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# Calculus & Analytic Geometry I MATH 131

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This 11 page Class Notes was uploaded by Dr. Cleo Wunsch on Thursday October 15, 2015. The Class Notes belongs to MATH 131 at New Mexico Institute of Mining and Technology taught by Lynda Ballou in Fall. Since its upload, it has received 11 views. For similar materials see /class/223624/math-131-new-mexico-institute-of-mining-and-technology in Mathematics (M) at New Mexico Institute of Mining and Technology.

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## Reviews for Calculus & Analytic Geometry I

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Date Created: 10/15/15

Final Exam Review Math 131 L Ballou 1 If fx3Txf1nd Whatis the domainof composite is not de ned at X 3 so the domain is all reals not equal to 0 or 3 2 Are the lines 2x y 1 and 2x y 1 perpendicular The slopes of the lines 2X y l and 2X 7 y l are 72 and 2 These are not negative reciprocals therefore they are not perpendicular to each other 3 True orFalse If limfxL then faL 0 sinx 1 x 5 DNE False ling 4 If andgx are differentiable then This is the productrule f39xgxg39xfx 5 If y l use the de nition of derivative to nd x 1 1 7 h ilslim 39h xlimx x lim h 21 dxx H0 h H0 xhxh HO xhxh x 6 Sketch a function f where f is continuous at x a but f is not differentiable at x a Any continuous function with a sharp comer at X a will work Final Exam Review Math 131 L Ballou 7 If gx f ux u3 5 u 393 2f5 3 andf 5 3 nd an equation ofthe tangent line to the graph of g x atx 3 The equation for the tangent line will be yg3g393x g3fu3 u g39lt 3gtf39ult 3gtgtu39lt 3gtf y EK fi mi ESIS yfu3 f u3u 3x3 y 3 6x3 y 6x 15 3 A UI V t I U V II I U V A N V II I o 8 Find the maximum and minimum values of f x AxB where A gt 0 and B are constants on 61 b fx AxB 3 f39xA A gt 0 3 fx is increasing The min is at a fa and the maX is at b fb 9 z m sin2x m 2sin2x 2m sin2x 236 x 2 36 2x 2 36 2x Or by L39Hospital hm sin2x 1m 2cos2x Xgt0 x Xgt0 1 10 Does the function g x lx 2 satisfy the hypotheses ofthe Mean Value Theorem on 14 The requirements of MVT are 1fx is continuous on ab yes 2fx is differentiable on a b no f392 DNE Final Exam Review Math 131 L Ballou 11 By the Intermediate Value Theorem if f is continuous on the interval a b and K is between fa andfbthen K fc for some 0 in a b 12 Find the equation of the tangent line to the graph of the equation tanxy y2 at n4 1 The tangent line is given by y 1 y39x To nd y39 we take the derivative implicitly sec2 xyy xy39 Zyy39 3 sec2 2y39 Do not substitute values until you have taken the derivative With a little algebra y39 4 so y 1 4 x 4 7r 4 7r 4 13 Evaluate the followin limits 2 m max lim H lim 90 x x xao x xao 3x xgt0 6x Ho 6 3 2 b ML Hlim7xHlim2xoo Hoe xlnx Hw11nx Hw Final Exam Review Math 131 L Ballou 14 Consider the function whose graph is 2 i a What is the value of fame 7i 39 4 the curve 39 portion ofthe curve is half ofa circle ofradius 1 or nZ b f39x0at x At x 0 a local max where the function is differentiable c f xgt0for ltslt l2ltxlt3 in d f39xfajls to exist at x lx1123Tnetangentisvenica1atx 11 quot 39 39 39 zannz e fxfajls to be continuous at l At x 39 and I39 39 L e at x 3 the limit does not exist f fxfajls to exist at x 7 A L A n39he t Final Exam Review Math 131 L Ballou 15 Find the rectangle of largest area that can be inscribed in a semicircle of radius R assuming one side of the rectangle lies on the diameter of the semicircle as shown A I39ll Let y IRZ x2 then given the rectangle shown in the picture above we have A 2xle2 x2 To maximize this we take the derivative and set it equal to zero This gives us 2x2 2R2 4x2 JER W x2 W xZ 3 x T 2 R2 JER 7 T A392 RZ xz y JER Dimensions JER by T 16 Let fx 27x 3 6 3 Calculate f X and use it to nd all critical points offX Classify all critical points and determine intervals where f X is increasing and decreasing 27 4 Thls functlon 1s contmuous for all reals f 39x 3 mx x 274 At X 0 the function goes vertical brie y at X 274 we have a local maX 00 274 T 274 00 l The critical values are at X 0 and at X Final Exam Review Math 131 L Ballou 1 For the graph of y 3 nd the in ection point l 2 Gwen yxmy39 x39m3yquotx3953 ForXlt0 yquotgt0forXgt0 yquotlt0AtX0 the second derivative is unde ned but the function is de ned the concavity changes at X 0 so 00 is the IP 2 Does the graph of y 2x sin x have a ve1t1cal asymptote at x 0 x 1i 2x sinx 113 3 the graph has a hole not a VA H x 3 Verify that the function f x J satis es the hypotheses of the Mean Value Theorem on 0 2 The requirements of MVT are 1 f x is continuous on 61 b yes radicals are continuous on their domain 2 f x is differentiable on a b yes radicals are continuous on their domain 4 If f x if then fx fx I xzdx f C 5 If f X 2 on 3 0 then the Riemann Sum for f X on the given interval is 20 7 3 6 6 The total distance traveled during the time interval IS I S 2 by an object with velocity vt 4t3 is units l7 2 Total distance traveled is Ilvlttgtldt I4t3dt t4 15 units a l Final Exam Review 7 The graph of fx Math 131 2 L has a slant asymptote of L Ballou yX2 Evaluate the following a Isecz 5xdx Isecz 5xdxtan5xCLetu5xdu5dx l 5 2 x b d i298 1 x 0 1 5 2 5 5 5 IZXSXHabcgln2x31 tgln3Letu23 1 Sdexgdu Ix1 x2 dx 2 298 x4 x l xzdx x 2x2x dxx C H gt I 3 or use the substitution below 1 xu x1 u dx du Ilt1 ugtu2duIu3u2du C1 x4 1 xgt3c Final Exam Review Math 131 L Ballou 4 71 e d dx i J l u J du 2J3 4 7 x 72 71 e 2 2 dx 2 e du 2 e du 26 1 i M i l 2 e e2 dh 1 9 A tree has been transplanted and after t years of grow1ng at a rate of E 1W ftyr At t1 two years it has reached a height of 5 ft How tall was the tree when it was planted 2 1 2 1 1 8 j 1 2 dtj1t1 2dtt g2 1 0 H1 0 11 3 3 In the two years since being transplanted it has grown 83 feet If it is 5 ft it was 73 ft when transplanted Ln 4 10 Evaluate 005 2xdx 0 37 T 3 3 Heoslt2xgtldx Icoslt2xgtdx cos2xdx 3Jcoslt2xgtdxw 0 3 0 0 0 E s1 s Final Exam Review Math 131 11 Find for the following x L Ballou a y tan6x y39 6sec2 6x b y sinz x y39 25inxcosx sin2x 0 y 3x5 4x2 15x4 8x 2W d y a y39 2x 1e 2quot e y x1 yxlquot gt1nyan x yj 211 21nxx m f y Zii y39 Wax l Wax 1 2 2e 13 Final Exam Review Math 131 L Ballou 12 A race of cial is watching a race car approach the nish line at a rate of 200 kmh Suppose the of cial is sitting at the nish line 20 In from the point Where the car Will cross and let 9 be the angle between the nish line and the of cial s line of sight to the car At What rate is 9 changing When the car crosses the nish line tan0 gt002secz0a50cos20 7 i lOOOOrad d 7 d dx 7 hr At90 Let X be the distance from the nish line for the car Adjust so that the units are consistent 20 meters is 002 km Letfx In Calculate f x and use it to nd intervals Where the graph of fx is X concave up and concave down Find all in ection points The domain offis x gt 0 fcM Menxemf x0 3 6 Wm x f clt0When0ltxltemf xgt0Whenxgte32 3 Hexare Final Exam Review Math 131 L Ballou Sketch a possible gmph of a continuous function y f x using the gmph of f 39x shown below if f0f30 yf39X max P f N4 1 6 n i D l f 4L1 2 3 4 5 6 f 4L1 3 4 a 6 n U H U When f39x gt 0fxT f39xlt 0fx i Ito indicate increase D to indicate decrease Whenf39xTfxU f39x vfx 1 occurs When concavity changes max and min occur When fx changes from increase to decrease or decrease to increase respectively

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