Math 140 Final Exam Fall 2008 1 20 Points p x cid 0 2 cid 0 1 x cid 0 3 a lim x 3 x cid 0 2 x2 cid 0 4 b lim x 2 2 20 Points a State the de cid 12 nition of the derivative a function f x at x b Using the de cid 12 nition above cid 12 nd the derivative of f x x2 4 3 20 Points a Find the equation of the tangent line of f x x3 cid 0 2x 3 at the point x 1 b Find the derivative of the following functions 1 3 x 1 45 i f x x2 ii y x2 6 tan x 4 20 Points Use implicit di cid 11 erentiation to cid 12 nd the slope of tangent line to the graph of the equation xy 6 at the point cid 0 2 cid 0 3 5 10 Points An open box has a square base and a cid 12 xed voulme of 2 cubic feet We want to cid 12 nd the dimension of this box with a minimum surface area Draw a diagram write down the primary equation secondary equation and its feasible domain to solve this problem 6 20 Points The critical numbers of f x 3x5 cid 0 5x3 are 0 1 and cid 0 1 Using the second derivative test to test relative extremas of f 1 and f cid 0 1 7 30 Points a Use the cid 12 rst Fundamental Theorem of Calculus to evaluate dx x2 cid 0 1 x 3 2 b Evaluate the derivative of F x x t cid 3 sin t2 dt 0 8 30 Points Use substitution to evaluate a x3 sec2 x4 dx b x2 sinh x3 dx 9 40 Points Evaluate the following integrals 3x2 4x 7 x3 2x2 7x 2 dx a c 1 dx x ln x 10 40 Points b d ln x 3 x dx 3x2 1 x6 dx dy dx p 1 y x b Solve the following initial value problem cid 0 x y dy dx y 3 cid 0 4 a Solve the following di cid 11 erential equation by separation of variables