xmyn, (m,n positive integers) b' zfy c) cos(x2 + y) f (x)g(y), for any differentiable f and g 2A-4 By using fxy = fy,, tell for what value of the constant a there exists a function f (x, y) for which f, = axy + 3y2, fy = + 6xy, and then using this value, find such a function by inspection. 2A-5 Show the following functions w = f (x,y) satisfy the equation w,, + wyY = 0 (called the two-dimensional Laplace equation): w = eax sin ay (a constant) b) w = ln(x2 + y2) 2B. Tangent Plane; Linear Approximation 2B-1 Give the equation of the tangent plane to each of these surfaces at the point indicated. a) z = xy2, (Ill,1) b) w = y2/x, (1,2,4) 2B-2 a) Find the equation of the tangent plane to the cone z = d w at the point Po : (xo , yo, zo) on the cone. b) Write parametric equations for the ray from the origin passing through Po,and using them, show the ray lies on both the cone and the tangent plane at Po. 2B-3 Using the approximation formula, find the approximate change in the hypotenuse of a right triangle, if the legs, initially of length 3 and 4, are each increased by .010 . 2B-4 The combined resistance R of two wires in parallel, having resistances R1 and R2 respectively, is given by 1 1 - - -+- 1 R R1 R2 If the resistance in the wires are initially 1 and 2 ohms, with a possible error in each of f .l ohm, what is the value of R, and by how much might this be in error? (Use the approximation formula.) 2B-5 Give the linearizations of each of the following functions at the indicated points: a) (x+ y +2)2 at (0,O); at (1,2) ex cosy at (0,O); at (017r/2)
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2 E. 18.02 EXERCISES a) (x+ y +2)2 at (0,O); at (1,2) b) excosy at (0,O); at (0, n/2) 2B-6 To determine the volume of a cylinder of radius around 2 and height around 3, about how accurately should the radius and height be measured for the error in the calculated volume not to exceed .1 ? 2B-7 a) If x and y are known to within .01, with what accuracy can the polar coordinates r and 8 be calculated? Assume x = 3, y = 4. b) At this point, are r and 8 more sensitive to small changes in x or in y? Draw a picture showing x, y, r, 8 and confirm your results by using geometric intuition. 2B-8* Two sides of a triangle are a and b, and 8 is the included angle. The third side is c. a) Give the approximation for Ac in terms of a, b, c, 8, and Aa, Ab, Ad. b) If a = 1, b = 2, 8 = n/3, is c more sensitive to small changes in a or b?
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2. Partial Differentiation - 2. Partial Differentiation 2A....

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