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MATH 110, Assignment 4 Make sure you justify all your work and include complete arguments and explanations. Please include your name and student number on the top of the first page of your assignment. Problem 1. Differentiate. (a) y = ex 1+x (b) y = eu (cos u + u) (c) f (x) = ex cos x Problem 2. The functions f and g are given as followings: Find the following values: (a) g 0 (1) (b) (f g)0 (3) (c) (f −1 )0 (3) Problem 3. How many tangent lines to the curve y = x x+1 pass through the point (1, 2). Problem 4. On what interval is the function f (x) = x3 ex increasing? Problem 5. Using product rule, prove that d (f (x))3 = 3(f (x))2 f 0 (x). dx (Hint: Show that (f gh)0 = f 0 gh + f g 0 h + f gh0 first and take f = g = h.) 1 Problem 6. A ladder 10m long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x changes with respect to θ when θ = π4 ? 2