Math 140 006 Fall 2020 1 20 points a lim x 3 x 2 1 x 3 x 2 x2 4 b lim x 2 Final Exam December 16th 2020 2 20 points a State the limit de nition of the derivative a function b Using the de nition above nd the derivative of f x x2 4 c Using the de nition above nd the equation of the tangent line of f x x2 4 at x 1 3 20 points Find the derivative of the following functions do not use the limit de nition of f cid 48 x here cid 16 cid 17 5 a f x x2 3x 1 b y x2 6e2x csc x 4 20 points Use implicit di erentiation to nd the slope of tangent line to the graph of the equation x 2 y 5 at the point 9 1 5 20 points A box with surface area 120 square feet has its length equal twice its width What dimension of this box has a maximum volume Give the Primary equation the secondary equation rewrite volume in term of only one variable and nd the feasible domain for that variable f x 3x5 5x3 6 30 points Given that a Find all the critical numbers of f b Determine on which interval f is increasing on which interval f is decreasing c Find all ppoi s of f d Determine on which interval f is concave up concave down e Use the 2nd Derivative Test to test all its non zero critical numbers i e you can tell except when x 0 whether it gives rise to a relative min or relative max 7 30 points Use substitution to evaluate the following inde nite integrals a 8x3 sec2 x4 dx cid 90 cid 90 x2 cid 16 2x3 cid 17 cid 90 ex 3x2 x3 ex dx dx b c Math 141 Final Exam Continued December 16th 2020 8 20 points Use the second Fundamental Theorem of Calculus to evaluate the deriva tive of F x a F x b F x et2 dt 1 s ds s cid 90 1 cid 90 x2 x 3 9 20 points a Solve the following di erential equation by separation of variables dy dx 1 y x b Solve the following initial value problem dy dx x y y 3 4 Page 2