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F97_First_Midterm-G.Bergman - at the point(1,l(b(8 points...

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I0/04/2o0L THU 16:43 FAX 64A4AA0 George M. Bcrgman 120 Latimer MOFFITT LIBRARY Fall 1997, Math 53M First Midterm Exam B oor 30 Sept., 1997 8: 10-9:30 AM 1. (25 points) Irt C be the plane curve defined by the parametric equations x = sin r, y = tan t (te(-n/Z, n/2)). (a) (7 points) SkeEh the Lurve C, showing any maxinra, minima, intercepts or asymptotcs it may have. (b) (7+ll points) Obtain formulas for dy/dx and, d.zy/dx2 on this curve C as functions of r. 2. (15 points) Compute the arc length of the space curve r(t) - (t2tZ- ln t)i + (2 sin t)i - Q cos f)k from r(1) to r(3). 3. (20 points) Suppose F is a differentiable function of two variables, whose domain includes (1,1), and we write f(1,l) = a, F*(l,t) = b, Fr(l,1) = c. (a) (12 points) Express
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Unformatted text preview: at the point (1,l). (b) (8 points) Find the directional derivative of (r+y)F(x,y) at the point (1,1) in the direction of the vector 4.4>-4, (25 points) Suppose F is a differentiable function of three variables, and we define a function G of two variables by G(x,y) = F(x*y, xy, x). (a) (13 points) Express the partial derivatives of G(a y) with respect to .r and y in terms of F and its partial derivatives FI, F2, F3. (b) (12 points) T-et (a,b) be a point of the plane, and C the level curve to the function G(x,y) (defined above) which passes through (a,b). Find the equation of the tangent line to C at (a,b) in terrrs of the value andpartial derivatives of .F. 5. (15 points) Suppose u and u are differentiable functions of two variables. Derive the formula Y (u/u) - (uY u - uY u)/uz....
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