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Date Created: 09/19/15
1 Equot 9quot 1 5 Math 143 1 Final Exam Review Find the following limits if they exist sin 7x 3x a 11m 7 1 11m 7 xrgt0 9x gt0 s1n4x b lim j lim 2x cot3x x7gt0 tan2x xrgt0 3 8 C lim x2 k 1m 2x Hzx 4 x7gt71x 3x2 Z Z d lim x 1 1 1 w Xigt1 x x7gt71 x sinh x2 2x 6 11m m 1m hgt0 h xgt1 x 1 2 f lim LOW n lim x we h H4 x2 1 1 2 g o IIgtI3XI 2x 1 h lim1 00 2x xgt0 x We know that lim2x 1 3 Give the largest value of 6 that works with 8 2 1100 in the proof of this XHZ limit Use the de nition ofthe derivative to compute the derivative of fx Vx 1 Find values ofA andor B so that the function is continuous x2 xlt1 Ax B xSl a fxA 3 gt1 b x 3x 1ltxS2 x x BxZ A xgt2 3x2 1 xlt4 C fx A x Bx l xgt4 Determine ifit is possible to find A so that fX is continuous and ifitis find A Ax xgt3 xZ l xlt3 a x b 8 x23 f x23 x33 fx Ax l xgt3 2 1x 11 d f Zx l x32 c x x x Ax2 3x xgt2 Ax Z xgt 1 Find the derivative of the following 3 x f x3 2x b ycos32x c fxxtanx d fx 3x cos2x e x 2 x2 3x f f x sinx2 2x g fx i x2 2 h y cosx2 3x5 i fx sin2x cos3x3 j fx 45in x cos5x5 k fx sin2x 1 fx cos3x m fx tan4x n fx c0tx 0 fx secx p f x csc2x q fx 2x3 4x2 7 r fx xsinx s fx m t fx 1x1 u fx tan2x x4 7 Use the intermediate value theorem to show that the function fx 2x5 3x 1 has a root on the interval 42 72x273x1 x4 and a minimum value on the interval L 1 2 9 Noticetha xy 12 is asolution to the equation xy3 y 10x Compute dydx at m 12 Give an equation for the tangent line to the graph of the function fx 2x2 a 3x 1 atthe pointwhere H o Xl W H 39 ueii an e rindthe following values x 1 z 3 4 x 3 2 1 4 x 1 4 2 3 gx 2 1 4 3 339le 4 2 3 1 0 M4 If hrfgr 17 M4 If hrfgr 5 M4 If hrgfr 01 4 If hrgfr 2 4 If hock1C f 4 If hrfrgr I 12 quot L L L 5 y j 39 j is increasing decreasing intervals of c a fxx 712x x0 b fxx3712x2x8 72M c fx d fxx x37 2 nntha interval 3 at x 1 with raspacttntha increment 110 x 15 Give the differential at fx 1s 17 Use differentials to approximate V63 18 Evaluate E1 114 3dt dx 3 2 3t 4dt 2x21 dx 1 id dx 3xt dx25x 1 I dt dx 3 Zt wf J oxsin t2 dt Ifsxsin t2 dt 19 The function fX given below is continuous find a formula for fX a b 2x4 3x2 6 J39ftdt 2 2x3 3x2 x 1Jftdt 71 20 Integrate a Jcscz x Vcotx dx Jx2 3 x dx 3 2x2 5dx d j dx 1 e 1 78 l x f Isin3 3x cos 3x dx 7 g va x2 de 2 h xi 2cosx3 6xdx i jzxdx j sin2xdx 1 j c0s3xdx j secz2xdx j cscz3xdx 0 jsec2xtan2xdx p Ixx1dx q I xx2 14 dx 21 Give the average value of fx x2 on the interval 02 22 Given the graph at x 74 With the area at regirnA tr eduai tr 73 regirn E is 343 and regirn c is 73 Find a The area rithe regirnhrundedhy x andthaxraxis between 72 and 4 b fxdx 2 23 Use the dermitidn di derivative a nd the derivative at b fx 7h 3x71 regidn a Fx x b Fx x2 Me n unctiandver the interval a b a Fx x2 e x 01 I Fxx23x 730 c F0 4 722 26 Write the aquatinn dithe tangent line a a y exy60attheprint 52 E L8 75xyy2 4 atthapnint 31 N gt1 N 00 W H W N W W wwww you14gt As a balloon in the shape ofa sphere is being blown up the volume is increasing at a rate of4 in3sec At what rate is the radius increasing when r 1 in Sand is falling offa conveyor onto a conical pile at a rate of 15 cubic feet per minute The diameter of the base ofthe cone is twice the altitude At what rate is the height of the pile changing when itis 10 feet high 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 ofthe 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 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 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 Suppose f is a differentiable function on the interval ab a Explain how to find the absolute maximum and absolute minimum values of f on the interval ab b Use a graph to demonstrate that a function can have its absolute maximum value occur at exactly 3 places Consider the function fx 3x4 20x3 42x2 36x on the interval 04 a Show that the critical numbers off are 1 and 3 b Give the intervals ofincrease and decrease of f c Give the values ofX at whichf has either a local minimum or a local maximum d Give the values of X at which f has an absolute minimum or an absolute maximum e Give the interval 5 where the graph of f is concave up f Give the intervals where the graph offis concave down g Give the values of X where the graph of f has inflection h Plot f State the extreme value theorem State the mean value theorem Give a geometric explanation of Newton39s method Graph a functionfwhich has a cusp at x 1 a vertical tangent line atx 2 a horizontal asymptote of 3 and vertical asymptotes atx 2 and x 3 38 List the domain critical numbers intervals ofincrease intervals of decrease in ection points intervals of 4 Ch 4 1 4 00 4 0 fl 0 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 ofinflection 3 fx x12x2 1 9 b x x4 2x2 f 4 4 c fx 2x3 3x2 12x 3 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 cvcis b yx2y6 x y cvcis Let fx x3 3x be defined on 1 1 Find 0 on 1 1 that satisfies the conclusion ofthe MeanValue Theorem Use two iterations of Newton39s Method to estimate the solution to fx 0 for fx x3 x 3 starting atx11 Estimate tan28 using differentials Find the net area bounded by the graph of fx x3 x2 and the X aXis on the interval 02 Find the area bounded by the graph of fx x3 x2 and the X aXis on the interval 02 Find the centroid ofthe region bounded by the line y 4 and the graph of fx x2 Revolve the region in problem 45 about the xaxis and give the integral resulting from using the method ofwashers to find its volume Do not compute the integral Revolve the region in problem 45 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 E 39r3 for the volume ofa sphere ofradius rby revolving the region bounded by a circle of radius r centered at the origin around either the XaXis or they axis Compute the Riemann sum for the function 2x on the interval 02 associated with the partition P012 given that the heights ofthe rectangles are created by using the midpoint of each subinterval 2 i 1 Give the definite integral associated with the Riemann sum Zilj i n 11 11 Final Exam Review Cal 1 121210 Limits expected value In order for a limit to exist the limit from the left must equal the limit from the right If the limit from the left right and the function equal it is continuous Examples of DNE absolute value graphs ij if the denominator 0 its in nity therefore it doesn39t exist oscillating behavior x Techniques for evaluating Limits 1 Direct substitution plug it in 2Functions that agree at all but on point a cancellation technique factor and cancel b rationalization technique rationalize the numerator ex Lim x1 quot1z 2 multiply both by numerator xgt3 x 0 nonzero 0 nonzero nonzero nonzero0 DNE 00 more work What is continuity no 0 es no vertical asymptotes no jumps no breaks Three Step Method to Prove Continuity 1 fc is defined 2 lim xgtc fx exists 3 fc lim fx Where are polynomials continuous inf inf Where are rational functions continuous On their own domain The Pinching Theorem Trig Limits As long as the limit is xgt0 Sinxx 1cosxx0 cosxtanxx1 sinxx Intermediate Value Theorem I 0 is between the two numbers there is a solution If not there isn39t The definition of the derivative A function fx is differentiable at x if and only if Lim hgt0 h If f is differentiable at xa then f is continuous at xa Not every continuous function is differentiable A function is not differentiable at 1 Points of discontinuity jump hole vertical asymptotes 2 Cusps 3 Sharp turns corners Product Rule 151d 2ml 2 101 d 1st Quotient Rule 1 lo d hi i1i d lo Loquot2 Rates of Change Shape Area C 39 quot Circle pi rquot2 2pi r Rectangle xy 2X2y Triangle bh 2 more info needed absinc 2 Trapezoid 172 h b1b2 more info needed Cube Xquot3 6Xquot2 Sphere 43 pi rquot3 4 pi rquot2 Cylinder pi rquot2 h 2 pi rquot2 2 pi r h Cone 13 pi rquot2h na Velocity and Acceleration Position Xt Velocity X39t vt Acceleration Xquott v39t at Speed at time t vt Free Fall of An Object Feet yt 16tquot2 v0 initial velocity t y0 initial position Meters yt 49tquot2v0 t y0 Chain Rule Y39 9 00 g X Differentiating Trig Functions in Xcos X Cos X sin X Tan X sec quot 2X Sec X sec X tanX Cot X cscquot2 X Csc X csc X cot X Implicit Differentiation Ru es 1 Differentiable both sides of the equation with respect to X 2 Collect all y39 terms 3 Factor our y39 4 Solve for y39 If solving in terms of X take the derivative as usual If solving in terms of y use the chain rule Related Rates How to 1 Draw a picture 2 What do you know 3 What do you need to find 4 Write an equation involving the variables whose rates of change either are given or are to to be determined This is an equation that relates the parts of the problem 5 Implicitly differentiate both sides of the equation with respect to time THIS FREEZES THE PROBLEM 6 Solve for what you need plug and chug NewtonRaphson Approximation D a h Fah fa f ah Newton39s Method Formula Xn1 Xn fXnfXn n012 Mean Value Theorem If f is continuous on the closed interval ab and differentiable on ab then there exists a number c in ab such that slope oftangent Iineslope ofsecantline Rolle39s Theorem Same as MVT but f c 0 This means if 1 F is continuous on ab and 2 Differentiable on ab and 3 Ha ffb slope0 Increasing and Decreasing Functions Where f x is positive fx is increasing Where f x is negative fx is decreasing Critical Numbers C N The numbers c in the domain of a function f for which either 1 c 0 or f c does not exist are called the critical number of f First Derivative Test How can we classify critical points as either local maximums or local minimums 1 If fx changes from negative to positive at c then fc is a local minimum of f 2 If fx changes from positive to negative at c then fc is a local maximum of f Second Derivative Test 1 If f39cgt0 then fc is a local minimum up like a cup 2 If f39clt0 then fx is a local maximum down like afrown 3 If f39c0 then the test fails In such cases you can use the First Derivative Testquot How to Use Determine the CN using the rst derivative Plug these numbers into the 2m1 derivative and get the value Then if value is relative minimum If value is relative maximum If value is 0 1St derivative test to determine local max or min Endpoint and Absolute Extreme Values ether or not a function f has a local or endpoint extreme value at some point depends entirely on the behavior of f for x close to that point Absolute extreme values depend on the behavior of the function on its entire domain the highest and lowest endpoints of all x39s in the domain Optimization Problems to maximize or minimize W to 1 Draw a picture 2 Primary function to be maxmin 3 Use secondary function if necessary to get primary in terms of one variable 4 Determine feasible domain 5 Find max min 6 Show that answer is max min 1St derivative test or 2ml derivative test Concavity and Points of ln ection POI Fquotgt0 gt f is increasing F39 is increasing gt f is concave up Fquotlt0 gt f is decreasin F39 is decreasing gt f is concave down A POI occur where the concavity changes where tquot changes signs Vertical Tangents and Cusps A vertical tangent exists at cfc if as x gtc then fx gtinf or in A vertical cusp exists at cfc ifas xgtc then fx gtinf and as x gtc then f xgt inf vice versa Using Calculus to Graph a Function 1 Domain restrictions on x 2 Asymptotes vertical and horizontal 3 End behavior a number 1 inf 4 Intercepts y intercepts 0y X intercepts x0 5 1St derivative inc dec max min find derivative nd CN test points 6 2m1 derivative concavity POI nd derivative nd CN test points 7 Graph Put it all together The Definite Integral If f is nonnegative on or above the xaxis on ab then the integral of ab gives the area below the graph of f If we consider an integral as an accumulation of areaquot then the derivative of the integral is a rate of changequot of an accumulation of an areaquot The Fundamental Theorem of Integral Calculus Let f be continuous on ab A function G is called an antiderivative for f on ab G is continuous on ab and G39x fx for all xEab How to integate Add one to exponent divide by the exponent antiderivative Some Antiderivatives Function AN Antiderivative Sinx cos x Cos x sin x Sec quot 2 x tan x Sec x tan x sec x Csc quot2 x cot x Cscxcotx cscx Remember numbers may go through an integral sign variables may NOT Area of a Region on a Graph 4 t es 1 Basic no intersection one function integral ab fX 2 Two nonintersecting curves integral ab fX gXdX 3 Two curves that intersect nd intersection ab integral ab fX gXdX 4 Two curves that intersect several times nd intersections abc integral ab fX gxdxintegral bc gX fxdx Integral from ab fX gXdX Top bottom dx Big Little dx Indefinite Integrals The general antiderivate of f with C at the end C is an arbitrary constant and FX is an antiderivative off X Difference between Inde nite and De nite Inde nite a family of functions De nite a value The USubstitution Change of Variabes quotundoing the chain rule U quot power du U quotpower 1 C Power1 How do we handle usubstitution with a de nite integral We change the limits of integration to re ect the substitution Mean Value Theorem for Integrals The number fc is called the average mean value of f on ab 1ba integral ab fXdXfc height 1width area Volumes of Known Cross Sections If the cross section is perpendicular to the XaXis and its area is a function of X say AX then the volume of the solid from a to b is given by V integral ab AX area dx width If the cross section is perpendicular to the yaXis and its area is a function of y say Ay then the volume of the solid from c to d is given by V integral cd Aydy Volume Revolving Around an Axis Multiple Methods 1 Disc Method Revolving about the X aXis V integral ab pifX quot2 dx Revolving about the yaXis V integral cd pigy quot2 dy 2 Washer Method Revolving about the XaXis Vintegral ab pi fX quot2 gX quot2 dX Revolving about the y axis Vintegral cd pi fy quot2 gy quot2 dy I Pi Rquot2rquot2 o R distance from axis of revolution to the points farthest away being rotated o r closest hole hollow 3 Shell Method Difference from Disc Disc the rectangle of revolution is perpendicular to axis of revolution Shell the rectangle of revolution is parallel to the axis of revolution Revolving about the yaxis or vertical V integral ab 2piPxhxdx Revolving about the xaxis or horizontal V integral cd 2piPyhydy V2pi r h area dx h height of rectangle p distance from axis of revolution to rectangle Centroids and Centers of Mass The center of mass nonhomogeneous material is always the balancing point The centroid is the balancing point when the region is treated as homogeneous x integral abx f x g x dx A y integralab fx quot2 g x quot2 dx A x height of rec tangle 12 topquot2bottomquot2 OTHER THINGS MIGHT NEED TO KNOW Factoring a cube SOAP Same OppositeAIways Positive 39 aquot3bquot3 abaquot2abbquot2 ii aquot3bquot3 abaquot2abbquot2 iii aquot2bquot2ab ab What is a ratio naIfunctio n Polynom al Polynomial Horizontal Asymp to tes BOBO BO TN Big on Bottom 0 Big on Top none Same exponents fraction or proportion of coef cients Vertical Asymp to tes X Denominator will equal 0 DOMAIN Trig Functions Tan sin cos Cot cossin
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