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# Note for MATH 115 at KU

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Date Created: 02/06/15

Sample Math 115 Final Exam Problems Fall Semester 2010 The nal is on December 15 Wednesday 430 700PM Contact your in structor for the exam room schedule Note that you have to bring a 2 pencil to the exam Calculators are allowed for all parts of the exam but some problems may ask for an exact value The problems below all multiple choice with exactly one correct answer choice are intended to be reasonably representative of what might appear on the actual exam which will have 30 problems The nal exam will cover 0 O O O 0 Chapter 2 Sections 21 22 23 24 25 26 Chapter 3 Sections 31 32 33 34 35 36 37 Chapter 4 Sections 41 42 43 44 45 Chapter 5 Sections 51 52 54 55 56 Chapter 6 Sections 61 62 63 64 65 Note Six of the rst twelve problems listed below will be on the nal i i 2x The domain of function f 2 7 4 is A 2700 B roof 0 OOWQl U 2300 D foo 72 U 2 00 None of the above Find the limit lim 1A0 m A 0 B 1 C 2 D limit does not exist None of the above Suppose that h 2x 1g and 91 72g 1 3 Find h 1 A 2 B 3 C 4 D 5 None of the above The average cost for manufacturing x units of a commodity is given by the function Cx and the demand equation is known to be p Find the marginal pro t A x we 7 6a B mac 7 6m c we e 0m z W e 0m D f mfx 7 Cz None of the above 10 11 Find the second order derivative of function f It is known that f is continuous in 70000 and f71 74 f0 72 and f2 4 Which of the following statements is True 7 A f must have a zero in 10 B f must have a zero in 2 1 C f must have a zero in 24 D f must have a zero in 02 E None of the above is true Let f Then A f is not de ned at z 0 B f has no limit at z 0 C f is not continuous at z 0 E None of the above D f has no derivative at z 0 The second derivative of function f 2x2 13 is A 12z2z212 B 6z2z212 C 1222110x2 1 D 62z2 18z2 1 3x 7 10 7 2x E None of the above for z lt 3 for z gt 3 Let f A The domain of f is 7oooo B f exists C f is continuous at z 3 D f is differentiable at z 3 E None of the above L i m239 74 z 2 74 x 2 74 x 2gt3 74 A z 24 B C D None of the above NH Cl Find d7y in terms of z and y when x and y are related by the equation xi 7 y 1 ow ow on on A sigy B zigy C xgy D 23y None of the above Find at point 2 2 when x and y are related by the equation y2 7 2 16 E 5 1 5 A B xg C 7 None of the above CT CT E 12 13 14 15 16 17 18 19 Find the differential dy of y 2 1 as x changes from 3 to 31 A 04 B 06 C 041 D 061 None of the above Let f and g Then7 g o is i m21 1 A C x E D x None of the above x2 1 B Suppose that fx2 17 f1 17 and f 1 3 Find F 1 D 6 None of the above The unit price p and the quantity z demanded are related by the demand equation 50 7 p2 1 0 Find the revenue function R x2 1 50 50x x2 1 50 z B C gt9521 gt9521 E None of the above A D Find the marginal revenue for the revenue function found in Problem 15 17z2 z 7100z 952 12 0 25 501 7 x2 x 1 A B D E None of the above Find 7 in terms of z and y when x and y are related by the equation 1 9 1 thwerthmeW E None of the above Find 7 at point 27 2M6 when x and y are related by the equation y2 7 2 16 MS 5 1 A 7 B xg C 7 D 7 E None ofthe above ltgtSltgt ltgt ltgtmltgt Let f The domain of f is A 7002 and 27oo B 70072 and 27oo C 7172 and 27oo D 717 00 None of the above 20 21 22 23 24 25 26 27 28 Let f ln2 7 The domain of f is A OO OO B 0700 C 0070 D 7002 None of the above Evaluate lim32 7 4 17gt A 23 B 5 C 4 D The limit does not exist None of the above Evaluate lim m7gt5 m A 3 B 8 C 0 D The limit does not exist None of the above Evaluate lim 32 m7gt14r m A 71 B 0 C 1 D The limit does not exist None of the above Evaluate lim 32 z7gt1 17 A 71 B 0 C 1 D The limit does not exist None of the above Find the horizontal asymptotes of function f 1 An1 Bu4 0271 D The function has no horizontal asymptotes None of the above Find the vertical asymptotes of function f 1232 A72Bz1Cy0 D The function has no vertical asymptotes None of the above The line tangent to y x2 7 3x through the point 17 2 has equation Ayx73 By22z73z71 Cy7z71 D y 7 2 2x 7 3 m 71 None of the above Find an equation of the tangent line to the graph of y z lnx at the point 170 Ayz1 Byx71 Cyz1lnx D y x 7 1 lnz None of the above 29 Find an equation of the tangent line to the graph of y lnx2 at the point 21n4 2x72ln4 2z727ln4 Cy Ayx27ln4 By D y z 7 2 ln4 None of the above 30 Find an equation of the tangent line to the graph of y ezm g at the point g 1 A y262m 3 B y274 C y2x72 D y 262m73 7 None of the above 31 Find an equation of the tangent line to the graph of y 67 at the point 116 2ltz1gt3 ltBgty773ltz71gt73 cm 2ltz71gt3 6 6 6 6 6 6 A y 1 D y 72ze m2x 7 1 E None of the above 32 The absolute maximum value and the absolute minimum value of the function f if 7 Q on 07 3 are 3 9 A absolute min value 7 5 absolute max value 5 7 23 B absolute min value 0 absolute max value 3 C absolute min value 0 no absolute max value D no absolute min value absolute max value 3 E None of the above 33 Find the absolute maximum value and the absolute minimum value7 if any7 of the function f A absolute min value 0 absolute max value 1 B absolute min value 0 no absolute max value C no absolute min value absolute max value 1 D no absolute min value no absolute max value E None of the above 34 Find the absolute extrema of function ft t6quot 1 A absolute min value 0 absolute max value E B absolute min value 0 no absolute max value C no absolute min value absolute max value 7 e D no absolute min value no absolute max value E None of the above 35 Find the absolute extrema of the function ft 11 on 17 2 l 2 A absolute min value 0 absolute max value L 2 B absolute min value 0 absolute max value C absolute min value 1 absolute max value 2 D absolute min value 0 absolute max value 6 E None of the above 36 Let fx zg 7 x2 z 7 6 Determine the intervals where the function is increasing and where it is decreasing A increasing on 7001 and on 17 00 B increasing on 7001 and decreasing on 17 00 C decreasing on 7001 and increasing on 17 00 D decreasing on 7001 and on 17 00 E None of the above 37 Let the function f be de ned in Problem 36 Find the intervals where f is concave upward and where it is concave downward A concave upward on 7001 and on 17 00 B concave upward on 7001 and downward on 17 00 C concave downward on 7001 and upward on 100 D Concave downward on 70071 and on 17 00 E None of the above 38 39 40 41 42 Let the function f be de ned in Problem 36 Find the in ection points7 if any 1 1 A 9671 17f1 B 9671 5 C 9671 07f0 D No in ection points None of the above Let f 67H Determine the intervals where the function is increasing and where it is decreasing A increasing on 7007 0 and on 07 oo gt B increasing on foo7 0 and decreasing on 07 00 C decreasing on 7000 and increasing on 07 00 000 and on 000 E None of the above D decreasing on 7 Let the function f be de ned in Problem 39 Find the relative extrema of f A relative min value 0 relative max value 1 B no relative min value relative max value 1 C relative min value 0 no relative max value D no relative min value no relative max value E None of the above Let the function f be de ned in Problem 39 Find the intervals where f is concave upward and where it is concave downward A concave upward on 7007 0 and on 07 00 B concave downward on foo7 0 and on 000 1 1 77 and on 1 D concave downward on 7007 7E and on 700 concave upward on 7 E None of the above C concave upward on 700 700 concave downward on 7 Let the function f be de ned in Problem 39 Find the in ection points7 if any 1 1 1 1 A 9571 07f0 B 9571 EJCPED and 9571 EJVWD C 71 i7f 1 D Ly None of the above l 43 Let f z ln x Determine the intervals where the function is increasing and where it is decreasing 1 A increasing on foo7 7 and decreasing on 77 oo 6 1 B decreasing on foo7 7 and increasing on 700 e e i i 1 i 1 C increasing on 07 7 and decreasing on 7700 e e i 1 i i 1 D decreasing on 07 7 and increasing on 77 oo 6 e E None of the above 44 Suppose that f is de ned in Problem 43 Determine the intervals of concavity for the function A concave upward on 07 00 B concave downward on 07 oo 1 1 C concave upward on 07 7 concave downward on 700 e 1 1 concave ownwar on 7 concave upwar on 700 D d d 0 6 d E None of the above 45 Suppose that f is de ned in Problem 43 Find the in ection points7 if any 1 1 A 9671 91 B 9671 17f1 C 9671 6416 D No in ection points None of the above 46 Find the derivative of function y x1 Hint use logarithmic differentiation 2 lnx i zhim A y lt1nxgt2 B 2 9 c y x1 D the derivative does not exist None of the above 47 Find the derivative of function y 10 Hint use logarithmic differentiation A y 10m ln 10 B 110w C 1 10wlne D the derivative does not exist None of the above 48 49 50 51 An open box is to be made from a square sheet of tin measuring 12 inches gtlt 12 inches by cutting out a square of side z inches from each corner of the sheet and folding up the four resulting aps To maximize the volume of the box7 take z A 1 B 2 0 3 D 4 None of the above A rectangular box is to have a square base and a volume of 20ft3 1f the material for the base costs 30 centssquare7 the material for the four sides costs 10 centssquare7 and the material for the top costs 20 centssquare7 determine the dimensions of the box that can be constructed at minimum cost See Fig 1 Azgtltzgtlth1gtlt1gtlt20 Bzgtltzgtlth2gtlt2gtlt5 Czgtltzgtlth25gtlt25gtlt32 D x gtlt z gtlt h 3 gtlt 3 gtlt 222 None of the above Figure 1 Problem 49 Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 108 inches Find the dimensions of the cylindrical package of greatest volume that may be sent through the mail In the answers7 r is the radius and l is the length rgtltl gtlt35 35 36 A l7 37 B l7 36 C rgtlt 7Tgtlt rgtlt 7Tgtlt 7T 38 Drgtltligtlt34 7T None of the above It costs an artist 1000 5x dollars to produce z signed prints of one of her drawings The price at which z prints will sell is 400E dollars per print How many prints should she make in order to maximize her pro t 7 A 1200 B 1400 C 1600 D 1800 E None of the above 52 53 54 55 56 57 58 The differential of function f 1000 is A 1000 B1000dz C 0 D dz E None of the above Use differentials to estimate the change in m when x increases from 2 to 2123 A 0083 B 0082 C 0081 D 0080 E None of the above The velocity of a car in feetsecond t seconds after starting from rest is given by the function ft NE Find the cars position at any time t 0gtg30 A gig20 B 3W2 0 Ell20 4 D gtlZ None of the above Evaluate f 7 26m dx 2 32 x 2 32 x 3 23 x A g2 726 B g2 726 0 C is 726 3 D ExZS 7 26m C None of the above Calculate 18 4x13 32 dz A 49 B 50 0 51 D 52 None of the above Evaluate 03 l1 7 zldx A 32 B 52 C 72 D 92 E None of the above Evaluate f xe m2dz 1 1 7eim2C B 7 eim0 0 172x2 2o D 7Ee 20 A E None of the above 10 59 Find the area of the region under the graph of function f 2 on the interval 07 1 A B c D E None of the above CH H 60 Find the area of the region under the graph of y 2 1 from x 71 to z 2 E None of the above 11

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