Exam 3 Review (Sections Covered: , 6.7topic and )

Transcription

1 2 ' Exam 3 Review (Sections Covered: 6166, 67topic and 8182) 1 Find the most general antiderivative of the following functions (Use C for the constant of integration Remember to use absolute values where appropriate) (a) Z 5 p x 5 +4e x 2x 5 +3x 2 1 x a 5 } x " 2 + 4e Z x 2 +7x 4 (b) dx x 3 y x dx 4+3 tgx 3 In l x I + C Qx"2+4e + 54tx31n1x# fly dx 1n1x17x"t2I2tCa Z 7e x +13 (c) dx e x f7 + Be dx (d) 7xtl3e # Z 2 x x 4 x 7 dx 2g + } x ' 4 x 'd x z he I a Is To 6t C 2ln1x1 x3+{ (e) Z 48 + u 2 8u du fj + tsu du 6 h I u I + 's + C a 6hrlultlu#

2 3 4 2 For the following functions, evaluate the integral (Use C for the constant of integration Remember to use absolute values where appropriate) Z (a) (x 4 +2)(10x + x 5 +1) 3 dx Ua1Oxtx5t1du4ot5x4JdxSlx4t2Tu3K_glfu3duFx4Tt5x4atsetyu4tCz@xs4ct7dxs Z im#yux5fxteuyxlheudu (b) x 4 e dx d : Z (c) (3x 3 9)e (3x4 36x) dx tseutcly#c NX fl3x#h de#tyfeududud2x336)dxtyeutclge3@x+c7dxdu* 36x Z (ln x) 36 (d) dx x 't fiy6xdufu36du 3 dut d Z 30e 6/x is?+c@p7+c7dxxd:6/xdu6fx2dx7dxx2du*#xoidu5seudu5eutc5@+c (e) x 2 2 Fall 2016, Maya Johnson

3 3 The speed of a runner increased steadily during the first twelve seconds of a race Her speed at twosecond intervals is given in the table Find lower and upper estimates for the distance that she traveled during these twelve seconds using a lefthand sum and a righthand sum with n 6 Xo Xc Xz Xz XY Xs X6 t(s) v(ft/s) [ 0,12 ], n 6 DX 12/62 Lb 2 ( o ) R6 22 ( ) h@ 4 Speedometer readings for a motorcycle at 12second intervals are given in 156 the table Xo Xc Xz Xs X4 X5 t(s) v(ft/s) (a) Estimate the distance traveled by the motorcycle during this time period using a lefthand sum with n 5 [0/60], n5 Bx 60/512 L 512 ( ) (b) Estimate the distance traveled by the motorcycle during this time period using a righthand sum with n 5 Rs 12 ( ) Fall 2016, Maya Johnson

4 Sun 5 Use a lefthand sum and a righthand sum with rectangles of equal width for the given value of n to approximate the integral Round the answers to two decimal places Z 13 1 (2x 2 +1)dx, n 3 [1/13], n 3 DX (131)/3 4 L } 4su/Seg(2x2H, X, 1,43 4), 4) )# R } 4 Sun (Seq (2 2+1,14+4), 1314 ))a2@ 6 Use a lefthand sum and a righthand sum with rectangles of equal width for the given value of n to approximate the integral Round the answers to four decimal places [ 1,10 ), n 3 D ( 1011/33 Z 10 1 x 2 ln(x) dx, n 3 L } 3 Sun ( Seq ( X2h( x ),, I, (103), R } 3 Sum ( Seq ( x2h( x ), X, (1+3), 10,3 )) Use a lefthand sum and a righthand sum with rectangles of equal width for the given value of n to approximate the integral Round the answers to two decimal places [ 0,12 ], n4 DX (120)/43 Z 12 0 (2x 3 + x) dx, n 4 Lot 3 Sun ( Seq ( 2 3 tx,, 0,42 3), 3 ))5@ Ry 3 ( Segkistx, x, (0+3312,3) Fall 2016, Maya Johnson

5 2x)KB3 8 Evaluate the following definite integrals: (a) Z 1 A 6 x dx Assume A<1 6 lnlxlljs 6411T 6 LIAI 6 (b) Z B 2 (3x 2 7x 3 +7x 2) dx Assume B>2 ( is 7 +7 ZBTHBI ZB (232, ) B3 72,1 +7 ZB ( to ) (c) Z B B3ly#7BI2B@@ex6sI+2x)lFl0ek6sBIt2B(10e0 x 6x 4 +2)dx Assume B> ( o ) ) (d) Z A x 2 +10xdx Assume A>1 1OeB6B ) IF ( NA AY 400 3aP+5NY 1oA3A3t5A2 9 If f(4) 18, f 0 is continuous, and Z 6 4 f 0 (x) dx 30,whatisthevalueoff(6)? 30 Sdf ' ( x ) dx f( 6) f (4) 30 f C 6) 18 fl 6) 5 Fall 2016, Maya Johnson

6 Zoo 10 Suppose the marginal cost function for a certain commodity is given by C 0 (x) 05x and C(0) 200, find the cost to make 12 units of this commodity 36 a ftp5xdx#ckx)dxc( 12 ) Cco ) 36 C ( 12 ) C ( 11 Suppose the marginal revenue function for a certain commodity is given by R 0 (x) 10x 6and R(1) 100, find the revenue when 10 units of this commodity are sold " dx{ Rkxsdx RC 6) RCI ) 441 R ( 10) 100 R ( 10 ) µ 12 Find the average value of the following functions on the given interval (Round answers to two decimal places as needed) (a) f(x) 6x +9x 2, [0, 5] (b) f(x) 12e 3x, [5, 7] atfj6xt9x2dxtsfnintl6xt9x3xos7o@baasjl2exdxtzfnintf12eyx57t5689jwy2x3tox2dxytcfnint42x3tox2x (c) f(x) 12x 3 10x 2, [2, 6] 7866, 2,6 ) 6 Fall 2016, Maya Johnson

7 13 The rate of sales of a certain brand of bicycle by a retailer in thousands of dollars per month is given by d S(t) 15t 057t2 dt (a) Find the amount of sales, in thousands of dollars, for the first six months after the start of the advertising campaign Give answer to three decimal places Ft (b) a57t2dtfnintfl5xi57x3xqlyg22896thousa@wm5tr57t2ttt6efnintf5xs7x2x Find the average sales per month for the second six month period of the advertising campaign Give answer to three decimal places, 6, 12 ) 871Zth usa 14 Suppose that copper is being projected to be extracted from a certain mine at a rate given by P 0 (t) 320e 008t where P (t) ismeasuredintonsofcopperandt is measured in years (a) How many tons of copper is projected to be extracted during the second four year period? Give answer to three decimal places fyzzoeittdt fnintfs2 e Yx, 4, (b) How many tons of copper is projected to be extracted during the third four year period? Give answer to three decimal places fglzzoe " " tdt fn Int ( ,12) Fall 2016, Maya Johnson

8 ) 15 Use properties of the definite integral and information listed below to solve the following problems: (Assume a and b are two real numbers such that a<b) Z b a Z b a Z 0 3 Z 3 0 (a) Evaluate f(x) dx 20 g(x) dx 12 u(x) dx 16 u(x) dx 50 Z 2a 2a f(x) dx (b) Evaluate Z a b g(x) dx a ( 12 (c) Evaluate Z b a [f(x) 3 g(x)] dx 2 20 El (d) Evaluate Z 3 3 u(x) sjulxldx {{ µ 16 Determine lxidxtfuhdx the area that is bounded by the graphs of the following equations y 64x, y x 3 X o ( x +8 0 Xx 0,8 8 a, x Area µ @ / 3 8 Fall 2016, Maya Johnson

9 6) tz 17 Determine the area that is bounded by the graphs of the following equations (Round answer to three decimal places) y 3x, y 9x x 2 :X ( x 9 x D X 0, 6 XZ X Area!x2+6 dx fniut(yz6x #, X, Q 6) 18 Determine the area that is bounded by the graphs of the following equations on the interval below (Round answer to three decimal places) y x 2 +7x, y 8x +56 ( X X 7,8 X XZ x 560 Area a dx 19 The graph of f is shown Use the graph to evaluate each integral (a) Z f(x) dx 5625 a ( 8) fl (b) Z 36 0 f(x) dx (81/4)+248 )(8) + (4 12) + } ( 12718) ( 12)(8) El4)(8) (8 8)320 (c) Z 12 0 f(x) dx ( 8)( 4) ttz 18 )( 8) Fall 2016, Maya Johnson

10 20 Calculate the producers surplus at the indicated price level for the supply equation below (Round answer to the nearest cent) p S(x) x 2, p o $ XE 324 X± H Fq 480 ( ) dx fn Int ( , 0, 1 8) $7@ 21 Calculate the consumers surplus for the demand equation at the given number of units demanded (Round answer to the nearest cent) p D(x) 27 2x 1/3, x o 343 Po D ( 343 ) 27 2 ( {3 If dxfniat( ,, 0,34 3) $12O@ 10 Fall 2016, Maya Johnson

11 5) Ist at ' 5 22 Determine the consumers surplus for the demand function below at the indicated price level 5Foo p D(x) x, p o $ Determine the indicated values of the following functions f(x, y) 4x 2 xy + y 9 g(x, y) x 4 5 2y dxfnint( ,6500 ) $1,26 OGX (a) f( 5, 2) 4155(5112) H 2) 9790 (b) g(1, 9) day i ' D (c) f(1, 3) 3g(7, 1) fl 1,314115(4/3) t( 3) 917, 111 4# 9 } ' f" ' ' 24 Determine the indicated value of the function (Round answer to one decimal place) D 3ft ) 5+1 # W (09, 6, 1, 5) for W (a, b, c, d) A 9 b 6,, 11 C, D 5 a(1 + b) d2 2c Wt 9, 6,1, 917k 25 Determine the indicated values of the function 3xy +6z f(x, y, z) 3xz 6y (a) f(1, 0, 1) (b) f(0, 1, 1) (c) f( 1, 1, 0) 3 ( 935 XO) +6 states D 3/0 )tdt6( 1) TEE D IXD task, D 11 Fall 2016, Maya Johnson

12 C 26 Macrosoft produces two versions of its popular gaming console: the Elite and the Casual The weekly demand and cost functions for the consoles are p 300 4x +2y q 225 x +9y C(x, y) x +120y where x represents the weekly demand for the Elite version; y represents the weekly demand for the Casual version; p and q represent the price (in dollars) of an Elite console and a Casual console, respectively; and C(x, y) isthecostfunction (a) Determine R(x, y), the weekly revenue function Rkiy) pxtqy3ooxch/2+xy+225y+9tplxiy)rlx (b) Determine P (x, y), the weekly profit function, y ) ( x, y ) 2lOx4 2txy+l05yt9yI3 (c) Find P (4, 2) p ( 4,4 $# 27 HeadsRock produces two versions of its popular headphones: the RockUrWorld and the SilentRocker The weekly demand and cost functions for the headphones are p 300 7x +2y q 225 x +8y C(x, y) x +120y where x represents the weekly demand for the RockUrWorld version; y represents the weekly demand for the SilentRocker version; p and q represent the price (in dollars) for a pair of Rock UrWorld headphones and a pair of SilentRocker headphones, respectively; and C(x, y) isthecost function (a) Determine R(x, y), the weekly revenue function R l, y ) pxtqy 3oox7x2txy+225y+8Ty 12 Fall 2016, Maya Johnson

13 (b) Determine P (x, y), the weekly profit function Plxsykrlky ) (c) Find P (7, 2) Clay) Zwx7x4xy+lo5y+8y440 P 17,4 28 Find the first partial derivatives of the function w 9z +10e xyz @y yze Y O (c) 9+1Oxye@ 29 Find the first partial derivatives of the function f(x, y) x 6 y 5 +7x 4 y (a) f x (x, y) (b) f y (x, y) 5x6y4+ 13 Fall 2016, Maya Johnson

14 30 Find all the second partial derivatives f(x, y) x 9 y 5 +3x 9 y + x 4 +30y 2 (a) f xx fx fxx 72 75> (b) f yy fyy 5 9 y Oy 2ox9y3# (c) f yx f xy fyx fag 45x8y4t2 31 Find all the second partial derivatives f(x, y) 200+3x 5 y 3 +2x 1 0y 2x 6 +12y 3 Ga (a) f xx fx 15 x 4g x9y 12 5 fxx (b) f yy fy t 3 6y 2 fyy (c) f yx f xy yx fxy 45x4y2t 14 Fall 2016, Maya Johnson