text stringlengths 1 3.55k | source stringlengths 24 207 | emb listlengths 1.02k 1.02k |
|---|---|---|
closer look at the initial velocity and initial position. in particular, suppose the object is thrown upward from the origin at an angle to the horizontal, with initial speed how can we modify the previous result to reflect this scenario? first, we can assume it is thrown from the origin. if not, then we can move the o... | openstax_calculus_volume_3_-_web | [
0.004649310372769833,
0.038104187697172165,
0.023306727409362793,
-0.03396761044859886,
0.014192154631018639,
0.010068533942103386,
-0.028739161789417267,
0.02145826257765293,
0.025274019688367844,
0.04974988102912903,
0.013023920357227325,
0.031180445104837418,
0.0018116956343874335,
-0.0... |
3. 4 • motion in space 281 figure 3. 17 the flight of a cannonball ( ignoring air resistance ) is projectile motion. solution we use the equation with and ft / sec. then the position equation becomes a. the cannonball reaches its maximum height when the vertical component of its velocity is zero, because the cannonball... | openstax_calculus_volume_3_-_web | [
0.00042411082540638745,
0.04851819574832916,
0.050736501812934875,
0.014043329283595085,
0.03545905277132988,
-0.011671073734760284,
-0.0021740510128438473,
0.013416128233075142,
0.02512214705348015,
0.037965115159749985,
0.01799107901751995,
0.053951676934957504,
0.002125299070030451,
-0.... |
3. 16 an archer fires an arrow at an angle of 40° above the horizontal with an initial speed of 98 m / sec. the height of the archer is 171. 5 cm. find the horizontal distance the arrow travels before it hits the ground. one final question remains : in general, what is the maximum distance a projectile can travel, give... | openstax_calculus_volume_3_-_web | [
0.025050658732652664,
0.05516747385263443,
0.031047634780406952,
0.0021577568259090185,
0.006539019290357828,
0.01552661880850792,
0.00502134021371603,
0.02367103099822998,
0.00751326372846961,
0.06553812325000763,
0.03629256784915924,
0.03085317090153694,
0.014667822979390621,
-0.06114086... |
3. 4 • motion in space 283 resistance, the best angle to fire a projectile ( to maximize the range ) is at a angle. the distance it travels is given by therefore, the range for an angle of is kepler ’ s laws during the early 1600s, johannes kepler was able to use the amazingly accurate data from his mentor tycho brahe ... | openstax_calculus_volume_3_-_web | [
-0.0017644144827499986,
0.060939665883779526,
0.03221617639064789,
-0.006234424188733101,
-0.016923515126109123,
0.03853955492377281,
-0.012038717977702618,
-0.0162555743008852,
0.022272031754255295,
0.08107506483793259,
0.04194440692663193,
0.009846635162830353,
-0.01325206458568573,
-0.0... |
. = 93, 000, 000 mi. since the time it takes for earth to orbit the sun is 1 year, we use earth years for units of time. then, substituting 1 year for the period of earth and 1 a. u. for the average distance to the sun, kepler ’ s third law can be written as for any planet in the solar system, where is the period of th... | openstax_calculus_volume_3_-_web | [
0.0027055558748543262,
0.06423977017402649,
0.036153990775346756,
-0.02840549871325493,
-0.023605169728398323,
0.04258904606103897,
-0.01915370300412178,
-0.025856981053948402,
0.03294961899518967,
0.06398852169513702,
0.04073301702737808,
-0.012018982321023941,
-0.005589685402810574,
-0.0... |
the earth times its acceleration. setting these two forces equal to each other, and using the fact that we obtain which can be rewritten as this equation shows that the vectors and r are parallel to each other, so next, let ’ s differentiate with respect to time : this proves that is a constant vector, which we call c.... | openstax_calculus_volume_3_-_web | [
0.004554897081106901,
0.05475231632590294,
0.02234724909067154,
-0.032594382762908936,
0.009758497588336468,
0.03596692159771919,
-0.010233961045742035,
0.006187682505697012,
0.004754256457090378,
0.04326473921537399,
0.02863023243844509,
0.03375183045864105,
0.0046979160979390144,
-0.0529... |
3. 4 • motion in space 285 however, therefore, equation 3. 29 becomes since c is a constant vector, we can integrate both sides and obtain where d is a constant vector. our goal is to solve for let ’ s start by calculating however, so since we have note that where is the angle between r and d. therefore, solving for wh... | openstax_calculus_volume_3_-_web | [
0.05184445530176163,
0.07422767579555511,
0.03044108860194683,
-0.018153531476855278,
-0.005667317193001509,
0.05399099364876747,
0.016177911311388016,
0.03108127787709236,
0.02256300114095211,
0.05281052738428116,
0.05766196921467781,
0.004337423015385866,
-0.011610524728894234,
-0.066626... |
the period of the moon as well : substitute all the data into equation 3. 30 and solve for a : analysis according to solarsystem. nasa. gov, the actual average distance from the moon to earth is 384, 400 km. this is calculated using reflectors left on the moon by apollo astronauts back in the 1960s. | openstax_calculus_volume_3_-_web | [
0.02379489317536354,
0.06797422468662262,
0.016474224627017975,
-0.0016176265198737383,
-0.03374670818448067,
0.03611943498253822,
-0.004361714236438274,
0.002634161151945591,
0.024819329380989075,
0.06576153635978699,
0.058023955672979355,
0.007978702895343304,
-0.022814221680164337,
-0.0... |
3. 17 titan is the largest moon of saturn. the mass of titan is approximately kg. the mass of saturn is approximately kg. titan takes approximately 16 days to orbit saturn. use this information, along with the universal gravitation constant to estimate the distance from titan to saturn. example 3. 18 chapter opener : h... | openstax_calculus_volume_3_-_web | [
0.018793288618326187,
0.036122120916843414,
0.009248938411474228,
0.013314355164766312,
0.0040721711702644825,
0.04849782958626747,
-0.020009789615869522,
-0.0007640387630090117,
0.008146652020514011,
0.04000922664999962,
0.016163567081093788,
-0.0003132765705231577,
-0.03158381208777428,
... |
3. 4 • motion in space 287 approximately mi. a natural question to ask is : what are the maximum ( aphelion ) and minimum ( perihelion ) distances from halley ’ s comet to the sun? the eccentricity of the orbit of halley ’ s comet is 0. 967 ( source : http : / / nssdc. gsfc. nasa. gov / planetary / factsheet / cometfac... | openstax_calculus_volume_3_-_web | [
0.04312188923358917,
0.03358423337340355,
0.025633158162236214,
-0.013008177280426025,
-0.0070357611402869225,
0.06298106908798218,
-0.030160581693053246,
0.018971556797623634,
0.00887333508580923,
0.041321057826280594,
0.035232506692409515,
-0.03305003419518471,
-0.04040342941880226,
-0.0... |
mathematics and physics necessary for answering questions such as this. a car of mass m moves with constant angular speed around a circular curve of radius r ( figure 3. 20 ). the curve is banked at an angle if the height of the car off the ground is h, then the position of the car at time t is given by the function fi... | openstax_calculus_volume_3_-_web | [
0.006426219828426838,
0.036632686853408813,
0.017964672297239304,
-0.002013831865042448,
-0.014390209689736366,
0.029477249830961227,
0.018866389989852905,
-0.011335302144289017,
-0.0003612737054936588,
0.05939863622188568,
0.06725571304559708,
0.017260467633605003,
-0.00797929335385561,
-... |
##etal force is the next three questions deal with developing a formula that relates the speed to the banking angle 5. show that conclude that 6. the centripetal force is the sum of the forces in the horizontal direction, since the centripetal force points toward the center of the circular curve. show that conclude tha... | openstax_calculus_volume_3_-_web | [
0.00814603827893734,
0.059751611202955246,
0.02143913321197033,
0.023107804358005524,
-0.034279629588127136,
0.03772212192416191,
0.032466549426317215,
-0.0019182580290362239,
-0.0005032846238464117,
0.049626268446445465,
0.0431557223200798,
0.010398058220744133,
-0.01259649358689785,
-0.0... |
3. 4 • motion in space 289 figure 3. 22 at the bristol motor speedway, bristol, tennessee ( a ), the turns have an inner radius of about 211 ft and a width of 40 ft ( b ). ( credit : part ( a ) photo by raniel diaz, flickr ) the coefficient of friction for a normal tire in dry conditions is approximately 0. 7. therefor... | openstax_calculus_volume_3_-_web | [
-0.007005901075899601,
0.06027630344033241,
0.036281708627939224,
0.01976577751338482,
-0.0222745630890131,
0.029507430270314217,
-0.0058775171637535095,
0.023330388590693474,
-0.03071597032248974,
0.06635908037424088,
0.06486639380455017,
0.03845740482211113,
-0.0051009864546358585,
-0.06... |
let find the velocity and acceleration vectors and show that the acceleration is proportional to | openstax_calculus_volume_3_-_web | [
-0.02280665747821331,
0.07976292073726654,
0.0028311964124441147,
-0.008987357839941978,
0.010210479609668255,
0.03357993811368942,
-0.03491237387061119,
0.008608823642134666,
0.027790529653429985,
0.05354723334312439,
-0.017410386353731155,
0.03951533883810043,
0.02831539884209633,
-0.041... |
3. 4 • motion in space 291 consider the motion of a point on the circumference of a rolling circle. as the circle rolls, it generates the cycloid where is the angular velocity of the circle : 165. find the equations for the velocity, acceleration, and speed of the particle at any time. a person on a hang glider is spir... | openstax_calculus_volume_3_-_web | [
0.013669949024915695,
0.04364918917417526,
0.027750277891755104,
0.01471538096666336,
0.01422103401273489,
0.014020767994225025,
-0.03195461630821228,
0.013140990398824215,
0.016322225332260132,
0.08009979873895645,
0.01850472018122673,
0.041431184858083725,
0.0044755032286047935,
-0.02883... |
range of the projectile. 180. a golf ball is hit in a horizontal direction off the top edge of a building that is 100 ft tall. how fast must the ball be launched to land 450 ft away? 181. a projectile is fired from ground level at an angle of 8° with the horizontal. the projectile is to have a range of 50 m. find the m... | openstax_calculus_volume_3_-_web | [
0.010026363655924797,
0.037689514458179474,
-0.015442825853824615,
-0.01931244693696499,
-0.010017197579145432,
0.03054238110780716,
0.003643233794718981,
0.020775161683559418,
0.01232102233916521,
0.08473262190818787,
0.04203581064939499,
0.042552314698696136,
0.013928310945630074,
-0.046... |
3. 4 • motion in space 293 for each of the following problems, find the tangential and normal components of acceleration. 187. the graph is shown here : 188. 189. 190. 191. 192. 193. 194. find the position vector - valued function given that and 195. the force on a particle is given by the particle is located at point ... | openstax_calculus_volume_3_-_web | [
0.00739227095618844,
0.05930142477154732,
0.032263148576021194,
-0.0020782900974154472,
0.0019132632296532393,
0.034322865307331085,
-0.03240540996193886,
0.009011531248688698,
-0.0096285967156291,
0.05967113748192787,
0.02632461115717888,
0.03257900848984718,
-0.015971308574080467,
-0.066... |
3. 4 • motion in space 295 chapter review key terms acceleration vector the second derivative of the position vector arc - length function a function that describes the arc length of curve c as a function of t arc - length parameterization a reparameterization of a vector - valued function in which the parameter is equ... | openstax_calculus_volume_3_-_web | [
-0.009527326561510563,
0.04895348474383354,
0.04392814263701439,
0.006088074762374163,
0.009969438426196575,
0.028055887669324875,
-0.028277285397052765,
-0.03997514769434929,
-0.005596222821623087,
0.06745036691427231,
0.02549024112522602,
-0.00009759705426404253,
-0.015947047621011734,
-... |
of ordered pairs together with their defining parametric equations and principal unit normal vector a vector orthogonal to the unit tangent vector, given by the formula principal unit tangent vector a unit vector tangent to a curve c projectile motion motion of an object with an initial velocity but no force acting on ... | openstax_calculus_volume_3_-_web | [
-0.010136455297470093,
0.03841728717088699,
0.04280724376440048,
-0.0021957862190902233,
-0.002537672407925129,
0.020029891282320023,
-0.03163910657167435,
-0.016733169555664062,
0.00007160638779168949,
0.05447112023830414,
0.013283885084092617,
-0.009222316555678844,
-0.002005724236369133,
... |
a space curve. • it is possible to represent an arbitrary plane curve by a vector - valued function. • to calculate the limit of a vector - valued function, calculate the limits of the component functions separately. | openstax_calculus_volume_3_-_web | [
-0.008922942914068699,
0.017219707369804382,
0.03688700124621391,
-0.0017737788148224354,
0.01781393773853779,
-0.03186081722378731,
0.008945312350988388,
-0.01750616915524006,
0.010929376818239689,
0.056461673229932785,
0.03506040945649147,
-0.03101247362792492,
-0.029305538162589073,
-0.... |
3. 2 calculus of vector - valued functions • to calculate the derivative of a vector - valued function, calculate the derivatives of the component functions, then put them back into a new vector - valued function. • many of the properties of differentiation from the introduction to derivatives ( http : / / openstax. or... | openstax_calculus_volume_3_-_web | [
-0.03669082373380661,
0.045173462480306625,
0.030340656638145447,
-0.028173698112368584,
-0.0047200508415699005,
0.0014047521399334073,
-0.003277418203651905,
-0.017273128032684326,
-0.004574224818497896,
0.06841479986906052,
-0.0009149074903689325,
0.017202964052557945,
0.016693025827407837... |
3. 3 arc length and curvature • the arc - length function for a vector - valued function is calculated using the integral formula this formula is valid in both two and three dimensions. • the curvature of a curve at a point in either two or three dimensions is defined to be the curvature of the inscribed circle at that... | openstax_calculus_volume_3_-_web | [
0.026240043342113495,
0.03005085699260235,
0.0635833591222763,
-0.015076861716806889,
-0.013228519819676876,
0.038381922990083694,
0.01751563511788845,
-0.0280561912804842,
-0.015945889055728912,
0.06154196336865425,
0.05645404011011124,
0.005848508328199387,
-0.02703869715332985,
-0.03607... |
3. 4 motion in space • if represents the position of an object at time t, then represents the velocity and represents the acceleration of the object at time t. the magnitude of the velocity vector is speed. • the acceleration vector always points toward the concave side of the curve defined by the tangential and normal... | openstax_calculus_volume_3_-_web | [
-0.005240441299974918,
0.051049645990133286,
0.03441765904426575,
-0.013337788172066212,
0.010248567909002304,
0.032149363309144974,
-0.0037724520079791546,
0.0005612274399027228,
0.0217641219496727,
0.06538566946983337,
0.005533052142709494,
0.007204111199826002,
0.003312501823529601,
-0.... |
the unit tangent vector, the unit normal vector, and the binormal vector for 223. find the tangential and normal acceleration components with the position vector 224. a ferris wheel car is moving at a constant speed and has a constant radius find the tangential and normal acceleration of the ferris wheel car. 225. the ... | openstax_calculus_volume_3_-_web | [
0.015439318493008614,
0.04647376388311386,
0.01695379428565502,
0.006825708318501711,
-0.0026365008670836687,
0.0176169965416193,
-0.01694735325872898,
0.002778412541374564,
0.004497077781707048,
0.05134586989879608,
0.018476087599992752,
0.032069381326436996,
-0.010395534336566925,
-0.041... |
4. 8 lagrange multipliers introduction in introduction to applications of derivatives ( http : / / openstax. org / books / calculus - volume - 1 / pages / 4 - introduction ), we studied how to determine the maximum and minimum of a function of one variable over a closed interval. this function might represent the tempe... | openstax_calculus_volume_3_-_web | [
0.0227743498980999,
0.05645358934998512,
0.028901465237140656,
0.02719121240079403,
-0.02270745486021042,
-0.009904506616294384,
-0.007749492302536964,
0.01605813205242157,
-0.0032319314777851105,
0.07647812366485596,
0.040231429040431976,
0.02796892449259758,
-0.003619919065386057,
-0.012... |
4. 1 functions of several variables learning objectives 4. 1. 1 recognize a function of two variables and identify its domain and range. 4. 1. 2 sketch a graph of a function of two variables. 4. 1. 3 sketch several traces or level curves of a function of two variables. 4. 1. 4 recognize a function of three or more vari... | openstax_calculus_volume_3_-_web | [
0.01650991477072239,
0.022012529894709587,
0.03236333653330803,
-0.008291101083159447,
-0.03845798969268799,
-0.0052276779897511005,
0.0009213355951942503,
0.029301699250936508,
-0.0034420175943523645,
0.08993858098983765,
0.051866088062524796,
0.023811716586351395,
-0.005607364699244499,
... |
at openstax. org solution to the equation or one such solution can be obtained by first setting which yields the equation the solution to this equation is which gives the ordered pair as a solution to the equation for any value of therefore, the range of the function is all real numbers, or ℝ b. for the function to hav... | openstax_calculus_volume_3_-_web | [
-0.023192163556814194,
0.02301483228802681,
0.03744909539818764,
0.009721124544739723,
-0.02812422811985016,
-0.014860174618661404,
-0.00804303027689457,
0.03193746134638786,
0.01534193754196167,
0.0967743769288063,
0.04723943769931793,
0.024901757016777992,
0.0016375590348616242,
-0.00973... |
4. 1 find the domain and range of the function graphing functions of two variables suppose we wish to graph the function this function has two independent variables and one dependent variable when graphing a function of one variable, we use the cartesian plane. we are able to graph any ordered pair in the plane, and ev... | openstax_calculus_volume_3_-_web | [
-0.00993349403142929,
0.02333638072013855,
0.03642095625400543,
-0.02414526976644993,
-0.029299519956111908,
-0.009295213036239147,
0.002618937287479639,
0.019614579156041145,
-0.008475102484226227,
0.08835456520318985,
0.03748214989900589,
0.01965765841305256,
-0.017412757501006126,
0.006... |
the given function of two variables. b. this function also contains the expression setting this expression equal to various values starting at zero, we obtain circles of increasing radius. the minimum value of is zero ( attained when when the function becomes and when then the function becomes these are cross - section... | openstax_calculus_volume_3_-_web | [
0.0025423974730074406,
0.01815222203731537,
0.07690173387527466,
-0.02347354032099247,
-0.0009126380318775773,
-0.0029841854702681303,
-0.004298295360058546,
0.017733179032802582,
-0.013024457730352879,
0.061868369579315186,
0.06082626059651375,
0.03957902640104294,
-0.017830008640885353,
... |
4. 1 • functions of several variables 305 figure 4. 5 a paraboloid is the graph of the given function of two variables. example 4. 3 nuts and bolts a profit function for a hardware manufacturer is given by where is the number of nuts sold per month ( measured in thousands ) and represents the number of bolts sold per m... | openstax_calculus_volume_3_-_web | [
0.031366150826215744,
0.032441094517707825,
0.035165540874004364,
-0.013724627904593945,
-0.00533689372241497,
0.000251057674176991,
0.006799993570894003,
0.02830098755657673,
0.0024227637331932783,
0.07998023927211761,
0.04866421967744827,
0.051046792417764664,
0.00909150019288063,
-0.023... |
a perspective photo of devil ’ s tower shows just how steep its sides are. notice the top of the tower has the same shape as the center of the topographical map. | openstax_calculus_volume_3_-_web | [
-0.00521842110902071,
-0.011735063046216965,
0.019578641280531883,
-0.02705269679427147,
0.02393614687025547,
0.04017023369669914,
-0.03236622363328934,
0.010425825603306293,
0.016957296058535576,
0.020500369369983673,
0.05000678077340126,
-0.012852613814175129,
0.02414061687886715,
-0.037... |
4. 1 • functions of several variables 307 definition given a function and a number in the range of level curve of a function of two variables for the value is defined to be the set of points satisfying the equation returning to the function we can determine the level curves of this function. the range of is the closed ... | openstax_calculus_volume_3_-_web | [
-0.00033897292451001704,
0.0044073802419006824,
0.02880631387233734,
0.019508248195052147,
-0.015278415754437447,
-0.011216886341571808,
0.004065528512001038,
0.02099752426147461,
-0.02105703577399254,
0.10741778463125229,
0.02843828871846199,
0.028824305161833763,
-0.013048222288489342,
-... |
: next, we factor the left - hand side and simplify the right - hand side : last, we divide both sides by this equation describes an ellipse centered at the graph of this ellipse appears in the following graph. figure 4. 9 level curve of the function corresponding to we can repeat the same derivation for values of less... | openstax_calculus_volume_3_-_web | [
0.019148854538798332,
0.023310348391532898,
0.0017045876011252403,
0.011901985853910446,
-0.023075483739376068,
-0.03718772158026695,
0.00044577167136594653,
0.02822537161409855,
-0.03919878974556923,
0.08939994871616364,
0.021722067147493362,
0.02330099418759346,
0.011002723127603531,
-0.... |
4. 2 find and graph the level curve of the function corresponding to another useful tool for understanding the graph of a function of two variables is called a vertical trace. level curves are always graphed in the but as their name implies, vertical traces are graphed in the - or definition consider a function with do... | openstax_calculus_volume_3_-_web | [
0.026689959689974785,
0.01919739507138729,
0.018493713811039925,
0.005178547464311123,
-0.00989574659615755,
-0.031185250729322433,
-0.008261846378445625,
0.008956601843237877,
0.0004373221891000867,
0.07682858407497406,
0.03978768736124039,
0.057663362473249435,
0.0032536883372813463,
-0.... |
4. 1 • functions of several variables 311 corresponding to and describe its graph. functions of two variables can produce some striking - looking surfaces. the following figure shows two examples. figure 4. 12 examples of surfaces representing functions of two variables : ( a ) a combination of a power function and a s... | openstax_calculus_volume_3_-_web | [
0.0014219582080841064,
0.02346763201057911,
0.04989324510097504,
-0.022431809455156326,
-0.013959711417555809,
-0.008517313748598099,
0.005154286045581102,
0.01320790033787489,
-0.002110318746417761,
0.08408200740814209,
0.06192457303404808,
0.0027269632555544376,
0.0034732110798358917,
0.... |
. the denominator cannot be zero. since the radicand cannot be negative, this implies and therefore that since the denominator cannot be zero, or which can be rewritten as, which are the equations of two lines passing through the origin. therefore, the domain of is 4. 4 find the domain of the function functions of two ... | openstax_calculus_volume_3_-_web | [
-0.016049349680542946,
0.020068729296326637,
0.031211160123348236,
-0.023093121126294136,
0.0005557917174883187,
-0.027066802605986595,
0.04290672019124031,
0.01204165443778038,
-0.03162636235356331,
0.07914435863494873,
0.007659998256713152,
0.0274643674492836,
0.002089393325150013,
-0.00... |
4. 5 find the equation of the level surface of the function corresponding to and describe the surface, if possible. 314 4 • differentiation of functions of several variables access for free at openstax. org section 4. 1 exercises for the following exercises, evaluate each function at the indicated values. 1. find 2. fi... | openstax_calculus_volume_3_-_web | [
0.010729560628533363,
0.02033340185880661,
0.05468851700425148,
0.0051711988635361195,
-0.02777678705751896,
-0.008101307787001133,
0.0062344674952328205,
0.025649605318903923,
-0.011308121494948864,
0.09246961027383804,
0.05596372112631798,
0.04969535395503044,
-0.017707407474517822,
-0.0... |
4. 1 • functions of several variables 315 find the domain of the following functions. 33. 34. 35. 36. 37. 38. for the following exercises, plot a graph of the function. 39. 40. 41. use technology to graph sketch the following by finding the level curves. verify the graph using technology. 42. 43. 44. 45. 46. 47. descri... | openstax_calculus_volume_3_-_web | [
0.008617449551820755,
0.036121830344200134,
0.026458799839019775,
-0.015318045392632484,
-0.01844228059053421,
0.009153989143669605,
0.01967996172606945,
0.0068014441058039665,
-0.009854290634393692,
0.07875789701938629,
0.01347844023257494,
0.023561622947454453,
0.02206847257912159,
-0.05... |
4. 2 limits and continuity learning objectives 4. 2. 1 calculate the limit of a function of two variables. 4. 2. 2 learn how a function of two variables can approach different values at a boundary point, depending on the path of approach. 4. 2. 3 state the conditions for continuity of a function of two variables. 4. 2.... | openstax_calculus_volume_3_-_web | [
0.004089245572686195,
0.0339902900159359,
0.06482388079166412,
0.030867593362927437,
-0.03701096028089523,
-0.024073883891105652,
0.008189613930881023,
0.012504631653428078,
-0.003621133044362068,
0.09025827795267105,
0.043488506227731705,
0.029053084552288055,
-0.012763332575559616,
0.003... |
idea of a disk appears in the definition of the limit of a function of two variables. if is small, then all the points in the disk are close to this is completely analogous to being close to in the definition of a limit of a function of one variable. in one dimension, we express this restriction as in more than one dim... | openstax_calculus_volume_3_-_web | [
-0.005666193086653948,
0.008245640434324741,
0.0692409947514534,
0.006638783495873213,
-0.0214256402105093,
-0.01978069543838501,
0.03684129938483238,
-0.0027050201315432787,
-0.003963278140872717,
0.07620430737733841,
0.05882372334599495,
0.03405504301190376,
-0.016094844788312912,
0.0070... |
4. 2 • limits and continuity 317 figure 4. 15 the limit of a function involving two variables requires that be within of whenever is within of the smaller the value of the smaller the value of proving that a limit exists using the definition of a limit of a function of two variables can be challenging. instead, we use ... | openstax_calculus_volume_3_-_web | [
0.004075381439179182,
0.043968770653009415,
0.065860316157341,
0.02864472009241581,
-0.013765382580459118,
-0.02265232242643833,
-0.0011563000734895468,
0.006235617212951183,
0.00972008891403675,
0.11179933696985245,
0.03503118082880974,
0.019745314493775368,
-0.0014429251896217465,
-0.030... |
law on the second limit : ( 4. 7 ) ( 4. 8 ) ( 4. 9 ) ( 4. 10 ) ( 4. 11 ) 4. 2 • limits and continuity 319 last, use the identity laws on the first six limits and the constant law on the last limit : b. before applying the quotient law, we need to verify that the limit of the denominator is nonzero. using the difference... | openstax_calculus_volume_3_-_web | [
-0.00417051138356328,
0.012705798260867596,
0.01881822571158409,
0.003035969566553831,
-0.01969069428741932,
-0.045622505247592926,
-0.005139511544257402,
-0.027143850922584534,
-0.004643544554710388,
0.09767182916402817,
0.04824850335717201,
0.02880830317735672,
0.03596695140004158,
-0.00... |
4. 6 evaluate the following limit : since we are taking the limit of a function of two variables, the point is in ℝ and it is possible to approach this point from an infinite number of directions. sometimes when calculating a limit, the answer varies depending on the path taken toward if this is the case, then the limi... | openstax_calculus_volume_3_-_web | [
0.0011972172651439905,
0.033752668648958206,
0.031998977065086365,
0.004327184520661831,
-0.018887070938944817,
-0.0028329913038760424,
0.008175824768841267,
0.02518559992313385,
0.01162426546216011,
0.11286820471286774,
0.05356159061193466,
0.02395780198276043,
-0.02876846306025982,
0.004... |
fixed at zero. the same is true for the suppose we approach the origin along a straight line of slope the equation of this line is then the limit becomes 4. 2 • limits and continuity 321 regardless of the value of it would seem that the limit is equal to zero. what if we chose a curve passing through the origin instead... | openstax_calculus_volume_3_-_web | [
-0.009142034687101841,
0.018089834600687027,
0.08120840787887573,
-0.017121126875281334,
0.001843336271122098,
-0.011505947448313236,
0.01839357055723667,
0.010026304982602596,
-0.008888944052159786,
0.0874791070818901,
0.0685751885175705,
-0.007460008841007948,
-0.022366326302289963,
0.00... |
4. 7 show that does not exist. interior points and boundary points to study continuity and differentiability of a function of two or more variables, we first need to learn some new terminology. definition let s be a subset of ℝ ( figure 4. 17 ). 1. a point is called an interior point of if there is a disk centered arou... | openstax_calculus_volume_3_-_web | [
-0.017768075689673424,
0.01806829683482647,
0.04348453879356384,
0.0064586251974105835,
-0.018224066123366356,
-0.03129023686051369,
0.002815570216625929,
0.0539964996278286,
-0.020659809932112694,
0.07853595167398453,
0.044863276183605194,
0.033090852200984955,
0.019593019038438797,
0.005... |
need only consider points that are inside both the disk and the domain of the function. this leads to the definition of the limit of a function at a boundary point. definition let be a function of two variables, and and suppose is on the boundary of the domain of then, the limit of as approaches is written if for any t... | openstax_calculus_volume_3_-_web | [
-0.019641883671283722,
0.03656618297100067,
0.04441678524017334,
0.006546972319483757,
-0.027274759486317635,
-0.02443867363035679,
0.011343545280396938,
0.030027462169528008,
-0.005981700960546732,
0.10241999477148056,
0.018461408093571663,
0.02347869612276554,
-0.010699569247663021,
0.01... |
4. 8 evaluate the following limit : continuity of functions of two variables in continuity ( http : / / openstax. org / books / calculus - volume - 1 / pages / 2 - 4 - continuity ), we defined the continuity of a function of one variable and saw how it relied on the limit of a function of one variable. in particular, t... | openstax_calculus_volume_3_-_web | [
-0.004416752140969038,
0.0181343425065279,
0.03157757222652435,
0.01429003942757845,
-0.028157243505120277,
-0.029215985909104347,
-0.0031521341297775507,
0.017029359936714172,
-0.0031851257663220167,
0.09772282093763351,
0.042879994958639145,
0.0016020730836316943,
0.006207616068422794,
0... |
4. 2 • limits and continuity 325 theorem 4. 3 the product of continuous functions is continuous if is continuous at and is continuous at then is continuous at theorem 4. 4 the composition of continuous functions is continuous let be a function of two variables from a domain ℝ to a range ℝsuppose is continuous at some p... | openstax_calculus_volume_3_-_web | [
-0.013627255335450172,
-0.0002239517925772816,
0.047238681465387344,
0.01398933120071888,
0.0026336233131587505,
-0.021622003987431526,
-0.03319681063294411,
-0.0008934199577197433,
-0.016030604019761086,
0.10446275025606155,
0.059848006814718246,
0.009437683038413525,
0.022211208939552307,
... |
4. 10 show that the functions and are continuous everywhere. functions of three or more variables the limit of a function of three or more variables occurs readily in applications. for example, suppose we have a function that gives the temperature at a physical location in three dimensions. or perhaps a function can in... | openstax_calculus_volume_3_-_web | [
0.007326236926019192,
0.03930282965302467,
0.05069826543331146,
0.010594546794891357,
-0.007207244168967009,
-0.015334030613303185,
0.011713732033967972,
-0.011081903241574764,
0.001760362763889134,
0.09348482638597488,
0.051648057997226715,
0.032069507986307144,
0.00390320154838264,
-0.01... |
law, difference law, constant multiple law, and identity law, last, applying the quotient law : 4. 11 find | openstax_calculus_volume_3_-_web | [
-0.00498242536559701,
0.029735377058386803,
-0.013702599331736565,
0.011743251234292984,
-0.02839747630059719,
-0.029894068837165833,
-0.013182512484490871,
-0.023420067504048347,
0.015440395101904869,
0.1152821034193039,
0.04271290451288223,
0.02353634126484394,
0.036202527582645416,
-0.0... |
4. 2 • limits and continuity 327 section 4. 2 exercises for the following exercises, find the limit of the function. 60. 61. 62. show that the limit exists and is the same along the paths : and and along for the following exercises, evaluate the limits at the indicated values of if the limit does not exist, state this ... | openstax_calculus_volume_3_-_web | [
0.002065533772110939,
0.026840778067708015,
0.02990568056702614,
0.021976780146360397,
-0.011037918739020824,
-0.014132886193692684,
-0.019125087186694145,
0.01554252952337265,
-0.024032719433307648,
0.1097903922200203,
0.048813723027706146,
0.031102051958441734,
0.01038178987801075,
-0.03... |
paths. ) 96. 97. determine whether is continuous at 98. create a plot using graphing software to determine where the limit does not exist. find where in the coordinate plane is continuous. 99. determine the region of the in which the function is continuous. use technology to support your conclusion. 100. determine the ... | openstax_calculus_volume_3_-_web | [
0.020526248961687088,
0.03576439991593361,
0.03328092396259308,
-0.005472404416650534,
-0.0008770345011726022,
-0.019625205546617508,
0.014477631077170372,
0.005227195098996162,
-0.009050941094756126,
0.10819531977176666,
0.020593218505382538,
0.02531258761882782,
0.015271790325641632,
-0.... |
4. 3 partial derivatives learning objectives 4. 3. 1 calculate the partial derivatives of a function of two variables. 4. 3. 2 calculate the partial derivatives of a function of more than two variables. 4. 3. 3 determine the higher - order derivatives of a function of two variables. 4. 3. 4 explain the meaning of a par... | openstax_calculus_volume_3_-_web | [
-0.01978759840130806,
0.022580601274967194,
0.04776696488261223,
0.009705163538455963,
-0.02139517478644848,
0.004734298679977655,
0.04385728761553764,
0.01861736923456192,
-0.0003599167102947831,
0.07156343013048172,
0.0396137610077858,
0.017509443685412407,
-0.01697065494954586,
0.016928... |
. second, we now have two different derivatives we can take, since there are two different independent variables. depending on which variable we choose, we can come up with different partial derivatives altogether, and often do. example 4. 14 calculating partial derivatives from the definition use the definition of the... | openstax_calculus_volume_3_-_web | [
-0.015559124760329723,
0.0366528257727623,
0.026056984439492226,
0.00500219501554966,
-0.015407443046569824,
-0.013051980175077915,
0.028543243184685707,
0.015779845416545868,
0.009716285392642021,
0.07304737716913223,
0.05141058191657066,
0.013724558986723423,
-0.0022746422328054905,
-0.0... |
4. 3 • partial derivatives 331 all differentiation rules from introduction to derivatives ( http : / / openstax. org / books / calculus - volume - 1 / pages / 3 - introduction ) apply. example 4. 15 calculating partial derivatives calculate and for the following functions by holding the opposite variable constant then ... | openstax_calculus_volume_3_-_web | [
0.009619222022593021,
0.034175317734479904,
0.04201239347457886,
-0.016718663275241852,
0.005709890276193619,
-0.02391660027205944,
0.01549494918435812,
0.038937173783779144,
-0.01918153092265129,
0.08874473720788956,
0.03795705735683441,
0.025736050680279732,
0.0029361462220549583,
-0.014... |
4. 13 calculate and for the function by holding the opposite variable constant, then differentiating. how can we interpret these partial derivatives? recall that the graph of a function of two variables is a surface in ℝ if we remove the limit from the definition of the partial derivative with respect to the difference... | openstax_calculus_volume_3_-_web | [
0.002533341059461236,
0.06059962138533592,
0.0013829958625137806,
-0.0017976189265027642,
-0.009258286096155643,
-0.005714320112019777,
0.03047391213476658,
0.007533524185419083,
-0.0046929409727454185,
0.09415233135223389,
0.013473473489284515,
0.020130468532443047,
0.014661983586847782,
... |
4. 3 • partial derivatives 333 figure 4. 22 contour map for the function using and corresponds to the origin ). the inner circle on the contour map corresponds to and the next circle out corresponds to the first circle is given by the equation the second circle is given by the equation the first equation simplifies to ... | openstax_calculus_volume_3_-_web | [
-0.005184974521398544,
0.045334313064813614,
0.037932656705379486,
-0.0388290099799633,
-0.002574061043560505,
-0.008473468013107777,
-0.021797241643071175,
0.01982399821281433,
-0.012885087169706821,
0.07936229556798935,
0.043096236884593964,
0.020994001999497414,
-0.00025132251903414726,
... |
4. 14 use a contour map to estimate at point for the function compare this with the exact answer. functions of more than two variables suppose we have a function of three variables, such as we can calculate partial derivatives of with respect to any of the independent variables, simply as extensions of the definitions ... | openstax_calculus_volume_3_-_web | [
0.022092537954449654,
0.03322012722492218,
0.0411868691444397,
0.0033348717261105776,
-0.017151599749922752,
-0.009188623167574406,
0.0464479997754097,
0.03797108680009842,
0.013256668113172054,
0.07458403706550598,
0.028387486934661865,
0.009206736460328102,
-0.006918294355273247,
-0.0251... |
derivatives to calculate for the function then find and by setting the other two variables constant and differentiating accordingly. example 4. 18 calculating partial derivatives for a function of three variables calculate the three partial derivatives of the following functions. a. b. solution in each case, treat all ... | openstax_calculus_volume_3_-_web | [
0.003136415034532547,
0.03403840586543083,
0.03800010308623314,
-0.028102852404117584,
-0.014881712384521961,
-0.008613456971943378,
0.03234190493822098,
0.03403247147798538,
0.002602162305265665,
0.056796785444021225,
0.041051123291254044,
0.018387412652373314,
0.001560709555633366,
-0.02... |
4. 16 calculate and for the function higher - order partial derivatives consider the function its partial derivatives are each of these partial derivatives is a function of two variables, so we can calculate partial derivatives of these functions. just as with derivatives of single - variable functions, we can call the... | openstax_calculus_volume_3_-_web | [
0.00008767913823248819,
0.020198127254843712,
0.002838663524016738,
-0.008675424382090569,
-0.02055937983095646,
-0.012211309745907784,
0.04679626598954201,
0.032344672828912735,
0.008511059917509556,
0.05770610272884369,
0.037754494696855545,
0.021357756108045578,
0.0027684017550200224,
-... |
4. 17 calculate all four second partial derivatives for the function at this point we should notice that, in both example 4. 19 and the checkpoint, it was true that under certain conditions, this is always true. in fact, it is a direct consequence of the following theorem. theorem 4. 5 equality of mixed partial derivat... | openstax_calculus_volume_3_-_web | [
-0.014919896610081196,
0.035633958876132965,
0.02098964713513851,
0.007096962071955204,
-0.027437284588813782,
-0.010713363066315651,
0.013778189197182655,
0.025985490530729294,
0.0009140868787653744,
0.09830128401517868,
0.040609780699014664,
0.031361114233732224,
0.020472832024097443,
-0... |
solution to the wave equation verify that is a solution to the wave equation solution first, we calculate and next, we substitute each of these into the right - hand side of equation 4. 20 and simplify : this verifies the solution. ( 4. 17 ) ( 4. 18 ) ( 4. 19 ) ( 4. 20 ) 4. 3 • partial derivatives 339 | openstax_calculus_volume_3_-_web | [
0.04293171316385269,
-0.000800643814727664,
0.020607810467481613,
-0.006427441257983446,
-0.004448005463927984,
0.021843796595931053,
0.01321863941848278,
0.028828619047999382,
0.013549028895795345,
0.05615580081939697,
0.02785845845937729,
0.0334649533033371,
0.018533295020461082,
0.01238... |
4. 18 verify that is a solution to the heat equation since the solution to the two - dimensional heat equation is a function of three variables, it is not easy to create a visual representation of the solution. we can graph the solution for fixed values of t, which amounts to snapshots of the heat distributions at fixe... | openstax_calculus_volume_3_-_web | [
0.03282708302140236,
0.05021325498819351,
0.061246324330568314,
-0.004392375238239765,
-0.005770114250481129,
0.021907633170485497,
-0.014273446053266525,
-0.01895601861178875,
0.011732051149010658,
0.08521012216806412,
0.03509311005473137,
0.016636623069643974,
0.020708071067929268,
-0.00... |
the same time, charles darwin had published his treatise on evolution. darwin ’ s view was that evolution needed many millions of years to take place, and he made a bold claim that the | openstax_calculus_volume_3_-_web | [
0.003139581298455596,
-0.005424797534942627,
0.03509253263473511,
-0.002916676690801978,
-0.0008128523477353156,
0.05396537482738495,
-0.007454370614141226,
-0.01376246102154255,
0.02510480210185051,
0.030934933573007584,
0.01904968172311783,
-0.022078104317188263,
-0.003679914865642786,
-... |
4. 3 • partial derivatives 341 weald chalk fields, where important fossils were found, were the result of million years of erosion. at that time, eminent physicist william thomson ( lord kelvin ) used an important partial differential equation, known as the heat diffusion equation, to estimate the age of earth by deter... | openstax_calculus_volume_3_-_web | [
0.023430895060300827,
0.04510701820254326,
0.054432809352874756,
0.013676175847649574,
-0.003013866487890482,
0.04540698230266571,
-0.0027353407349437475,
0.004231948871165514,
-0.00003090625978074968,
0.047577112913131714,
0.05096174776554108,
-0.0050263116136193275,
0.040916938334703445,
... |
functions containing each variable separately. in this case, we would write the temperature as 1. substitute this form into equation 4. 13 and, noting that is constant with respect to distance and is constant with respect to time show that 2. this equation represents the separation of variables we want. the left - hand... | openstax_calculus_volume_3_-_web | [
0.02980826050043106,
0.06113522872328758,
0.047542788088321686,
-0.019776834174990654,
-0.04431964084506035,
0.019906792789697647,
0.01651165448129177,
0.010807165876030922,
-0.02139105647802353,
0.07888583838939667,
0.039738740772008896,
0.027728406712412834,
0.01947634108364582,
-0.03972... |
we assume that all of earth was at an initial hot temperature ( kelvin took this to be about the application of this boundary condition involves the more advanced application of fourier coefficients. as noted in part b. each value of represents a valid solution, and the general solution is a sum of all these solutions.... | openstax_calculus_volume_3_-_web | [
0.01999504305422306,
0.0555478073656559,
0.046834252774715424,
-0.020133057609200478,
-0.02790917456150055,
0.016275253146886826,
-0.0015926826745271683,
-0.000658985401969403,
-0.005806603468954563,
0.04040506109595299,
0.0452209897339344,
-0.0037156296893954277,
0.037482935935258865,
-0.... |
4. 3 • partial derivatives 343 in for trouble at the last part of my speech dealing with the age of the earth, where my views conflicted with his. to my relief, kelvin fell fast asleep, but as i came to the important point, i saw the old bird sit up, open an eye and cock a baleful glance at me. then a sudden inspiratio... | openstax_calculus_volume_3_-_web | [
0.02469056285917759,
0.04829563945531845,
0.046602729707956314,
0.005089849233627319,
-0.024780822917819023,
0.019598519429564476,
0.014856374822556973,
0.0038351111579686403,
-0.006858875043690205,
0.051984868943691254,
0.07689496129751205,
-0.014763168059289455,
0.022276999428868294,
-0.... |
show that the rate of change of the volume of the cylinder with respect to its radius is the product of its circumference multiplied by its height. c. show that the rate of change of the volume of the cylinder with respect to its height is equal to the area of the circular base. 134. calculate for find the indicated hi... | openstax_calculus_volume_3_-_web | [
0.01311919279396534,
0.01683790422976017,
0.08571065217256546,
-0.006577420514076948,
0.008940710686147213,
-0.00311654363758862,
-0.017532849684357643,
0.018689097836613655,
-0.005102895200252533,
0.08698739111423492,
0.07510548830032349,
0.03351319953799248,
-0.014654726721346378,
-0.015... |
4. 3 • partial derivatives 345 145. given find all points at which simultaneously. 146. given find all points at which and simultaneously. 147. given find all points on at which simultaneously. 148. given find all points at which simultaneously. 149. show that satisfies the equation 150. show that solves laplace ’ s eq... | openstax_calculus_volume_3_-_web | [
0.034090470522642136,
0.04131929948925972,
0.036127589643001556,
0.008737102150917053,
-0.01886584423482418,
-0.007840822450816631,
0.0035375538282096386,
0.01641950011253357,
0.023300351575016975,
0.07673729956150055,
0.05738421157002449,
0.055675238370895386,
0.02755660191178322,
-0.0329... |
4. 4 tangent planes and linear approximations learning objectives 4. 4. 1 determine the equation of a plane tangent to a given surface at a point. 4. 4. 2 use the tangent plane to approximate a function of two variables at a point. 4. 4. 3 explain when a function of two variables is differentiable. 4. 4. 4 use the tota... | openstax_calculus_volume_3_-_web | [
0.021979009732604027,
0.04560530185699463,
0.02597348764538765,
-0.017252448946237564,
-0.010607504285871983,
0.0019475651206448674,
0.0064772251062095165,
0.011983920820057392,
0.027977369725704193,
0.06415973603725433,
0.0011561205610632896,
0.006163205020129681,
0.005195780657231808,
0.... |
4. 4 • tangent planes and linear approximations 347 figure 4. 27 the tangent plane to a surface at a point contains all the tangent lines to curves in that pass through for a tangent plane to a surface to exist at a point on that surface, it is sufficient for the function that defines the surface to be differentiable a... | openstax_calculus_volume_3_-_web | [
0.010644877329468727,
0.03292430192232132,
0.03818106651306152,
-0.016701487824320793,
0.013634661212563515,
-0.0046506887301802635,
0.00005049271203461103,
0.016235169023275375,
0.026952089741826057,
0.05632922053337097,
0.004542620852589607,
0.025253834202885628,
-0.00023716130817774683,
... |
4. 4 • tangent planes and linear approximations 349 example 4. 22 finding another tangent plane find the equation of the tangent plane to the surface defined by the function at the point solution first, calculate and then use equation 4. 24 with and then equation 4. 24 becomes a tangent plane to a surface does not alwa... | openstax_calculus_volume_3_-_web | [
0.006879337131977081,
0.03635379672050476,
0.03292030841112137,
-0.013692440465092659,
0.008963553234934807,
-0.0194475706666708,
-0.006894668564200401,
0.022797269746661186,
0.028474004939198494,
0.0686354860663414,
0.03742024302482605,
0.019085902720689774,
-0.004640656989067793,
0.01185... |
tangent line can be used as an approximation to the function for values of reasonably close to when working with a function of two variables, the tangent line is replaced by a tangent plane, but the approximation idea is much the same. definition given a function with continuous partial derivatives that exist at the po... | openstax_calculus_volume_3_-_web | [
0.012165690772235394,
0.023732416331768036,
0.038813065737485886,
0.004628090187907219,
-0.024741966277360916,
0.0005138293490745127,
0.006743998732417822,
0.010010740719735622,
0.04180686175823212,
0.0678587406873703,
0.028164369985461235,
-0.006728374864906073,
-0.0009072131360881031,
0.... |
4. 20 given the function approximate using point for what is the approximate value of to four decimal places? differentiability when working with a function of one variable, the function is said to be differentiable at a point if exists. furthermore, if a function of one variable is differentiable at a point, the graph... | openstax_calculus_volume_3_-_web | [
0.008216873742640018,
0.03792203217744827,
0.039995890110731125,
-0.007744703907519579,
-0.017308015376329422,
-0.010914544574916363,
0.023021642118692398,
0.004502827767282724,
0.011301656253635883,
0.059715624898672104,
0.04099322855472565,
0.004504329524934292,
0.010733414441347122,
0.0... |
##s 353 next, we calculate since for any value of the original limit must be equal to zero. therefore, is differentiable at point | openstax_calculus_volume_3_-_web | [
-0.0198393352329731,
0.06810092180967331,
0.019324351102113724,
0.007505799178034067,
0.0008966567693278193,
-0.020803628489375114,
0.005915431771427393,
0.032702572643756866,
0.0007537289639003575,
0.06651908159255981,
0.031168844550848007,
0.03242266923189163,
0.007038998417556286,
-0.00... |
4. 21 show that the function is differentiable at point the function is not differentiable at the origin. we can see this by calculating the partial derivatives. this function appeared earlier in the section, where we showed that substituting this information into equation 4. 26 using and we get calculating gives depen... | openstax_calculus_volume_3_-_web | [
-0.027237122878432274,
0.021992124617099762,
0.03440062329173088,
-0.01959953084588051,
0.014701739884912968,
-0.03236519172787666,
0.0038204011507332325,
0.009258588775992393,
-0.009400510229170322,
0.08369060605764389,
0.04212966933846474,
0.01752542145550251,
0.02317808009684086,
0.0040... |
4. 4 • tangent planes and linear approximations 355 was not differentiable at the origin. let ’ s calculate the partial derivatives and the contrapositive of the preceding theorem states that if a function is not differentiable, then at least one of the hypotheses must be false. let ’ s explore the condition that must ... | openstax_calculus_volume_3_-_web | [
-0.0072114476934075356,
0.005643444135785103,
0.033030007034540176,
0.018557630479335785,
0.0005053284112364054,
-0.015126013197004795,
0.004854933824390173,
0.0068121859803795815,
0.012421920895576477,
0.08754003047943115,
0.020975179970264435,
-0.0009829303016886115,
0.012972268275916576,
... |
quantity, the differential can be used to estimate the error in the total volume of the gadget. example 4. 25 approximation by differentials find the differential of the function and use it to approximate at point use and what is the exact value of solution first, we must calculate using and then, we substitute these q... | openstax_calculus_volume_3_-_web | [
-0.0022330363281071186,
0.02989458665251732,
0.05692390725016594,
0.011107527650892735,
-0.05208370462059975,
-0.028948765248060226,
-0.0004122356476727873,
-0.00016303971642628312,
0.02088267169892788,
0.07478486746549606,
0.031409334391355515,
0.002113166032359004,
0.003364189062267542,
... |
4. 4 • tangent planes and linear approximations 357 definition a function is differentiable at a point if for all points in a disk around we can write where the error term e satisfies if a function of three variables is differentiable at a point then it is continuous there. furthermore, continuity of first partial deri... | openstax_calculus_volume_3_-_web | [
0.007005168590694666,
0.05084233731031418,
0.04920559749007225,
0.034825023263692856,
0.010353589430451393,
-0.001972778467461467,
0.0001554409827804193,
0.00648741889744997,
0.0007111412123776972,
0.08223771303892136,
0.033351052552461624,
0.04752161726355553,
0.0010641497792676091,
-0.01... |
approaches 194. find the total differential of the function where changes from and changes from 195. let compute from to and then find the approximate change in from point to point recall and and are approximately equal. 196. the volume of a right circular cylinder is given by find the differential interpret the formul... | openstax_calculus_volume_3_-_web | [
0.02252235636115074,
0.012216364033520222,
0.04682635888457298,
0.012589693069458008,
-0.01773427054286003,
-0.016507618129253387,
-0.005126661621034145,
0.009124943055212498,
-0.000008810983672447037,
0.1281757652759552,
0.07276056706905365,
0.007353499531745911,
-0.03442159667611122,
-0.... |
4. 4 • tangent planes and linear approximations 359 198. use the differential to approximate the change in as moves from point to point compare this approximation with the actual change in the function. 199. let find the exact change in the function and the approximate change in the function as changes from and changes... | openstax_calculus_volume_3_-_web | [
0.013085272163152695,
0.026857873424887657,
0.007545252330601215,
0.015561796724796295,
-0.004832223057746887,
0.009170113131403923,
0.010901357047259808,
-0.01119094155728817,
0.017327608540654182,
0.08952555060386658,
0.0316205732524395,
0.025152575224637985,
0.008873018436133862,
-0.002... |
. use differentials to approximate the maximum percentage error in the calculated value of 206. electrical power is given by where is the voltage and is the resistance. approximate the maximum percentage error in calculating power if is applied to a resistor and the possible percent errors in measuring and are and resp... | openstax_calculus_volume_3_-_web | [
-0.0061353896744549274,
0.026671001687645912,
0.0011201222660019994,
0.005052410531789064,
-0.005805335007607937,
-0.009676637127995491,
0.03341681882739067,
0.0016624779673293233,
0.01987585611641407,
0.06255672127008438,
0.00022504988010041416,
0.03915979340672493,
0.02424044907093048,
-... |
4. 5 the chain rule learning objectives 4. 5. 1 state the chain rules for one or two independent variables. 4. 5. 2 use tree diagrams as an aid to understanding the chain rule for several independent and intermediate variables. 4. 5. 3 perform implicit differentiation of a function of two or more variables. in single -... | openstax_calculus_volume_3_-_web | [
-0.011586055159568787,
0.04363081976771355,
0.03286748379468918,
-0.0015625037485733628,
-0.018440790474414825,
-0.003100571222603321,
-0.012270228005945683,
0.0014777459437027574,
0.009740696288645267,
0.09750935435295105,
0.051643867045640945,
0.014307484962046146,
-0.006884825881570578,
... |
4. 5 • the chain rule 361 next, we divide both sides by then we take the limit as approaches the left - hand side of this equation is equal to which leads to the last term can be rewritten as as approaches approaches so we can rewrite the last product as since the first limit is equal to zero, we need only show that th... | openstax_calculus_volume_3_-_web | [
0.0062281000427901745,
0.03218436986207962,
0.03293607011437416,
-0.013296611607074738,
-0.014082430861890316,
-0.00735150882974267,
-0.010020153596997261,
0.007582498714327812,
-0.007094365078955889,
0.09925591945648193,
0.05429532378911972,
0.02735973708331585,
-0.003915437962859869,
-0.... |
chain rule, we again need four quantities — and we substitute each of these into equation 4. 29 : 4. 5 • the chain rule 363 to reduce this to one variable, we use the fact that and therefore, to eliminate negative exponents, we multiply the top by and the bottom by again, this derivative can also be calculated by first... | openstax_calculus_volume_3_-_web | [
0.012061047367751598,
0.05438642576336861,
0.014583391137421131,
-0.020633984357118607,
0.008015288971364498,
-0.0049948785454034805,
-0.003912155982106924,
0.03880612924695015,
-0.009326810948550701,
0.08115395158529282,
0.06049779802560806,
-0.010085299611091614,
-0.02889235131442547,
-0... |
4. 23 calculate given the following functions. express the final answer in terms of it is often useful to create a visual representation of equation 4. 29 for the chain rule. this is called a tree diagram for the chain rule for functions of one variable and it provides a way to remember the formula ( figure 4. 34 ). th... | openstax_calculus_volume_3_-_web | [
-0.0115352226421237,
0.016274278983473778,
0.05224885419011116,
-0.017744721844792366,
0.016546085476875305,
-0.023420648649334908,
0.009794516488909721,
0.01933949999511242,
-0.006975460331887007,
0.09169407933950424,
0.045385997742414474,
0.013588444329798222,
-0.01745385304093361,
0.013... |
4. 5 • the chain rule 365 add the terms that appear at the end of those branches. for the formula for follow only the branches that end with and add the terms that appear at the end of those branches. there is an important difference between these two chain rule theorems. in chain rule for one independent variable, the... | openstax_calculus_volume_3_-_web | [
-0.024212174117565155,
0.015961363911628723,
0.03032156452536583,
-0.027280127629637718,
-0.030507000163197517,
-0.007044188678264618,
-0.0009627607651054859,
0.022489124909043312,
0.0017394409514963627,
0.09683194011449814,
0.02639041282236576,
0.01414940319955349,
-0.009203046560287476,
... |
4. 24 calculate and given the following functions : the generalized chain rule now that we ’ ve see how to extend the original chain rule to functions of two variables, it is natural to ask : can we extend the rule to more than two variables? the answer is yes, as the generalized chain rule states. 366 4 • differentiat... | openstax_calculus_volume_3_-_web | [
0.000824113201815635,
0.03387843817472458,
0.05389579385519028,
-0.022180894389748573,
-0.004309128038585186,
-0.003957221284508705,
0.002584462985396385,
0.033360328525304794,
0.019029458984732628,
0.10017424821853638,
0.03592279925942421,
0.010093572549521923,
-0.012101322412490845,
-0.0... |
4. 26 create a tree diagram for the case when and write out the formulas for the three partial derivatives of implicit differentiation recall from implicit differentiation ( http : / / openstax. org / books / calculus - volume - 1 / pages / 3 - 8 - implicit - differentiation ) that implicit differentiation provides a m... | openstax_calculus_volume_3_-_web | [
0.005744722206145525,
0.03968292847275734,
0.028972802683711052,
-0.0022412531543523073,
0.0034617469646036625,
-0.03290165960788727,
0.012365108355879784,
-0.010326419025659561,
-0.0026378778275102377,
0.07503533363342285,
0.0157698392868042,
0.0035711724776774645,
0.012763667851686478,
-... |
4. 5 • the chain rule 369 as long as equation 4. 34 is a direct consequence of equation 4. 31. in particular, if we assume that is defined implicitly as a function of via the equation we can apply the chain rule to find solving this equation for gives equation 4. 34. equation 4. 35 can be derived in a similar fashion. ... | openstax_calculus_volume_3_-_web | [
0.0034871790558099747,
0.0441083200275898,
0.04484470188617706,
-0.00098323670681566,
-0.018072541803121567,
-0.011689137667417526,
-0.018598997965455055,
0.004530931822955608,
-0.0053912680596113205,
0.11927318572998047,
0.020195558667182922,
0.03163953125476837,
0.0016332445666193962,
-0... |
4. 5 • the chain rule 371 for the following exercises, find using the chain rule and direct substitution. 221. 222. 223. 224. 225. 226. 227. let and express as a function of and find directly. then, find using the chain rule. 228. let where and find 229. let where and find when and for the following exercises, find usi... | openstax_calculus_volume_3_-_web | [
0.015443489886820316,
0.055414456874132156,
0.05392342060804367,
0.01638183183968067,
-0.024846522137522697,
0.00936284102499485,
-0.032802172005176544,
0.02663644216954708,
-0.0021525523625314236,
0.13506245613098145,
0.06498158723115921,
0.03972834348678589,
-0.01434387918561697,
-0.0316... |
the formula where is the radius of the smaller circle, is the radius of the larger circle, and is the height of the frustum ( see figure ). find the rate of change of the volume of this frustum when if ( all in / min ). 255. a closed box is in the shape of a rectangular solid with dimensions ( dimensions are in inches.... | openstax_calculus_volume_3_-_web | [
-0.006837522145360708,
0.016828924417495728,
0.05586524307727814,
0.008235311135649681,
0.004945310764014721,
0.008192927576601505,
0.012577461078763008,
0.00391781609505415,
-0.007793046534061432,
0.10300938785076141,
0.06980569660663605,
0.029727641493082047,
-0.008914348669350147,
-0.02... |
4. 6 directional derivatives and the gradient learning objectives 4. 6. 1 determine the directional derivative in a given direction for a function of two variables. 4. 6. 2 determine the gradient vector of a given real - valued function. 4. 6. 3 explain the significance of the gradient vector with regard to direction o... | openstax_calculus_volume_3_-_web | [
0.017208589240908623,
0.06575116515159607,
-0.0005121213034726679,
-0.003319517243653536,
-0.01362165343016386,
0.00032165730954147875,
0.030867593362927437,
0.010074193589389324,
0.013011316768825054,
0.08535243570804596,
0.0020826885011047125,
0.02169126830995083,
0.015592275187373161,
0... |
secant line is 374 4 • differentiation of functions of several variables access for free at openstax. org to find the slope of the tangent line in the same direction, we take the limit as approaches zero. definition suppose is a function of two variables with a domain of let and define then the directional derivative o... | openstax_calculus_volume_3_-_web | [
0.004859487526118755,
0.06202762946486473,
0.0020537886302918196,
-0.0072931027971208096,
-0.037702526897192,
-0.008905034512281418,
0.04659661650657654,
0.003281916258856654,
-0.0027107747737318277,
0.08601225912570953,
0.005657890345901251,
0.021954823285341263,
-0.00025082900538109243,
... |
4. 6 • directional derivatives and the gradient 375 ( see the following figure. ) figure 4. 40 finding the directional derivative in a given direction at a given point on a surface. the plane is tangent to the surface at the given point another approach to calculating a directional derivative involves partial derivativ... | openstax_calculus_volume_3_-_web | [
-0.0015502620954066515,
0.03722095116972923,
0.006327867042273283,
-0.0073692407459020615,
-0.010454335249960423,
0.0018019068520516157,
0.04232526943087578,
0.004674495663493872,
-0.015880724415183067,
0.08104371279478073,
-0.006931658834218979,
0.02994469366967678,
0.005671454127877951,
... |
is given for the direction of the derivative is not a unit vector, then it is only necessary to divide by the ( 4. 38 ) ( 4. 39 ) 4. 6 • directional derivatives and the gradient 377 norm of the vector. for example, if we wished to find the directional derivative of the function in example 4. 32 in the direction of the ... | openstax_calculus_volume_3_-_web | [
0.0037828017957508564,
0.021159134805202484,
-0.017858460545539856,
-0.012071899138391018,
-0.03427138179540634,
0.005961000453680754,
0.0418609082698822,
-0.0027535499539226294,
-0.041672833263874054,
0.08105350285768509,
0.017437826842069626,
0.01734110154211521,
0.027213525027036667,
-0... |
4. 29 find the gradient of the gradient has some important properties. we have already seen one formula that uses the gradient : the formula for the directional derivative. recall from the dot product that if the angle between two vectors and is then therefore, if the angle between and is we have the disappears because... | openstax_calculus_volume_3_-_web | [
0.012401202693581581,
0.03912835568189621,
-0.013016295619308949,
0.005049861967563629,
-0.010486535727977753,
0.0035882177762687206,
0.04646503925323486,
0.010769477114081383,
-0.023526256904006004,
0.08145549148321152,
0.005839819088578224,
0.014393463730812073,
0.028404250741004944,
0.0... |
quadrant. therefore, the maximum value of the directional derivative at is ( see the following figure ). 4. 6 • directional derivatives and the gradient 379 figure 4. 42 the maximum value of the directional derivative at is in the direction of the gradient. | openstax_calculus_volume_3_-_web | [
0.03722018003463745,
0.01992577500641346,
-0.0072884876281023026,
0.03994383662939072,
-0.02941668964922428,
-0.01393693033605814,
0.07046414166688919,
0.03696802258491516,
-0.024936405941843987,
0.08644434064626694,
0.003787682391703129,
0.04710537567734718,
0.008700252510607243,
0.017373... |
4. 30 find the direction for which the directional derivative of at is a maximum. what is the maximum value? figure 4. 43 shows a portion of the graph of the function given a point in the domain of the maximum value of the gradient at that point is given by this would equal the rate of greatest ascent if the surface re... | openstax_calculus_volume_3_-_web | [
0.0156521275639534,
0.03133312985301018,
-0.004495005123317242,
0.016330793499946594,
0.003407358191907406,
-0.011818948201835155,
0.02523738332092762,
0.026169858872890472,
-0.030215200036764145,
0.08721021562814713,
-0.017489058896899223,
0.03304534777998924,
0.039310455322265625,
-0.027... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.