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Math 231 Final Exam A Fall 2012 page 1 of 13 1 Find the slope of the line tangent to the curve x tan y 2 at the point 1 2 xy 2 A F B G 2 2 2 2 2 2 2 2 2 2 C D E 2 2 2 Find the exact area bounded by the curve the x axis x 1 and x 3 y x 2 1 A F 26 3 B G 4 C 6 D 16 3 20 3 8 3 E 8 Math 231 Final Exam A Fall 2012 page 2 of 13 3 Use L Hospital s Rule to determine the limit A and C are constants C A 0 4 Use logarithmic differentiation to find the derivative of y 1 x 21 x ln eA xC x lim x 2C A Ce F 0 B 1 G C C D E A A 1 2 x 21 x 1 1 x 1 1 x B 1 4 x x 21 x 1ln 1 x 1ln 1ln 2 x x 1 1 1 x C 1 2 x 21 x 1 x D 1 2 x 21 x 1 x E 1ln 2 1 x x dy dx dy dx dy dx dy dx dy dx Math 231 Final Exam A Fall 2012 page 3 of 13 6 Given the graph of f x on 5 2 answer the following questions about the graph of xf 5 If xg f t dt x ln x find 1 f 2 x 0 A 2 B 1 F 1 1 G 0 2 C 2 2 D E 1 Answers to the nearest integer f x x xf simultaneously Where is the graph of increasing and concave down Math 231 Final Exam A Fall 2012 page 4 of 13 4 6 f f 0 2 xf is differentiable everywhere and 7 Suppose a function xf Apply the Mean Value Theorem to the interval 0 6 to determine how large can possibly be A 4 F 24 B 26 G none of these C 2 D 22 E 6 8 Find y if 1 x sin y 3 2 A F 1 3 2 x 12 x 3 B G none of these 3 2 x 1 x 2 1 3 2 x 12 x 2 1 3 2 x 12 x 3 3 2 x 1 x 3 C D E 1 3 2 x 12 x 2 Math 231 Final Exam A Fall 2012 page 5 of 13 9 Find B such that lim 2 x 3 x 2 x 4 7 Bx 10 Consider a function For what value s of x does xg with derivative xg have a local maximum xx 2 2 3 xg x 9 8 A 0 F 8 B 0 2 G 0 8 C 0 2 8 D 2 E 2 8 A 24 B 15 C 7 D 1 E 20 F 1 G 7 11 Given g 1 6 g 1 1 find when 1 x d dx 2 2x xg 1 Math 231 Final Exam A Fall 2012 page 6 of 13 12 The following limit represents the derivative of some function f x at some number a State such an f x and a lim h 0 1 h 8 1 h f x a 13 Consider the graph of xf B a constant Bx 2 B x 2 At what value s of x does a hole removable discontinuity in the graph occur F B 2B G B 2B A B C D B E no holes 2B Math 231 Final Exam A Fall 2012 page 7 of 13 xf that has exactly one discontinuity and 14 Sketch the graph of a function satisfies the following conditions 4 does not exist 2 f 2 f x x f f lim x 0 on 1 2 1 0 on 2 0 on 1 2 1 xf 1 xf 3 xf f x 2 lim 1 x lim 1 x y 1 1 1 x 2 3 x xf 8 a L x 15 Given the function a Find the linear approximation L x to the function at b Use your result from part a to approximate the value of the function at Leave answer in fraction form Do not use your calculator 7 2 3 7 x Math 231 Final Exam A Fall 2012 page 8 of 13 16 A particle moves along a path described by and y changing at the same rate A 1 2 15 4 F 0 2 y 4 x 2 At what point along the curve are x 17 A rectangle with base on the x axis has its upper vertices on the curve Find the maximum area of such a rectangle 12 x y 2 0 2 G 3 1 B C 1 2 15 4 D always at the same rate E never at the same rate A 144 F 32 B 2 G 16 C 8 D 12 E 64 Math 231 Final Exam A Fall 2012 page 9 of 13 18 a Express the limit of a Riemann sum on 2 5 as a definite integral n 5 x 2 i x lim n i 1 b Find the horizontal asymptote for the graph of xf A B C D constant 4 Ax 2 Bx 4 C Dx y 19 A side of a square field is measured to be 10 feet with a possible error of 1 12 ft Use differentials to approximate the maximum error in the calculated area of the field 2 ft F 2 100 ft 20 ft G none of these 2 A 25 3 B C D E 5 3 5 6 2 ft 2 ft 2 121 ft Math 231 Final Exam A Fall 2012 page 10 of 13 20 If Newton s method is used to solve 2 3 x x 2 01 with first approximation 1x 1 what is the second approximation 2x 21 On a certain planet a ball is thrown upward with a speed of 33 feet per second from the edge of a cliff 44 feet above the ground When does it hit the ground use ta 22 3 1 F 4 5 G 8 7 A 2 B 8 3 C 8 3 D 5 E 4 1 s1 F s11 s4 G none of these A B C D 3 s2 s55 E 3 41 4 s Math 231 Final Exam A Fall 2012 page 11 of 13 22 Determine the absolute minimum value of the function Show all work Exact value only No credit for calculator approximations xf e cos x absolute min value 23 Evaluate the definite integral Exact answer only no calculator approximations Show work 24 x dx 3 1 x 3 x Math 231 Final Exam A Fall 2012 page 12 of 13 cos x 2 sin x 2 0 25 Evaluate the definite integral dx Show all work for credit Exact answer 24 Evaluate the integral e x 5 2 sec3 dx x only no calculator approximations Math 231 Final Exam …


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UNC-Chapel Hill MATH 231 - Final Exam

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