MAT 21A, practice problems for Midterm 2 1. Find the derivatives of the following functions: x a) f (x) = sin ln x b) f (x) = xecos x c) f (x) = eln(2+x)ln(1+x) . d)* f (x) = (sin x)cos x e) q x1 . x+1 2. Find the derivative of y(x) using implicit differentiation, if a) 3x2 + 2y 2 = 10 b) cos(x) + cos(y) = 15 c) x y y x =1 3. Find the equation of the tangent line to the graph of f (x) = x4 ex at a point (1, e1 ). 4. Find the maximal and minimal values of a given function on a given interval: a) f (x) = x + sin x, [0, 4] b) f (x) = x3 27x + 1, [5, 5] c) ln x , x [1, 2]. 5. (ex. 11 on p.270) You are designing a rectangular poster to contain 50 square inches of printing with a 4-in. margin at the top and bottom and 2-in. margin at each side. What overall dimensions will minimize the amount of paper used? 6. A ship is moving east with speed 1, another ship is moving north with speed 2. At some moment, their coordinates were (1, 0) and (0, 1), respectively. What will be the minimal distance between these ships? 7. (ex. 58 on p. 275) The 800-room Mega Motel chain is filled to capacity when the room charge is $50 per night. For each $10 increase in the room charge, 40 fewer rooms are filled each night. What charge per room will result in the maximum revenue per night? 1