Math 140 Final Exam 1 20 Points Evaluate the following limits Name analytically p 6 cid 0 x cid 0 2 2 cid 0 x x2 cid 0 25 x cid 0 5 a lim x 2 b lim x 5 2 10 Points Using the de nition of deriva tive nd the rst derivative of f x x3 F x et2 dt 3 30 Points 10 30 Points Use substitution to evaluate 8 10 Points Suppose we need tosketch the Analyze its domain graph of f x x2 x2 3 intercepts symmetry and asymptote 9 20 Points a Evaluate 2e 2 x cid 0 1 x dx b Evaluate the derivative of 1 x a b x2 sec2 x3 dx x sinh 3x2 dx 11 40 Points Evaluate the following inte ii y x x cid 0 2 sec x grals a b c d 3x2 4x 7 x3 2x2 7x 2 dx ln x 2 dx x 7x 1 x4 dx e3x 1 cid 0 e6x dx 12 30 Points a Solve the following di erential equation x2 2 3y2 dy dx cid 0 x y y 3 4 b Solve dy dx a Find the equation of the tangent line of f x x3 cid 0 x 2 at the point x 1 b Find the derivative of the following func tions i f x x3 1 2 x 1 10 4 20 Points Find the derivative of implicit function x2 y3 3x2y 5 10 Points A spherical snowball is melt ing at the rate of 2 cid 25 cubic inches per minute At what rate is the radius decreasing when the radius is 3 inches Hint volume of sphere 4 3 cid 25 r3 6 10 Points The sum of two positive num bers is 14 what is the smallest of their sum of squares give feasible domain primary and secondary equations 7 10 Points Fill in all the blanks The function is f x 2x cid 0 x2 and we di vide the interval 0 2 into 5 sub intervals of equal width Denoting by R the sum of all the area of rectangles evaluated using the right end point of each sub intervals then R x f f f f f