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# Review Sheet for MATH 1431 with Professor Caboussat at UH

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Date Created: 02/06/15

Math 1431 Final Exam Review 1 Find the following limits if they exist sin 7x 5x a 1 b 1m W 9 we tan2x 3 2 0 mi 8 d limx 1 ix 4 gt1 x1 e 1 Slug f H 1 005k hgt0 1 him g hm h 1 1 00252x 1 x 1 xgt0 x i lim 3x J 1113 2x cot3x gt0 sin4x x 1 2 k 11m 2 1 hm gt4 x 3x 2 gt71 x2 1 x2 2x x2 x 2 In 11m 11 11m 2 xgt1 x 1 W71 x 1 o limex2 2x 1 xgt2 2 We know that lim2x 1 3 Give the largest value of 5 that works with 2 8 1100 in the proof of this limit 3 Find values of A andor B so that the function is continuous x2 xlt1 a39 fxAx 3 xgt1 Ax B x51 b fog 3x 1ltx52 BxZ A xgt2 3x2 1 xlt4 c fx A x4 Bx l xgt4 4 Determine if it is possible to find A so that fX is continuous and if it is find A f Ax xgt3 a x x2 3 x53 xZ l xlt3 b fx x3 Ax l xgt3 x22 xlt 1 c fx1 x 1 Ax Z xgt 1 d f 2x 1 x52 x Ax2 3x xgt2 5 Find the derivative of the following 3 a mm b y cos3 2x c fx xtanx d fx 3x cos2x 9 x zzxz f fxsinx22x x 3x g fx 2x h y cosx2 3x5 x 2 1 fx sin2x cos3x3 j fx 4 sin x cos5x5 k fx sin2x 1 fx cos3x m fx tan4x n fx cotx 0 fx secx p fx csc2x q fx 2x3 4x2 7 r fx xsinx s fx m t fx 1x1 fx tan2x x4 s Use the intermediate value theorem to show that the function 2x5 3x 1 has a root on the interval 12 2x2 3x 1 Use the extreme value theorem to show that the function has 4 both a maximum and a minimum value on the interval 12 Notice that Xy 12 is a solution to the equation 993 y 10x Compute dydx at Xy 12 Give an equation for the tangent line to the graph of the function 2x2 3x 1 at the point where X 391 10 Suppose we are given the data in the table about the functions and g and their derivatives Find the following values x 1 2 3 4 fx 3 2 1 4 f x 1 4 2 3 gltxgt 2 1 4 3 g39 x 4 2 3 1 0 h4 if hxfgx b W4 if hxfgx 6 h4 if hxgf 16 1 W4 if MIC gfx 9 W4 if MIC 1 W4 if MIC f 90806 11 Each ofthe graphs below are graphs off DeterInine where f is increasing decreasing intervals concave up and concave dow i 12 The value Xa is a critical number for fX Classify a as a local maximum local minimum or neither a fxx3 12x2x0 b fxx3 12x2x8 c fxx2 2x1x1 l 3 d x 7x4x3 7x2 x4x1 f 4 2 13 Giveni yfxgt a Give the intervals ofincrease of f 1 Give the intervals of decrease of f c Give the critical numbers of f d Give the values of x where fhas a local maximum e Give the values of x where f has a local minimum f Give the intervals where the graph of f is concave up g Give the intervals where the graph of f is concave down h Give the alueis39 sf 1 whexe the staphe39rrhas 1hhectxeh 39 14 Use the plat arthe uhstmh Dn the mtexval 723 Ln gAve a geemetm depmuun er the mean value themem y 200 1a Gwe the diffexenualnf fxx2 3 at x 1 mthxespectm the Inclement 110 15 Use the guess x a and D112 nexatmn UfNEWLDn s methed Ln appmxlm ate 810039 at fxx2 710 17 Use diffexenuals Ln appmxlmate J63 lE Evaluate 19 The function t given below is continuous find a formula for 1 63 20 a 2x4 3x2 6fftdt 2 b 2x3 3x2 x 1fftdt 71 Integrate 2 2 a f xdx b x d Jcotx x c f3x3 2x2 57Ix 4 d NM 1 0 l ei dx f fsin3 3x cos 3x dx 7 g fxxlxz 2dx 2 h fecz 2gt0s6c3 6xix 2x 139 f d j 172x dx xm k f sin2x1x l f cos3x1x In fsecz2xix n fcscz 3xdx o f sec2xtan2xdx p fdx1dx q fxx2 1dx r flwll xzdx 0 21 Give the average value of x2 on the interval 02 22 Gwen the graph offx Wlth the area ofreglon A to equal to 73 reglon B 5 343 and reglon c S 73 Flnd a The area of the reglon bounded by for and the X39aXJS between 392 and 4 o b fxdx 23 Use the deflnltnon of derlvatlve to flnd the denvatlve of 24 Gwen Fx for each problem graph the runehon and shade the area between Fx and the X39axls nd the X39coordlnate ofthe centrold ofthe shaded region and flnd the y39coordlnate of the centrold of the shaded region Fm Fx x2 4 25 Gwen Fx and the lnterval a b graph Fx over the lnterval nd the average value ofFx on that lnterval and nd the value of c that verlfles the concluslon of the mean value theorem for lntegrals for the functlon Fover the lnterval a b Fx a an 01 39 Fx 3 736 Fx 16274 2 26 Write the equation of the tangent line to a y2 xy 6 0 at the point 52 b 2x2 5xy y2 4 at the point 31 27 As a balloon in the shape of a sphere is being blown up the volume is increasing at a rate of 4 in3sec At what rate is the radius increasing when r 1 in 28 Sand is falling off a conveyor onto a conical pile at a rate of 15 cubic feet per minute The diameter of the base of the cone is twice the altitude At what rate is the height of the pile changing when it is 10 feet high 29 You have a square piece of cardboard 6 inches wide and 16 inches long that you wish to fold into a box It occurs to you that you can cut an equal square from each corner of the cardboard make a crease along each side and fold the sides up to make the box How much should you cut from the corners to form the box with maximum volume 30 A man is walking away from a light pole at a rate of 5 feet per second If the light pole is 20 feet tall and the man is 6 feet tall how fast is his shadow growing when the man is 30 feet from the light pole 31 A rectangle is drawn in the first quadrant so that its base is on the x axis and its left side is on the y axis What is the maximum area of this rectangle ifits upper right vertex lies on the line segment connecting the points 40 and 08 32 Suppose f is a differentiable function on the interval abl a Explain how to find the absolute maximum and absolute minimum values of f on the interval abl b Use a graph to demonstrate that a function can have its absolute maximum value occur at exactly 3 places 33 Consider the function 3x4 20x3 42x2 36x on the interval 04 a Show that the critical numbers of f are 1 and 3 b Give the intervals of increase and decrease of f c Give the values of X at which 139 has either a local minimum or a local maximum d Give the values of X at which 139 has an absolute minimum or an absolute maximum e Give the intervals where the graph of f is concave up f Give the intervals where the graph of fis concave down g Give the values of X where the graph of f has in ection h Plot 139 34 State the extreme value theorem 35 State the mean value theorem 36 Give a geometric explanation of Newton s method 37 Graph a function fwhich has a cusp at x 1 a vertical tangent line at x 2 a horizontal asymptote of 3 and vertical asymptotes at x 392 and x 3 38 List the domain critical numbers intervals of increase intervals of decrease in ection points intervals of concave up and intervals of concave down for the function given Then graph the function and carefully label any local maximums local minimums or points of in ection a fxx12x 2 l 9 b x x4 2x2 f 4 4 0 fx 2x3 3x2 12x 3 39 40 41 42 43 44 45 46 47 48 R is the region bounded by the given graphs and the given aXis Sketch each graph then find the area of R the volume when R is revolved about the xaXis and the volume when R is revolved about the yaXis a yx2y6 x x axz39s b yx2y6 x y axz39s Let x3 3x be defined on 1 1 Find 0 on 391 1 that satisfies the conclusion of the Mean Value Theorem Use two iterations of Newton s Method to estimate the solution to 0 for fx x3 x 3 starting at x1 1 Estimate tan28 using differentials Find the net area bounded by the graph of x3 x2 and the raids on the interval 0 2 Find the area bounded by the graph of x3 x2 and the raids on the interval 02 Find the centroid of the region bounded by the line y 4 and the graph of f x x2 Revolve the region in problem 15 about the xaXis and give the integral resulting from using the method of washers to find its volume Do not compute the integral Revolve the region in problem 15 about the yaXis and give the integral resulting from using the method of cylindrical shells to find its volume Do not compute the integral 4 Derive the formula V 371 r3 for the volume of a sphere of radius I39by revolving the region bounded by a circle of radius 139 centered at the origin around either the X aXis or the y aXis 49 Compute the Riemann sum for the function 2x on the interval 02 associated with the partition P012 given that the heights of the rectangles are created by using the midpoint of each subinterval 50 Give the definite integral associated with the Riemann sum 2 L 1 1 n

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