18 02 Problem Set 3 Due Thursday March 5 at 1 00 PM in 2 106 Reading The material for this problem set is covered in section SD of the 18 02 Notes Exercises and Solutions and sections 19 4 19 5 19 6 19 7 of Simmons Calculus with Analytic Geometry 2nd Edition Part A 10 points Exercises from the Supplementary Notes and Exercises Please turn in the 10 underlined problems for 1 point each the others are for more practice 2C 2C 1 2C 2 2C 3 2C 4 2D 2D 1 2D 2 2D 3 2D 4 2D 6 2D 7 2E 2E 1 2E 2 2E 4 2F 2F 1 2F 2 2F 5 2H 2H 1 2H 3 2H 4 2H 5 Part B 25 points 1 8 points lecture 10 Consider a triangle in the plane with angles Assume that that the radius of its incircle the circle tangent to all three sides is 1 a By decomposing the triangle into six right triangles having the incenter as a common vertex express the area A of the triangle in terms of and The use your result to show that A can be expressed as a function of the two variables and by the formula A cot cot tan 2 2 2 b What is the set of possible values for and Find all the critical points of the function A in this region c By computing the values of A at the critical points and its behavior on the boundary of the region where it is defined find the maximum and the minimum of A justify your answer Describe the shapes of the triangles corresponding to these two situations d Use the second derivative test to confirm the nature of the critical points you found in b 2 3 points lecture 11 The electrical resistance of a piece of wire is given by the formula R L S where the resistivity is a constant depending only on the material L is the length and S is the cross section area a Show that dR R 1 1 1 dL dS L S b The electrical resistance of a copper wire of length 100 m and cross section 1 mm2 is equal to 1 5 ohm Using the differential approximate the resistance when the length becomes 105 m and the cross section becomes 1 1 mm2 Compare your approximation with the actual value 3 5 points lecture 11 Let x uv y 12 v 2 u2 and f f x y a Use the chain rule to derive the change of variables formula in matrix form fu f A x fv fy b Invert the matrix A to obtain a formula expressing fx and fy in terms of fu and fv c If f u v ln u2 v 2 express fx and fy interms of u and v 4 5 points lecture 12 Consider the function f x y x x 1 x 2 y 1 x y a Find the maximum and minimum values of the direction derivative u varies df ds u at 1 3 2 as b Say in which direction u the maximum and minimum occur c Find the direction s u for which df ds u 0 5 5 points lecture 12 a Find the direction from 4 2 3 in which g x y z x2 y 2 6z decreases fastest b Follow the line in the direction you found in part a to estimate using linear approximation the location of the point closest to 4 2 3 at which g 0 Do not use a calculator Express your answer using fractions Next use a calculator to evaluate g at your point The value should be reasonably small 2
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