context stringlengths 545 71.9k | questionsrc stringlengths 16 10.2k | question stringlengths 11 563 |
|---|---|---|
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | cis-trans ( geometric ) isomers cis-trans ( geometric ) isomers , on the other hand , have their atoms connected in the same order , but differ in the configuration of atoms around these bonds . cis-trans isomerism does not apply to linear molecules that have only single carbon-carbon bonds , as these bonds can rotate ... | why ca n't double bonds rotate ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | for example , in nature , only the l forms of amino acids are typically used to make proteins , although the d forms of amino acids are occasionally found in the cell walls of bacteria . similarly , the d enantiomer of the sugar glucose is the main product of photosynthesis , while the l form is rarely seen in nature .... | is the levo form of glucose dangerous for you and if yes how ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | all of these types of structural differences influence the three-dimensional shape , or conformation , of a hydrocarbon molecule . in the context of macromolecules ( large biological molecules such as dna , proteins , and carbohydrates ) , structural differences in the carbon skeleton often affect how the molecule func... | under the subtitle `` hydrocarbons are diverse '' , can somebody explain why `` structural differences in the carbon skeleton often affect how the molecules function '' ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | the hydrocarbons ethane , ethene , and ethyne provide an example of how different types of bonds affect the geometry of a molecule . ethane ( $ \text c_2 \text h_6 $ ) , with a single bond between carbons , forms a two-tetrahedron shape ( one tetrahedron about each carbon ) , and can rotate freely around the carbon-car... | how large can a single carbon get ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | the hydrocarbons ethane , ethene , and ethyne provide an example of how different types of bonds affect the geometry of a molecule . ethane ( $ \text c_2 \text h_6 $ ) , with a single bond between carbons , forms a two-tetrahedron shape ( one tetrahedron about each carbon ) , and can rotate freely around the carbon-car... | can molecule with multiple carbon-carbon bounds rotate freely ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | in contrast , the double bond of ethene ( $ \text c_2 \text h_4 $ ) gives it a flat ( planar ) configuration and prevents rotation about the carbon-carbon bond . this is a general feature of carbon-carbon double bonds , so anytime you see one of these in a molecule , remember that the portion of the molecule with a dou... | does it just depend on which side of the molecule the atom happened to be before bonding or.. ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . | behaviour of electrons ) seems to play some kind of important role in explaining reactions ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | the hydrocarbons ethane , ethene , and ethyne provide an example of how different types of bonds affect the geometry of a molecule . ethane ( $ \text c_2 \text h_6 $ ) , with a single bond between carbons , forms a two-tetrahedron shape ( one tetrahedron about each carbon ) , and can rotate freely around the carbon-car... | because carbon has 4 valence electrons , could it not technically have a quadruple bond with another carbon , or is this impossible ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | you can learn more about enantiomers and the r/s naming system in the organic chemistry section . the difference between a pair of enantiomers may seem very small . in some cases , though , only one isomer may be produced by the body , or the two isomers may have very different biological effects . | is there any difference between the properties of methane , ethane , propane , butane , and ... ... ... ... ... ... all ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | cis-trans ( geometric ) isomers cis-trans ( geometric ) isomers , on the other hand , have their atoms connected in the same order , but differ in the configuration of atoms around these bonds . cis-trans isomerism does not apply to linear molecules that have only single carbon-carbon bonds , as these bonds can rotate ... | but does n't linear molecules only form in triple carbon-carbon bonds ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | the hydrocarbons ethane , ethene , and ethyne provide an example of how different types of bonds affect the geometry of a molecule . ethane ( $ \text c_2 \text h_6 $ ) , with a single bond between carbons , forms a two-tetrahedron shape ( one tetrahedron about each carbon ) , and can rotate freely around the carbon-car... | is it possible to have single carbon-carbon bond and be a linear molecule ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | molecules that share the same chemical formula but have their atoms connected differently , or arranged differently in space , are known as isomers . structural isomers structural isomers ( like butane and isobutane , shown at right ) actually have their atoms bonded together in different orders : both molecules have f... | in the structural isomer paragraph , should n't it be 'differences in physical properties '' rather than 'differences in chemical properties ' ? |
introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . if you could zoom in to the molecular level , though , you ’ d see that gasoline is actually made up of a striking range of different molecules , most of them hydrocarbon... | introduction if you put gasoline into a car , what does it look like ? to the naked eye , gasoline is a pretty uninteresting yellowish-brown liquid . | what ever number of hygoren is on the outside is the number of the exponent ? |
one of the most famous images of political authority from the middle ages is the mosaic of the emperor justinian and his court in the sanctuary of the church of san vitale in ravenna , italy . this image is an integral part of a much larger mosaic program in the chancel ( the space around the altar ) . a major theme of... | who 's in front ? closer examination of the justinian mosaic reveals an ambiguity in the positioning of the figures of justinian and the bishop maximianus . overlapping suggests that justinian is the closest figure to the viewer , but when the positioning of the figures on the picture plane is considered , it is eviden... | why is justinian stepping on the foot of the person to his right ? |
one of the most famous images of political authority from the middle ages is the mosaic of the emperor justinian and his court in the sanctuary of the church of san vitale in ravenna , italy . this image is an integral part of a much larger mosaic program in the chancel ( the space around the altar ) . a major theme of... | this can perhaps be seen as an indication of the tension between the authority of the emperor and the church . essay by allen farber additional resources mosaics of san vitale in ravenna by dr. allen farber byzantium on the metropolitan museum of art 's timeline of art history 360-degree panorama of the apse of san vit... | does the stance look similar to others art work ? |
one of the most famous images of political authority from the middle ages is the mosaic of the emperor justinian and his court in the sanctuary of the church of san vitale in ravenna , italy . this image is an integral part of a much larger mosaic program in the chancel ( the space around the altar ) . a major theme of... | justinian is thus christ 's vice-regent on earth , and his army is actually the army of christ as signified by the chi-rho on the shield . who 's in front ? closer examination of the justinian mosaic reveals an ambiguity in the positioning of the figures of justinian and the bishop maximianus . | who was the maker of the mosaic in which justinian is portrayed on the front ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . | what 's umbra and penumbra ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | this question was tackled over 2300 years ago by aristarchus of samos ( 310-230 bc ) . his measurement begins with the observation of a total lunar eclipse . he correctly assumed that this was the result of the earth casting a shadow on the moon . | if the lunar eclipse happens how would the shadow be red or would that be light pollution ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | in the previous article we determined that this was about 1 hour . so he ’ s left with two numbers : time it takes the moon to travel 1 moon diameter = 1 hour time it takes the moon to travel 1 earth diameter = 2.6 hours how could we figure out the size of the moon from this ? if the times were equal than it would impl... | will the moon always be there ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . | why are all planets round ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . | and why is our planet the only planet that we can breathe on ( that we found so far.. ) and why are there exo planets ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | though it was close enough for a nice estimation . he was actually timing how long between the entry and exit of the darker ( umbral ) shadow . which to him appeared to be around 2.6 hours . | re : `` how long between the entry and exit '' do we know if he was timing from the beginning of the entry to the beginning of the exit or from the moment of full eclipse to full exit ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | this question was tackled over 2300 years ago by aristarchus of samos ( 310-230 bc ) . his measurement begins with the observation of a total lunar eclipse . he correctly assumed that this was the result of the earth casting a shadow on the moon . | why does the moon get red in a lunar eclipse ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | we also know the actual size of the earth from a previous calculation . finally we can determine the approximate size of the moon ! moon diameter = earth diameter / 3.7 moon diameter = 12742/3.7 = 3444 km this is very close to the actual diameter of the moon : 3474.8 km we determined the size of the earth and moon usin... | how did aristarchus determine the size of the moon ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | this question was tackled over 2300 years ago by aristarchus of samos ( 310-230 bc ) . his measurement begins with the observation of a total lunar eclipse . he correctly assumed that this was the result of the earth casting a shadow on the moon . | what is the lunar the eclipse moon ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | how much larger ? we need to setup a basic proportion : time # 1 / time # 2 = moon diameter / earth diameter 1 / 2.6 = moon diameter / earth diameter this led him to claim that the earth was about 8/3 the diameter of the moon . this is pretty close to the actual difference : the earth is about 3.7 times bigger than the... | how can i calculate the distance between the earth and the moon ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | how much larger ? we need to setup a basic proportion : time # 1 / time # 2 = moon diameter / earth diameter 1 / 2.6 = moon diameter / earth diameter this led him to claim that the earth was about 8/3 the diameter of the moon . this is pretty close to the actual difference : the earth is about 3.7 times bigger than the... | what has the knowledge about the earth ( eratosthenes theory ) provided for the astronomers after his time ? |
relative sizes we previously determined that the diameter of the earth is 12742 km . now we want to determine the size of the moon relative to the earth . let ’ s start with an even simpler question : is the moon bigger or smaller than the earth ? how could you prove this ? remember this question wouldn ’ t seem silly ... | first aristarchus timed how long the moon took to travel through the earth ’ s shadow . as eratosthenes did , he also made a simplifying assumption that the sun ’ s light reaches us in perfectly parallel lines . which we know isn ’ t exactly true as two cone shaped shadows are produced : umbral and penumbral . | what new discoveries were found based on eratosthenes 's statement ? |
you might know marcel duchamp because of his fountain—the urinal that he signed ( as r. mutt ) and submitted to the society of independent artists art show in 1917 . this `` readymade '' was a whimsical and iconoclastic gesture—in keeping with the spirit of the new york dada circle . like his dada compatriots , includi... | though the large glass is essentially a flat , two-dimensional object , it is emphatically not a painting , as it is mostly transparent—you can walk around it and view it from both sides—and duchamp avoided using traditional materials like canvas and oil paint . instead , he concocted the imagery on the glass surface o... | did the glass break intentionally , or was it by accident ? |
the southeast asian country of indonesia consists of more than 17,000 tropical and volcanic islands that straddle the equator between the indian and pacific oceans . among indonesia ’ s principal regions are the islands of java , bali , and sumatra , as well as large parts of borneo and new guinea ( a contested region ... | today , indonesia is home to more than three hundred ethnic groups with approximately five hundred spoken languages and dialects . eighty-seven percent of the population , or some 200 million people , is islamic , making indonesia the largest muslim nation in the world . for thousands of years indonesians developed com... | what is the written language of indonesia given that it is the most populous muslim nation ? |
the southeast asian country of indonesia consists of more than 17,000 tropical and volcanic islands that straddle the equator between the indian and pacific oceans . among indonesia ’ s principal regions are the islands of java , bali , and sumatra , as well as large parts of borneo and new guinea ( a contested region ... | among indonesia ’ s principal regions are the islands of java , bali , and sumatra , as well as large parts of borneo and new guinea ( a contested region ) . today , indonesia is home to more than three hundred ethnic groups with approximately five hundred spoken languages and dialects . eighty-seven percent of the pop... | when the first time muslim came to indonesia ? |
the southeast asian country of indonesia consists of more than 17,000 tropical and volcanic islands that straddle the equator between the indian and pacific oceans . among indonesia ’ s principal regions are the islands of java , bali , and sumatra , as well as large parts of borneo and new guinea ( a contested region ... | today , indonesia is home to more than three hundred ethnic groups with approximately five hundred spoken languages and dialects . eighty-seven percent of the population , or some 200 million people , is islamic , making indonesia the largest muslim nation in the world . for thousands of years indonesians developed com... | `` indonesia the largest muslim nation in the world '' does it mean indonesia is an islamic country like saudi arabia ? |
right-hand rule ever wonder how physicists identify one another ? the right-hand rule is the closest thing that physics has to a gang sign . make a gun shape with the thumb and first two fingers of your right hand , and then point the middle finger to the left—or straight down , depending on how you hold your gun . now... | one way to remember these two coiling right-hand rules is that straight magnetic field lines are caused by circles of current , and straight lines of current cause circular magnetic fields . the right-hand rule allows us to remember both cases with a single hand gesture . consider the following : the magnetic field in ... | what is right hand rule ? |
right-hand rule ever wonder how physicists identify one another ? the right-hand rule is the closest thing that physics has to a gang sign . make a gun shape with the thumb and first two fingers of your right hand , and then point the middle finger to the left—or straight down , depending on how you hold your gun . now... | as long as you use your right hand . using your left hand will get you kicked out faster than you can say “ may the mass-times-acceleration be with you ” . in this article , we ’ ll talk about the way that this sign is used to help physicists ’ remember the direction of magnetic forces . | can somebody get me out of the confusion as many people are having ? |
right-hand rule ever wonder how physicists identify one another ? the right-hand rule is the closest thing that physics has to a gang sign . make a gun shape with the thumb and first two fingers of your right hand , and then point the middle finger to the left—or straight down , depending on how you hold your gun . now... | right-hand rule ever wonder how physicists identify one another ? the right-hand rule is the closest thing that physics has to a gang sign . | what is ampere 's swimming law ? |
right-hand rule ever wonder how physicists identify one another ? the right-hand rule is the closest thing that physics has to a gang sign . make a gun shape with the thumb and first two fingers of your right hand , and then point the middle finger to the left—or straight down , depending on how you hold your gun . now... | the right-hand rule can also be used to remember the direction of the axes in a standard $ x , y , z $ coordinate system : the thumb points in the positive $ x $ direction , the first finger in the positive $ y $ direction , and the middle finger in the positive $ z $ direction . magnetic field caused by current in a w... | if a wave is propagating in ( 1,1,0 ) direction how many possible combinations of electric and magnetic fields directions can it be ? |
right-hand rule ever wonder how physicists identify one another ? the right-hand rule is the closest thing that physics has to a gang sign . make a gun shape with the thumb and first two fingers of your right hand , and then point the middle finger to the left—or straight down , depending on how you hold your gun . now... | one way to remember these two coiling right-hand rules is that straight magnetic field lines are caused by circles of current , and straight lines of current cause circular magnetic fields . the right-hand rule allows us to remember both cases with a single hand gesture . consider the following : the magnetic field in ... | where is fleming 's left hand rule used and where is the right hand rule used ? |
right-hand rule ever wonder how physicists identify one another ? the right-hand rule is the closest thing that physics has to a gang sign . make a gun shape with the thumb and first two fingers of your right hand , and then point the middle finger to the left—or straight down , depending on how you hold your gun . now... | one way to remember these two coiling right-hand rules is that straight magnetic field lines are caused by circles of current , and straight lines of current cause circular magnetic fields . the right-hand rule allows us to remember both cases with a single hand gesture . consider the following : the magnetic field in ... | how do i calculate the force of the right hand rule ? |
right-hand rule ever wonder how physicists identify one another ? the right-hand rule is the closest thing that physics has to a gang sign . make a gun shape with the thumb and first two fingers of your right hand , and then point the middle finger to the left—or straight down , depending on how you hold your gun . now... | one way to remember these two coiling right-hand rules is that straight magnetic field lines are caused by circles of current , and straight lines of current cause circular magnetic fields . the right-hand rule allows us to remember both cases with a single hand gesture . consider the following : the magnetic field in ... | why right hand rule is used in place of fleming 's left hand rule ? |
right-hand rule ever wonder how physicists identify one another ? the right-hand rule is the closest thing that physics has to a gang sign . make a gun shape with the thumb and first two fingers of your right hand , and then point the middle finger to the left—or straight down , depending on how you hold your gun . now... | one way to remember these two coiling right-hand rules is that straight magnetic field lines are caused by circles of current , and straight lines of current cause circular magnetic fields . the right-hand rule allows us to remember both cases with a single hand gesture . consider the following : the magnetic field in ... | what hand ( left or right ) do we use for the grip rule in a convectional current ? |
right-hand rule ever wonder how physicists identify one another ? the right-hand rule is the closest thing that physics has to a gang sign . make a gun shape with the thumb and first two fingers of your right hand , and then point the middle finger to the left—or straight down , depending on how you hold your gun . now... | this last case represents what happens in an electromagnet in which current is run through a wire wrapped in the shape of a coil . this coil generates magnetic field lines that point in the direction of the coil ’ s long axis . one way to remember these two coiling right-hand rules is that straight magnetic field lines... | why is it that when a wire is coiled , the magnetic field is projected upwards or `` through the coil '' ? |
right-hand rule ever wonder how physicists identify one another ? the right-hand rule is the closest thing that physics has to a gang sign . make a gun shape with the thumb and first two fingers of your right hand , and then point the middle finger to the left—or straight down , depending on how you hold your gun . now... | one way to remember these two coiling right-hand rules is that straight magnetic field lines are caused by circles of current , and straight lines of current cause circular magnetic fields . the right-hand rule allows us to remember both cases with a single hand gesture . consider the following : the magnetic field in ... | what is the right hand grip rule for fields and poles ? |
in the center of manhattan island lies a great expanse of sculpted nature . this large swath of greenery—central park—was the first great manifesto of a new urban vision that sought to introduce nature into the heart of commercial and industrial cities in the united states . as a measure of its tremendous success , the... | new york engineer egbert viele—who was responsible for the first , unbuilt plan for central park in 1856—described the land as a “ pestilential spot ” with “ miasmatic odors ” emanating from the untended ground . unhealthy and unsightly , the land was ripe for reform as projections for manhattan ’ s future growth pushe... | that we play and dance on land that was `` stolen '' from immigrants and minorities ( likely for very small pay ) ? |
hi , i 'm colin fuller ! what do you work on ? i 'm a software engineer on the data science team at khan academy ( ka ) . this means that i help write the software that lets ka do things like figure out what content to recommend next , display your progress , and determine how effective we are at helping people learn .... | there were very particular rules that must be governing what happens . i did n't know what they could possibly be , and i was decades away from understanding them , but i was hooked by the idea . in college , i studied chemistry and did some research on diverse topics including interactions between laser pulses and par... | what techniques could be used in order to memorize those formulas of the factors that affect chemical equilibrium ? |
hi , i 'm colin fuller ! what do you work on ? i 'm a software engineer on the data science team at khan academy ( ka ) . this means that i help write the software that lets ka do things like figure out what content to recommend next , display your progress , and determine how effective we are at helping people learn .... | what do you work on ? i 'm a software engineer on the data science team at khan academy ( ka ) . this means that i help write the software that lets ka do things like figure out what content to recommend next , display your progress , and determine how effective we are at helping people learn . | do you have any tips or advice on how to succeed in making my dream of becoming a software engineer a reality ? |
overview in the early 1970s , the post-world war ii economic boom began to wane , due to increased international competition , the expense of the vietnam war , and the decline of manufacturing jobs . unemployment rates rose , while a combination of price increases and wage stagnation led to a period of economic doldrum... | indeed , contemporary observers commented that the postwar united states was in the midst of `` the greatest prosperity the world has ever known . `` $ ^1 $ the american gross national product ( gnp ) , a measure of all goods and services produced by a country 's citizens , increased from \ $ 200,000-million in 1940 to... | `` the american gross national product ( gnp ) , a measure of all goods and services produced in the country '' is n't this the definition of gdp instead of gnp ? |
overview in the early 1970s , the post-world war ii economic boom began to wane , due to increased international competition , the expense of the vietnam war , and the decline of manufacturing jobs . unemployment rates rose , while a combination of price increases and wage stagnation led to a period of economic doldrum... | the oil-rich nations of the middle east , already angry with the united states for devaluing the dollar ( the currency used to purchase oil ) determined to exact their revenge with an oil embargo . led by saudi arabia , the organization of the petroleum exporting countries ( opec ) announced an oil shipping embargo aga... | which member nation of opec ( organization of the petroleum exporting countries ) , was the founding member ? |
overview in the early 1970s , the post-world war ii economic boom began to wane , due to increased international competition , the expense of the vietnam war , and the decline of manufacturing jobs . unemployment rates rose , while a combination of price increases and wage stagnation led to a period of economic doldrum... | led by saudi arabia , the organization of the petroleum exporting countries ( opec ) announced an oil shipping embargo against the united states as well as israel 's european allies. $ ^6 $ the effects were immediate and dire . the price of oil shot up to \ $ 11.65 per barrel , an increase of 387 % . lines miles-long f... | the price of oil has decreased a lot during the last years but as opec still extists , what explains this lack of power compared to the 70s ? |
overview in the early 1970s , the post-world war ii economic boom began to wane , due to increased international competition , the expense of the vietnam war , and the decline of manufacturing jobs . unemployment rates rose , while a combination of price increases and wage stagnation led to a period of economic doldrum... | for example , many companies have moved manufacturing jobs out of the united states in order to save on labor costs . today , 80 % of all american jobs are in the service industry. $ ^8 $ since the oil embargo , the united states also has worked to reduce its dependence on foreign oil through a variety of means , inclu... | improvements of vehicle fuel-efficiency , investments in renewable energy , and increasing domestic oil production explain all ? |
overview in the early 1970s , the post-world war ii economic boom began to wane , due to increased international competition , the expense of the vietnam war , and the decline of manufacturing jobs . unemployment rates rose , while a combination of price increases and wage stagnation led to a period of economic doldrum... | the price of oil shot up to \ $ 11.65 per barrel , an increase of 387 % . lines miles-long formed at gas stations . the united states consumed one third of the world 's oil , and its citizens quickly discovered just how much of daily life depended on cheap oil . | in the image , it has cars lining up for gas ; if the prices were high , then why were there more cars lining up for gas ? |
overview in the early 1970s , the post-world war ii economic boom began to wane , due to increased international competition , the expense of the vietnam war , and the decline of manufacturing jobs . unemployment rates rose , while a combination of price increases and wage stagnation led to a period of economic doldrum... | what do you think ? what caused the economic problems of the 1970s ? were they avoidable ? | what caused the economic problems of the 1970s ? |
overview in the early 1970s , the post-world war ii economic boom began to wane , due to increased international competition , the expense of the vietnam war , and the decline of manufacturing jobs . unemployment rates rose , while a combination of price increases and wage stagnation led to a period of economic doldrum... | the united states consumed one third of the world 's oil , and its citizens quickly discovered just how much of daily life depended on cheap oil . families living in far-flung suburbs depended on automobiles to get everywhere . even after the embargo ended in march 1974 , prices for oil remained about 33 % higher than ... | i still do n't get how a war causes inflation ... who can help me with this ? |
overview in the early 1970s , the post-world war ii economic boom began to wane , due to increased international competition , the expense of the vietnam war , and the decline of manufacturing jobs . unemployment rates rose , while a combination of price increases and wage stagnation led to a period of economic doldrum... | more and more american jobs were in the service sector , which had lower wages and fewer benefits than manufacturing jobs . individuals born on the tail end of the baby boom found themselves competing in a very crowded labor market , especially as more women and immigrants entered the workforce. $ ^4 $ the oil embargo ... | the article discusses richard nixon 's attempts to stop stagflation , but what about gerald ford 's actions ? |
overview in the early 1970s , the post-world war ii economic boom began to wane , due to increased international competition , the expense of the vietnam war , and the decline of manufacturing jobs . unemployment rates rose , while a combination of price increases and wage stagnation led to a period of economic doldrum... | the crisis was compounded when oil-rich nations in the middle east declared an embargo against the united states in retaliation for its support of israel . the oil embargo had a lasting effect on energy prices . economic woes of the 1970s during the twenty-five years after world war ii , the economic power of the unite... | in the paragraph about the oil embargo , what was the economic impact on opec ? |
overview in the early 1970s , the post-world war ii economic boom began to wane , due to increased international competition , the expense of the vietnam war , and the decline of manufacturing jobs . unemployment rates rose , while a combination of price increases and wage stagnation led to a period of economic doldrum... | president nixon tried to alleviate these problems by devaluing the dollar and declaring wage- and price-freezes . the crisis was compounded when oil-rich nations in the middle east declared an embargo against the united states in retaliation for its support of israel . the oil embargo had a lasting effect on energy pri... | did the middle eastern countries suffer anything because of their oil embargo ? |
overview in the early 1970s , the post-world war ii economic boom began to wane , due to increased international competition , the expense of the vietnam war , and the decline of manufacturing jobs . unemployment rates rose , while a combination of price increases and wage stagnation led to a period of economic doldrum... | in october 1973 , the united states supported israel after a surprise attack by egypt and syria in the yom kippur war . the oil-rich nations of the middle east , already angry with the united states for devaluing the dollar ( the currency used to purchase oil ) determined to exact their revenge with an oil embargo . le... | so did the oil crisis cause because of the vitenam war or something else ? |
christianity becomes legal by the middle of the fourth century christianity had undergone a dramatic transformation . before emperor constantine 's acceptance , christianity had a marginal status in the roman world . attracting converts in the urban populations , christianity appealed to the faithful 's desires for per... | in both scenes the principal figure is flanked by two other figures . the importance of peter and paul in rome is made apparent that two of the major churches that constantine constructed in rome were the church of st. peter and the church of st. paul outside the walls . the site of the church of st. peter has long bel... | what was st. peter doing in rome to have founded this church ? |
christianity becomes legal by the middle of the fourth century christianity had undergone a dramatic transformation . before emperor constantine 's acceptance , christianity had a marginal status in the roman world . attracting converts in the urban populations , christianity appealed to the faithful 's desires for per... | peter and paul were martyred under roman rule . the remaining two scenes on the sarcophagus represent sts . peter and paul being lead to their martyrdoms . | does anyone know if this sarcophagus was painted at one time ? |
christianity becomes legal by the middle of the fourth century christianity had undergone a dramatic transformation . before emperor constantine 's acceptance , christianity had a marginal status in the roman world . attracting converts in the urban populations , christianity appealed to the faithful 's desires for per... | old and new together we can determine some intentionality in the inclusion of the old and new testament scenes . for example the image of adam and eve shown covering their nudity after the fall was intended to refer to the doctrine of original sin that necessitated christ 's entry into the world to redeem humanity thro... | who were adam and eve ? |
the resistor-capacitor $ ( \text { rc } ) $ circuit is one of the first interesting circuits we can create and analyze . understanding the behavior of this circuit is essential to learning electronics . forms of this circuit can be found everywhere . sometimes you will create this circuit on purpose , and other times i... | the solution to a differential equation is some sort of function , in our case , some function of voltage with respect to time , $ v ( t ) $ . $ v ( t ) $ is a solution if it makes the differential equation true . $ \text c\ , \dfrac { dv } { dt } + \dfrac 1 { \text r } \ , { v } = 0 $ ( differential equation ) where d... | are differential equations truly necessary ? |
the resistor-capacitor $ ( \text { rc } ) $ circuit is one of the first interesting circuits we can create and analyze . understanding the behavior of this circuit is essential to learning electronics . forms of this circuit can be found everywhere . sometimes you will create this circuit on purpose , and other times i... | then we will plug our solution into the equation and work out a few constants specific to the circuit . ( this part takes math . ) if we find constants that make the equation true , then the proposed function is a solution to the equation , and we win . | could n't this material become more accessible ( to lower math backgrounds ) if things were introduced in the frequency domain rather than the time domain ? |
the resistor-capacitor $ ( \text { rc } ) $ circuit is one of the first interesting circuits we can create and analyze . understanding the behavior of this circuit is essential to learning electronics . forms of this circuit can be found everywhere . sometimes you will create this circuit on purpose , and other times i... | current begins flowing out of the positive terminal of the battery , through $ \text r $ and $ \text c $ . charge accumulates on the capacitor . the accumulating charge generates a rising voltage across the capacitor ( $ v_\text c = q / \text c $ ) . | how can i calculate the time it takes to charge a capacitor ? |
the resistor-capacitor $ ( \text { rc } ) $ circuit is one of the first interesting circuits we can create and analyze . understanding the behavior of this circuit is essential to learning electronics . forms of this circuit can be found everywhere . sometimes you will create this circuit on purpose , and other times i... | this is called the natural response . the time constant for an $ \text { rc } $ circuit is $ \tau = \text { r } \cdot { \text c } $ the circuit we will study is a resistor in series with a capacitor . how does this circuit respond to an applied voltage ? first , use intuition to predict what happens the circuit we expl... | what about in a circuit without a resistor ? |
the resistor-capacitor $ ( \text { rc } ) $ circuit is one of the first interesting circuits we can create and analyze . understanding the behavior of this circuit is essential to learning electronics . forms of this circuit can be found everywhere . sometimes you will create this circuit on purpose , and other times i... | charge accumulates on the capacitor until $ v_\text c $ rises to the same value as the battery voltage : $ v_\text c = \text v_ { \text { bat } } $ . at that point , the voltage across the resistor is $ 0 $ volts , so current in the resistor stops flowing ( ohm 's law ) . that also means current ( charge ) stops flowin... | so if i am not wrong , v ( t ) is voltage across resistor ? |
the resistor-capacitor $ ( \text { rc } ) $ circuit is one of the first interesting circuits we can create and analyze . understanding the behavior of this circuit is essential to learning electronics . forms of this circuit can be found everywhere . sometimes you will create this circuit on purpose , and other times i... | charge accumulates on the capacitor until $ v_\text c $ rises to the same value as the battery voltage : $ v_\text c = \text v_ { \text { bat } } $ . at that point , the voltage across the resistor is $ 0 $ volts , so current in the resistor stops flowing ( ohm 's law ) . that also means current ( charge ) stops flowin... | but i want to know whether this v ( t ) is voltage across resistor or capacitor ? |
the resistor-capacitor $ ( \text { rc } ) $ circuit is one of the first interesting circuits we can create and analyze . understanding the behavior of this circuit is essential to learning electronics . forms of this circuit can be found everywhere . sometimes you will create this circuit on purpose , and other times i... | $ v ( t ) $ is a solution if it makes the differential equation true . $ \text c\ , \dfrac { dv } { dt } + \dfrac 1 { \text r } \ , { v } = 0 $ ( differential equation ) where do ode solutions come from ? one way is to make an informed guess at a solution , and try it out . | at model the circuit you write the equation i = c*dv/dt and i = v/r then you formulate the equation c*dv/dt + ( 1/r ) *v = 0 which is essentially i + i = 2i = 0 should n't the correct kcl be c*dv/dt - ( 1/r ) *v = i - i = 0 ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . | can somebody please tell me the topics covered in honors algebra ii for the common core curriculum ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | $ \begin { align } { f ( x ) } + { g ( x ) } & amp ; = ( { x+1 } ) + ( { 2x } ) \\ & amp ; = x+1+2x \\\ & amp ; =3x+1 \end { align } $ let 's call this new function $ h $ . so we have : $ { h ( x ) } = { f ( x ) } + { g ( x ) } { =3x+1 } $ part 2 : evaluating a combined function we can also evaluate combined functions ... | is algebra ii part of a high school curriculum or is it college level ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . | is there anyway i could take geometry over the summer , for high school credit , on khan academy ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . | how am i going to go to college if my sat score is going to be 800 at the highest ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | problem 3 the graphs of $ y=f ( x ) $ and $ y=g ( x ) $ are shown below . other ways to combine functions all of the examples we 've looked at so far create a new function by adding two functions , but you can also subtract , multiply , and divide two functions to make new functions ! for example , if $ f ( x ) =x+3 $ ... | in the first graph example , how do you know that the y intercept is one after adding the functions together ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | in problems 1 and 2 , let $ f ( x ) =3x+2 $ and $ g ( x ) =x-3 $ . problem 1 problem 2 a graphical connection we can also understand what it means to add two functions by looking at graphs of the functions . the graphs of $ y=m ( x ) $ and $ y=n ( x ) $ are shown below . | what happens if you add the functions of two parabolas , or even multiple graphs ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | other ways to combine functions all of the examples we 've looked at so far create a new function by adding two functions , but you can also subtract , multiply , and divide two functions to make new functions ! for example , if $ f ( x ) =x+3 $ and $ g ( x ) =x-2 $ , then we can not only find the sum , but also ... ..... | can you reduce `` ( x+3 ) / ( x-2 ) '' to 3/-2 because x/x=1 ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | problem 3 the graphs of $ y=f ( x ) $ and $ y=g ( x ) $ are shown below . other ways to combine functions all of the examples we 've looked at so far create a new function by adding two functions , but you can also subtract , multiply , and divide two functions to make new functions ! for example , if $ f ( x ) =x+3 $ ... | what are the uses of adding/subtracting/multiplying/dividing functions ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . | can somebody explain how graphically we can add , subtract , multiply or divide the function ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | so we have : $ { h ( x ) } = { f ( x ) } + { g ( x ) } { =3x+1 } $ part 2 : evaluating a combined function we can also evaluate combined functions for particular inputs . let 's evaluate function $ h $ above for $ x=2 $ . below are two ways of doing this . | how do you identify the domain and range of a quadratic function ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | now let 's try some practice problems . in problems 1 and 2 , let $ f ( x ) =3x+2 $ and $ g ( x ) =x-3 $ . problem 1 problem 2 a graphical connection we can also understand what it means to add two functions by looking at graphs of the functions . | f ( x ) = 2f ( 1/x ) + 5x + 1 then f ( 3 ) = ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | now let 's try some practice problems . in problems 1 and 2 , let $ f ( x ) =3x+2 $ and $ g ( x ) =x-3 $ . problem 1 problem 2 a graphical connection we can also understand what it means to add two functions by looking at graphs of the functions . | please explain f ( x ) =2f ( 1/x ) +5x + 1 then f ( 3 ) = ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | so we have : $ { h ( x ) } = { f ( x ) } + { g ( x ) } { =3x+1 } $ part 2 : evaluating a combined function we can also evaluate combined functions for particular inputs . let 's evaluate function $ h $ above for $ x=2 $ . below are two ways of doing this . | how does one know what point a function on a graph specifies ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | in the second graph , notice that $ n ( 4 ) =5 $ . let $ p ( x ) =m ( x ) +n ( x ) $ . now look at the graph of $ y=p ( x ) $ . notice that $ p ( 4 ) =\blued 2+\maroond 5=\purpled7 $ . | how does one locate the x and y in a function on a provided graph ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | now look at the graph of $ y=p ( x ) $ . notice that $ p ( 4 ) =\blued 2+\maroond 5=\purpled7 $ . challenge yourself to see that $ p ( x ) = m ( x ) + n ( x ) $ for every value of $ x $ by looking at the three graphs . | p ( 3 ) *q ( 3 ) *r ( 3 ) -p ( 3 ) =5*2*3-5=30-5 how does 30-5=5 and not 25 ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | } } } \end { align } $ ... the quotient . $ \begin { align } f ( x ) \div g ( x ) & amp ; =\dfrac { f ( x ) } { g ( x ) } \\ & amp ; =\dfrac { ( x+3 ) } { ( x-2 ) } ~~~~~~~~~~~~~~~~~~~~~\small { \gray { \text { substitute . } } } \end { align } $ in doing so , we have just created three new functions ! | what 's the point of `` ( f+g ) ( x ) = f ( x ) +g ( x ) '' ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | problem 3 the graphs of $ y=f ( x ) $ and $ y=g ( x ) $ are shown below . other ways to combine functions all of the examples we 've looked at so far create a new function by adding two functions , but you can also subtract , multiply , and divide two functions to make new functions ! for example , if $ f ( x ) =x+3 $ ... | what is the purpose for combining functions ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | } } } \end { align } $ ... the quotient . $ \begin { align } f ( x ) \div g ( x ) & amp ; =\dfrac { f ( x ) } { g ( x ) } \\ & amp ; =\dfrac { ( x+3 ) } { ( x-2 ) } ~~~~~~~~~~~~~~~~~~~~~\small { \gray { \text { substitute . } } } \end { align } $ in doing so , we have just created three new functions ! | how would i solve the equations given f ( x ) =2x-3 and g ( x ) =0.5x+4 find f of g of x ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | problem 3 the graphs of $ y=f ( x ) $ and $ y=g ( x ) $ are shown below . other ways to combine functions all of the examples we 've looked at so far create a new function by adding two functions , but you can also subtract , multiply , and divide two functions to make new functions ! for example , if $ f ( x ) =x+3 $ ... | why would the quotient of the two functions just be substituted and not go any further ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | what would combining ( adding ) the two equations/graphs result in ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | $ \begin { align } h ( x ) & amp ; =3x+1\\ h ( 2 ) & amp ; =3 ( 2 ) +1\\ & amp ; =\greend { 7 } \end { align } $ method 2 : find $ f ( 2 ) $ and $ g ( 2 ) $ and add the results . since $ h ( x ) =f ( x ) +g ( x ) $ , we can also find $ h ( 2 ) $ by finding $ f ( 2 ) +g ( 2 ) $ . first , let 's find $ f ( 2 ) $ : $ \beg... | for problem three , would you just multiply the 'g ' and 'f ' to find the solution ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | the graphs of $ y=m ( x ) $ and $ y=n ( x ) $ are shown below . in the first graph , notice that $ m ( 4 ) =2 $ . in the second graph , notice that $ n ( 4 ) =5 $ . | if sonic is the fastest thing in the world , why in , sonic advance on the boss fight for the first stage , is egg man able to catch up to sonic and exceed him in speed ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | now let 's try some practice problems . in problems 1 and 2 , let $ f ( x ) =3x+2 $ and $ g ( x ) =x-3 $ . problem 1 problem 2 a graphical connection we can also understand what it means to add two functions by looking at graphs of the functions . | how would you factor x^2 + 5x - 50 ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | so we have : $ { h ( x ) } = { f ( x ) } + { g ( x ) } { =3x+1 } $ part 2 : evaluating a combined function we can also evaluate combined functions for particular inputs . let 's evaluate function $ h $ above for $ x=2 $ . below are two ways of doing this . | how do i find the range and domain of a function ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | other ways to combine functions all of the examples we 've looked at so far create a new function by adding two functions , but you can also subtract , multiply , and divide two functions to make new functions ! for example , if $ f ( x ) =x+3 $ and $ g ( x ) =x-2 $ , then we can not only find the sum , but also ... ..... | is it possible to divide ( x=3 ) / ( x-2 ) even further ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | in the second graph , notice that $ n ( 4 ) =5 $ . let $ p ( x ) =m ( x ) +n ( x ) $ . now look at the graph of $ y=p ( x ) $ . | in the topic , `` a graphical connection '' , what are the variables : m and n when they use m , in y= m ( x ) and when they use n , in y = n ( x ) ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | problem 3 the graphs of $ y=f ( x ) $ and $ y=g ( x ) $ are shown below . other ways to combine functions all of the examples we 've looked at so far create a new function by adding two functions , but you can also subtract , multiply , and divide two functions to make new functions ! for example , if $ f ( x ) =x+3 $ ... | if we multiply or divide the two functions , what would we define those two function ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | notice that $ p ( 4 ) =\blued 2+\maroond 5=\purpled7 $ . challenge yourself to see that $ p ( x ) = m ( x ) + n ( x ) $ for every value of $ x $ by looking at the three graphs . let 's practice . | can you please explain what is the value of y and x on the graph part and how do i find it ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | $ \begin { align } h ( x ) & amp ; =3x+1\\ h ( 2 ) & amp ; =3 ( 2 ) +1\\ & amp ; =\greend { 7 } \end { align } $ method 2 : find $ f ( 2 ) $ and $ g ( 2 ) $ and add the results . since $ h ( x ) =f ( x ) +g ( x ) $ , we can also find $ h ( 2 ) $ by finding $ f ( 2 ) +g ( 2 ) $ . first , let 's find $ f ( 2 ) $ : $ \beg... | how do you find f ( a ) + f ( h ) for the function f ( x ) = x^2 - x - 3 ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | } } } \end { align } $ ... the quotient . $ \begin { align } f ( x ) \div g ( x ) & amp ; =\dfrac { f ( x ) } { g ( x ) } \\ & amp ; =\dfrac { ( x+3 ) } { ( x-2 ) } ~~~~~~~~~~~~~~~~~~~~~\small { \gray { \text { substitute . } } } \end { align } $ in doing so , we have just created three new functions ! | why do the graphs differ between f ( x ) and g ( x ) ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | in the first graph , notice that $ m ( 4 ) =2 $ . in the second graph , notice that $ n ( 4 ) =5 $ . let $ p ( x ) =m ( x ) +n ( x ) $ . | in the graphical connection , how does m ( 4 ) =2 and n ( 4 ) =5 ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | now look at the graph of $ y=p ( x ) $ . notice that $ p ( 4 ) =\blued 2+\maroond 5=\purpled7 $ . challenge yourself to see that $ p ( x ) = m ( x ) + n ( x ) $ for every value of $ x $ by looking at the three graphs . | what functions are used to arrive to 2 and 5 ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | in the second graph , notice that $ n ( 4 ) =5 $ . let $ p ( x ) =m ( x ) +n ( x ) $ . now look at the graph of $ y=p ( x ) $ . | now on the graphical connections , how do you get an answer such as p ( x ) =m ( x ) +n ( x ) ? |
just like we can add , subtract , multiply , and divide numbers , we can also add , subtract , multiply , and divide functions . the sum of two functions part 1 : creating a new function by adding two functions let 's add $ { f ( x ) =x+1 } $ and $ { g ( x ) =2x } $ together to make a new function . $ \begin { align } ... | } } } \end { align } $ ... the quotient . $ \begin { align } f ( x ) \div g ( x ) & amp ; =\dfrac { f ( x ) } { g ( x ) } \\ & amp ; =\dfrac { ( x+3 ) } { ( x-2 ) } ~~~~~~~~~~~~~~~~~~~~~\small { \gray { \text { substitute . } } } \end { align } $ in doing so , we have just created three new functions ! | why do i find f ( x ) + g ( x ) = ( f+g ) ( x ) written at some places in my textbook ? |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.