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# HW 5

Course Number: MATH E-21a, Fall 2009

College/University: Harvard

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Math E-21a Fall 2009 HW #5 problems Problems to turn in on Thurs, Oct 8: Section 11.2: In problems 10 and 11, find the limit, if it exists, or show that the limit does not exist. 6 x3 y xy 10. lim 11. lim 4 4 ( x , y ) (0,0) 2 x y ( x , y ) (0,0) x2 y 2 Section 11.3: 37. Find the indicated partial derivative: f ( x, y, z ) x ; yz f z (3, 2,1) 68. Show that the Cobb-Douglas production function P ( L, K )...

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E-21a Math Fall 2009 HW #5 problems Problems to turn in on Thurs, Oct 8: Section 11.2: In problems 10 and 11, find the limit, if it exists, or show that the limit does not exist. 6 x3 y xy 10. lim 11. lim 4 4 ( x , y ) (0,0) 2 x y ( x , y ) (0,0) x2 y 2 Section 11.3: 37. Find the indicated partial derivative: f ( x, y, z ) x ; yz f z (3, 2,1) 68. Show that the Cobb-Douglas production function P ( L, K ) bL K satisfies the equation L P K P ( ) P L K Section 11.4: 4. Find an equation of the tangent plane to the given surface at the specified point: z y ln x , (1, 4, 0) 28. Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0.1 cm thick and the metal in the sides is 0.05 cm thick. 31. If R is the total resistance of three resistors, connected in parallel, with resistances R1, R2, and R3, then 11 1 1 R R1 R2 R3 If the resistances are measured in ohms as R1 = 25, R2 = 40, and R3 = 50, with a possible error of 0.5% in each case, estimate the maximum error in the calculated value of R. 38. Suppose you need to know an equation of the tangent plane to a surface S at the point P(2,1,3). You dont have an equation for S but you know that the curves r1 (t ) 2 3t ,1 t 2 ,3 4t t 2 and r2 (u ) 1 u 2 , 2u 3 1, 2u 1 both lie on S. Find an equation of the tangent plane at P. Section 11.5: 2. Use the Chain Rule to find dz : z x ln( x 2 y ), x sin t , y cos t dt 32. The radius of a right circular cone is increasing at a rate of 1.8 in/sec while its height is decreasing at a rate of 2.5 in/sec. At what rate is the volume of the cone changing when the radius is 120 inches and the height is 140 inches? 34. The voltage V in a simple electrical circuit is slowly decreasing as the battery wears out. The resistance R is slowly increasing as the resistor heats up.. Use Ohms Law, V = IR, to find how the current I is changing at the moment when R = 400, I = 0.08A, dV 0.01 V/sec , and dR 0.03 /sec . dt dt Section 11.6: In problems 8 and 10, (a) find the gradient of f ; (b) Evaluate the gradient at the given point P ; (c) Find the rate of change of f at P in the direction of the vector u. 236 43 8. f ( x, y ) y ln x , P (1, 3) , u 5 , 5 10. f ( x, y, z ) x yz , P(1,3,1), u 7 , 7 , 7 In problems 12 and 14, find the directional derivative of the function at the given point in the direction of the vector v. x 12. f ( x, y ) ln( x 2 y 2 ), (2,1), v 1, 2 14. f ( x, y, z ) , (4,1,1) , v 1, 2,3 yz 1 30. Suppose that you are climbing a hill whose shape is given by the equation z 1000 0.005 x 2 0.01y 2 and you are standing at point a with coordinates (60, 40, 966). a) If you walk due south, will you start to ascend or descend? At what rate? b) If you walk northwest, will you start to ascend or descend? At what rate? c) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin? Additional practice problems: Section 11.2: 13. Find the limit, if it exists, or show that the limit does not exist: 2x2 y ( x , y ) (0,0) x 4 y 2 lim Section 11.4: 1. Find an equation of the tangent plane to the given surface at the specified point: z 4 x 2 y 2 2 y , (1, 2, 4) . 13. Find the linear approximation of the function f ( x, y ) 20 x 2 7 y 2 at (2, 1) and use it to approximate f (1.95,1.08) . 15. Find the linear approximation of the function f ( x, y, z ) x 2 y 2 z 2 at (3, 2, 6) and use it to approximate the number (3.02) 2 (1.97) 2 (5.99) 2 . 25. The length and width of a rectangle are measured as 30cm and 24cm, respectively, with an error in measurement of at most 0.1cm in each. Use differentials to estimate the maximum error in the calculated area of the rectangle. Section 11.5: dz : z sin x cos y, x t , y t dt 29. The temperature at a point (x, y) is T ( x, y ) , measured in degrees Celsius. A bug crawls so that its position 1. Use the Chain Rule to find after t seconds is given by x 1 t , y 2 1 t , where x and y are measured in centimeters. The 3 temperature function satisfies Tx (2,3) 4 and Ty (2,3) 3 . How fast is the temperature rising on the bugs path after 3 seconds? 33. The length l, width w, and height h of a box change with time. At a certain instant the dimensions are l = 1 meter, and w = h = 2 meters, and l and w are increasing at a rate of 2 meters/sec while h is decreasing at a rate of 3 meters/sec. At that instant find the rates at which the following quantities are changing: (a) The volume (b) The surface area (c) The length of the diagonal. Section 11.6: In problem 7, (a) find the gradient of f ; (b) Evaluate the gradient at the given point P ; (c) Find the rate of change of f at P in the direction of the vector u. 5 7. f ( x, y ) 5 xy 2 4 x 3 y , P (1, 2) , u 13 , 12 13 13. Find the directional derivative of the function at the given point in the direction of the vector v: g ( s, t ) s 2 et , (2, 0) , v i j 29. Suppose that over a certain region of space the electrical potential V is given by V ( x, y, z ) 5 x 2 3 xy xyz a. Find the rate of change of the potential at P(3, 4,5) in the direction of the vector v i j k . b. In which direction does V change most rapidly at P? c. What is the maximum rate of change at P? 2

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