1. Consider f {1, Jr, a} = 13* + I: subject to the constraint a:2 + y] + 12 = 32. Find the absolute minimum and absolute maximum value of f . 2. Minimize {trait} = 2m" + Zrtrh suhiettto the constraint r'zlt = 54.wherer :- lJ and it 2:- El. 3. A company manufactures two trees of bicycles. a racing bicycle and a mountain bicycle. The total revenue from x; units of racing bicycles and 12 units of mountain bicycles is R = -dx12 - 10x3: - 211 x: +3211 4-3:; where cyand x; are in thousandsof units. Find :1 and x: to maximize the revenue. 4. One way to measure species diversity is to use the Shannon diversity index ii. If a habitat consists of three species. A. B. and E. then its Shannon diversity index is = xiru: - yin): - alrta where x is the percent of species A in the habitat, y is the percent of species B in the habitat. and z is the percent of species it in the habitat. Use the fact that x + y + a = 1 to show that the maximum value of H occurs when x = 3r = a = i. What is the maximum value of H? 5. Find the limit [if It exists! and discuss the continuity of the function. llm F, Uri{L13 I +.r E. Find all first partial derivatives of shy} = Jr: 3r: 21. 1 The possible error involved in measuring each dimension of a right circular cone is 1:: in. The radius is 2 inches and the height is 5 inches. Approximate the propagated error and the relative error in the calculated volume of the cone. 3. Find the partial derivative of a impiicity: 1'22 - yarn: = i]. 9. Find the directional derivative of the function fifx, Jr) = is: - x: at [1.41: in the direction {2.1}. m. Find the gradient of the function 2 = x1)! and the maximum value of the directional derivative at the point {2.1}. 11. Find an equation of the tangent plane to the surface 93:2 + y: + 42" = 25 at the point [all '3: 21- 12. Find all relative extrema and saddle points off[x,3r] = 1:2 + 31y + 3r: 5x