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# CALCULUS I MATH 170

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This 7 page Class Notes was uploaded by Oswald Boyle on Monday October 12, 2015. The Class Notes belongs to MATH 170 at Idaho State University taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/222166/math-170-idaho-state-university in Mathematics (M) at Idaho State University.

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Date Created: 10/12/15

MATH 170 Review Problems 1 2 OJ q 01 a March 18 2009 The following is the graph of a function y x On the axes below the graph indicate where f x and f are positive negative zero or unde ned The following is the graph of the derivative of a function y x1 x2 x3 x4 x7 a Where is the function x increasing and where is it decreasing 10 Where is the function x concave up and where is it concave down c Where are the extrema and in ection points in the graph of the function x 7 Determine the absolute extrema if they exist of a y 2x3 7 3x2 712x 1 on 71 4 y73x25x2 on 02 yx3x27x1on712 d y 2 sinx cos2x on 0 27139 y ir5 2 x2 x on 72 2 Compute the following limits lim 2 7 3 x700 3x2 x2x 371 4 7 3x2 C 111711127 4x 7 x3 b a 131711 2x2 x71 2x71 sinx 7 tanx i lim 7 x70 30 7 1 3 d lim LE 0270 e lim f 0270 Sketch the following curves showing all critical points and in ection points mi412353 b gnaw c y712m d yx25 e yxlnx f y Find the critical points of x x5 7 5x4 5x3 2 and determine if they give local minima or maxima MATH 170 Review Problems Page 2 lt2ezgtlt1zgt mes N 147230 7 If x W then f W and f W Sketch the graph of 1 Be sure to determine all intercepts asymptotes extrema in ection points and intervals on which the function is increasing decreasing concave up or concave down 8 The graph of a function y x is given below For each of parts a through 1 determine if the speci ed quantity is positive negative zero or unde ned y p l l O 1 1 1 1 equot 1 m mg mg 3 39 4 z5 6 m7 9 1 zquot 9quot a HM 0 lim He C Hm d NM 8 NM f 954 g Mm h 133 1 i Nee 0 f k 1W7 1 NM 9 What point on the line segment from 03 to 20 is closest to the origin H 0 A wire 48 inches long is to be cut into two pieces One piece is to be bent into a circle the other into a square How should the wire be cut in order to minimize the sum of the areas of the circle and the square H H A eld along the bank of a river is to be enclosed by a fence with no fence being required along the river If the area of the eld is to be 4840 square yards an acre what dimensions will minimize the amount of fence required H to A window consists of a rectangular lower half together with a semicircular upper half If the perimeter of the window is required to be 15 ft what dimensions will maximize the total area of the window H 03 The following graph of a function y fm has several asymptotes For each asymptote write a limit involving the function and plus or minus in nity that describes the behavior of the function near the asymptote 14 State the Mean Value Theorem and give at least one important corollary to the theorem MATH 170 Review Problems 03 53F gt777 DHOKDOO H 03 H F Page 3 ANSWERS 7 i x I I H l39 M 1 r 7 7 0 7 Slgnf 7 7 gtk 0 0 gtk 0 gtk 7 0 39 Slgnf a Increasing z lt m4 z gt me Decreasing 4 lt z lt 6 b Concave up z lt m1 23 mam Concave down 12 mgm5 z gt 7 c Max at 4 min at 6 neither but critical at 2 ln ection at m1 m2 m3 m5 and 7 a Global min 2719 Global max 4 33 b Global max at z g No global min y0 2 but wli rgi73m2 5x 2 O c Global max 211 Global min at z d Global min 3717 Global max g or 5 e Global min at z 712 No global max limJr fz gt 772 a 7 b 0 c 7 d 71 e 7 f y 5m4 7 20 3 15 2 5x2 m 71m 7 3 So the critical points are 0 1 and 3 3 is a local minimum 1 is a local maximum and 0 is neither a local max or min a b i C 0 d e f g 0 h i i i k i 1 18 12 39 Minimize m 2 73x 1 32 Closest point E E Minimize 1035 g Minimize fz 2x Optimal dimensions V2420 x 22420 measured in yards 48w WM the length of the circle part 1 4847 Minimum at z Let the semicircular part of the window have radius r and let the height of the rectangular part 1 be y We want to maximize the total area A 5w r2 2ry subject to the constraint 15 7rr2y2r Eliminating y we have A 15r7iwr272r2 Solving A 1577rr74r 0 gives r 74 After simpli cation the corresponding height is also given by y lim2 00 lim mg 71 Let f be continuous on the closed interval 11 and differentiable on the open interval a 1 Then there exists 0 in ab with fc w lmportant corollaries include the fact that if f z 0 on an interval then f is constant on the interval Also if f m gt 0 on an interval respectively lt 0 then f is increasing respectively decreasing on that interval 1 7 1 Efw 0 fltgt 00 lim fm 7 05x 71 0 MATH 170 Review Problems April 177 2009 1 The following gure gives the graphs of two functions7 y x and y Compute values for the various integrals based on information in the gure and properties of de nite integrals The framed numbers inside the regions of the gure represent areas of the regions 9 y 95 3 g 3 12 w 171 yg a mm b 3fltzgtdz e mm d igltzgtldz e fxgxdx f lfmgmldm g g mdz h 3fx2dx lt1 fltzgtfltzgtdz 2 Evaluate the following inde nite integrals a z34z7 dz b dx 0 sin 3x 2zdz 3 dz d 3z74dz e 2x7 1 dz f lt52 Fgt dz g xxx21dx h dz 1 2jdx j z3 2z2 1 dz k dx 1 sin2x cosx dz 3 Determine x if f x x and 4 0 and f 4 0 4 Determine x if f x sin2x and 0 2 and f 0 3 5 Determine x if f x 6x72 and 1 1 and 2 3 6 Determine x if f x 53 71 and 0 1 and f 0 0 T b a What exactly is meant by xde b c d What is meant by the linearity and additivity properties of the de nite integral State the Fundamental Theorem 0f Calculus Explain the difference between de nite and inde nite integrals 00 Compute the area of the region that is above the x axis and below the curve y 2 x 7 x2 MATH 170 Review Problems Page 2 9 Differentiate the following functions as 02 2a m t2 cos tdt 9z 1253 1dt W sint2dt 1 30 10 Evaluate the following de nite integrals 2 7r2 a 113m22z71dz b 0 sin2z3coszdm c Wm d f 7521 dz 1 1 7r4 idm f tanzdm 111 o 0 H 11 An electronic monitor measures the rate at which pollutants are being released from a plant7 resulting in the following graph of the rate of release in lbshr as a function of time Estimate the total amount of pollutants released during the 24 hr period lbsghr 0 3 0 2 4 6 81012141618202224 12 Let f be a function such that f0 20 and whose derivative 1quot is given by the following graph Note the additional information in the gure about the ve points marked by 0 s and about the areas of the four regions ya Hm 117 740 Where is 1 increasing and where is f decreasing Determine the z y coordinates of any local minima and maxima in the graph of f c Does the graph of 1 have any in ection points If so7 where 13 Find the area of the region bounded by the curves y 2 3x 2 and y 5 z 5 14 Find the area of the region bounded by the curves y and y z3 MATH 170 Review Problems 1 1 00 Page 3 ANSWERS 723910 b 73f19fmdz730 725946 f 259420 f fzdm7ff2dz33972624 1472m27z0 b gx522z32C c 32175C e 2m7132C f e2 z327 1nm0 2127132C h5743m7122723m7 7252z2gt328z72120 j 2782E81n2EC 1 C093 VV 3 OJ CL 7 301 A 2 m2 71W2 21702312 71 2 C sinm3 C AAA 1 AAA VVVV 147 W V A U WwM H ooh Hp DA Hmwe m f z 7 cos2 z 7 101 7 sin2 z gm 2 f x3z272x0 fzz37m20xD Solving f11CD and f2 42CD gives 072 and D3 so fzz37m2i2m3 271531471 fltzgt75egw7az2ezg a f fmdm is the limit ofRiemann sums 2171 Axl as n a 00 or as H 0 where the norm of a partition 73 0 1 mn is de ned by 1mlaltxnAzi z 7 M4 b The Fundamental Theorem 0f Calculus has two parts 239 If f is continuous on the interval 1 1 then the function 9 de ned by gz f ft it for 1 g x g 1 is differentiable on 1 1 and g m 2391 If f is continuous on 1 1 and g is any antiderivative of f on 1 1 then f 1 195 95 9a c The inde nite integral f fmdm represents all antiderivatives of the function f ie all functions 9 with g m x for all m The de nite integral f fmdm is a number that represents the signed area under the graph y x between z 1 and z 1 By the signed area under the graph regions below the graph and above the m axis are viewed as having positive signed area while regions below the m axis and above the graph have negative signed area d Linearity is the statement that for any continuous functions f and g on 1 1 and for any real constants c and d we have bcfzdgmdzcabfzddIbgmdz a Additivity is the statement that for any continuous function f and any real numbers 1 1 f mmf fzdxfzdx and c we have ffl2zix2dz f z 2 cosz g m 2zxz6 173 V27 3 1 h m 6H sin4 67 23 sinm a m3 12 72 31 1071 9 b 7 cos2z 3 sinz 32 g 4 2eee d lnm 5672 ln3 5 6 7 e f 71nlteosltzgtgt 34 alum2 1nlt2gt 7 2 c f211 gdu 7 g 1132 7 232 etw4wnhg7ga7g MATH 170 Review Problems 11 Page 4 We need f024 ft dt Counting squares there are 32 squares worth of area under the graph each of which has an area of 2 hr 201123hr 401123 Thus the total amount of pollutants released is approximately 32 40 12801123 Alternatively do a midpoint Riemann sum with 6 subintervals f0 ftdt z f2 f6 f10 f14 f18 f22 At z 64 68 28 24 60 80 4 12961123 Actually the number of squares is probably closer to 325 which would give an estimate of 13001123 So the two estimates are quite close a f is increasing on 06 and is decreasing on 614 b By using fz 7 f0 for f t dt along with f0 20 we see f2 60 f6 130 f11 20 and f14 760 The endpoints give local minima while the point 6130 is a local maximum c f has in ection points at 2 60 and at 1120 because those are the points where 1quot changes increasing vs decreasing d 1 04 MD f0 Hm dm 60 The curves intersect at z 71 and z 3 so the area bounded by the curves is ff15x57 m23m2dx7z3x23ml 1 79997 173 The curves intersect at z 0 and z 9 so the area between the curves is f09Ei zdm gxg i x g 1872727 g

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