18652 questions

Differentiate x^x and sketch it

Write y = x^xln(y) = xln(x)1/y dy/dx = (1+ln(x)) by applying the chain rule to the LHS and product rule to the RHSdy/dx = y(1+ln(x)) Rearranging dy/dx = x^x (1+ln(x)) Substituting y=x^x into the equationNote that we cannot sketch y = x^x in general for values of x less than 0, as for every non-integer value of negative x we have to find roots of a negative number which will lie in the complex plane. However we can calculate (for example) (-1)^(-1) = -1 or (-2)^(-2) = 1/4 so it is possible for a few specific values! If you sketch this graph on a computer you will see it plot only a few points for x negative - this is exactly why this happens!
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James N.

Answered by James, Oxbridge Preparation tutor with MyTutor

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If f(x) =x^2 - 5x + 7 what are the coordinates of the minimum of f(x-2)?

The easiest way to do this is to notice that f(x-2) is just f(x) translated by two in the positive x-direction - it is not necessary to calculate what f(x-2) is at any point!
So we calculate f'(x) = 2x - 5Set f'(x) = 0 = 2x-5 to find that x = 2.5
So the corresponding value that minimises f(x-2) will be 4.5 as f(x-2) has been translated by two in the positive x-direction.
The value of y at this point will be 2.5^2 - 5 * 2.5 + 7 = 6.25 - 12.5 + 7 = 0.75
So our final answer is (4.5, 0.75) or (9/2, 3/4) in fraction form.

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James N.

Answered by James, MAT tutor with MyTutor

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Why does integration by parts work?

Recall the product rule for differentiation: the derivative of uv is equal to u'v+uv'.If we use the fact that integration reverses differentiation (so the integral of f' is f), then we calculate that uv is equal to integral of u'v+uv'. We can then rearrange this to get that the integral of u'v is equal to uv minus the integral of uv'.
The reason integration by parts is useful is that if we may not know how to integrate u'v, but if we do know how to integrate uv', we can find the solution. A good example is how should we integrate x cos(x)?Lets choose u'=cos(x), v=x. Then we know that u=sin(x), v'=1.So the integral of x cos(x) is equal to x sin(x) minus the integral of sin(x)*1=sin(x). Hence the integral of x cos(x) is equal to x sin(x) +cos(x).
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Thomas B.

Answered by Thomas, Maths tutor with MyTutor


How influential are pressure groups in the USA

Pressure groups have been very influential in the USA because they are able to shape debate, shape agenda and determine outcomes. However others have argued that party and voters have more influence than pressure groups at doing the above.
Pressure groups have shaped public agenda. This can be seen with NAACP who arguably made civil rights a national issues when they brought the case Brown V board. This shows how pressure groups can bring cases to the supreme court and then they can become a central issue. 
In addition, pressure groups can also shape public debate, this can be seen with Big Ag who have spent $480 million in the last 20 years to create iron triangles. Iron triangles are mutually beneficial relationships between legislators, pressure groups and the bureau. They help to boost Pressure groups influence dramatically which can be seen with the food stamps created in favor of Big Ag. Poorer pressure groups, however, are not able to compete with the resources of Big Ag and so are unable to have the same influence as they cannot afford to form iron triangles or get lobbyist help. Which suggests that funds make richer pressure groups have significant influence however others have little influence. 
Pressure groups can also have a significant influence because candidates are self financing and the best way to donations is through super PACs. Larry Lessig argues that Pressure groups are able to be very influential as they are able to make legislators beholden to them. This can be seen with Hiedi Hietkamp who was beholden to the NRAs wishes as seen in her vote on background checks despite the fact 90% of her constituency had voted in support of background checks in polls. This shows that Pressure groups are able to be very significant if they can get legislators to become beholden.
On the other hand, they do not determine political outcomes as the US politics have become increasingly partisan. This can be seen as 70% of votes were with the party and this has become even stronger as in the 113 congress 90% were with the party. This suggests that pressure groups do not determine outcomes rather they seek to fund candidates who agree with there ideology this can be seen with the NRA. 
Overall pressure groups play an important role in electing candidates who support there views rather than changing their views to suit the pressure groups. Pressure groups do have a very significant role in setting the agenda to social issues however not financial issues. They do also play an important role in the debate especially when iron triangles are formed and when they have greater funds however they can be ignored as seen with heath care. Thus pressure groups play an important role but party and voters also play an important role. 
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Sophia H.

Answered by Sophia, Government and Politics tutor with MyTutor


How are the characters from at least two literary works representational of people from the time and place in which they were written?

(for the sake of this example, I will focus on one text - The Great Gatsby)
Characters are moulded to embody the context of the novel in terms of ideals, standards and norms, creating vehicles for providing cultural insight, as exemplified in the novel 'The Great Gatsby' by F. Scott Fitzgerald. One of the key protagonists, Gatsby, is characterised to be a living representation of the post World War 1 United States, encapsulating the infamous 'roaring '20s' era in his extravagance and blind chasing of the 'American Dream'. His lack of connection from present reality and out-of-touch lifestyle ironically paints Gatsby as the epitome of a man of his time; with an insatiable appetite for more in a society hierarchically organised by materialism, his ambitious nature is what eventually consumes him, as it did America as a whole in the Great Depression of 1929, merely 4 years after the book's publishing. Fitzgerald guides the reader through the life of Gatsby to portray the materialism of the 1920's as an infection that even the best succumbed to, and the corruption in blurring the line between dreams and reality in a self-indoctrinated society. The narrator's bias in demanding he was an exception from the crowd of empty personalities only serves to highlight that Gatsby slowly lost himself to a past he could not recreate and a present he would never come to accept. Gatsby's life in ways acts as a metaphorical representation of the roaring '20's; immoderate and paved by delusion.
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Claudia H.

Answered by Claudia, English Literature tutor with MyTutor


What are the steps contained in action potential production?

The first step of the production of an action potential is a stimulus, which is a signal that excites the cell membrane causing Na+ ion channels to open. These channels are called voltage gated ion channels as it depends on the electrical membrane potential across the cell membrane. The opening of the channels means the membrane is more permeable to Na+ so Na+ ions diffuse into the neurone down their concentration gradient, as there is a higher concentration of Na+ ions outside than inside the cell. The inside becomes less negative as a result.If the potential difference created reaches a threshold potential, which is the level needs for a stimulus creating the generator potential to reach, the Na+ ion channels open and more Na+ diffuses into the neurone. This is known as depolarisation.The inside of the neurone becomes less negative to a point of around +30mV, where the Na+ ion channels close and the voltage-gated K+ ions open. K+ ions diffuse out of the neurone down their concentration gradient as there is more K+ inside than outside the neurone. This brings the membrane potential back down to the resting potential, the process is called repolarisation.However, hyperpolarisation can occur where K+ ion channels are too slow to close and too many K+ ions diffuse out of the neurone, so the potential difference is lower than the resting potential.Finally, the resting potential is restored as the ion channels are reset, the sodium-potassium pump returns membrane to its resting potential by pumping Na+ out and K+ in.
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Megan M.

Answered by Megan, who has applied to tutor Biology with MyTutor


Vous décrivez là où vous habitez pour votre blog. Décrivez ...

In order to gain 10 out of 10 for the "content" side of this question, there are certain things a student must do. I would say the easiest way to gain a marker's confidence is responding to all four bullet points listed below the question in your answer, and even though it might be hard to respond to them all equally, as long as you reference each bullet point, you have a chance of getting full marks. You want to make sure that you convey a lot of information, which is do-able within the ~90 word limit. But importantly, the marker is also looking for opinions, which is very easy to do also. Make sure that every or almost every statement is backed up by your personal thoughts; an easy way to ensure this occurs is including signpost statements like "Je pense" and "A mon avis".In terms of the 6 marks allocated for "quality of language", it is okay to make mistakes, as long as they are generally quite minor and not too common. What is important to bear in mind is that what you are trying to say comes across clearly. This means it is best to stick to set structures; for example, I would recommend using a structure like "je voudrais" + an infinitive verb, e.g. "je voudrais habiter aux Etas-Unis", because it is quite easy to stick this structure in and it also shows an ability to use different tenses, such as the future tense. Again, it would be easy to back this up with an opinion by saying "parce que" or "car...".
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Saoirse G.

Answered by Saoirse, French tutor with MyTutor


‘Much American literature is characterised by the importance of hope in adversity.’ By comparing at least two texts prescribed for this topic, discuss how far you have found this to be the case.

It is always worth bearing in mind the grounds upon which an answer responding to this question will be marked and I will meditate upon the use of context, which the criteria specifies, and this in two senses. Primarily, an answer must bear in mind the context in which the text was written. This means considering what was happening in the country where it was written/set at the time when it was written/set, in terms of the political, social and economic conditions. For example, for a text like John Steinbeck's 'The Grapes of Wrath', the student must acknowledge the Great Depression and how this occurred after the 'Jazz Age' years in which a text like Fitzgerald's 'The Great Gatsby' is set. A student should be able to tie this contextual knowledge to their response to the question, so in this case the suffering of the Joads can be situated among the issues that were facing thousands of farming families in the Dust Bowl states and that contributed to an immense migration away from these states. Furthermore, the mark scheme details that an answer should consider how the time in which the text is received is important, which ties into modern critics' perspectives because reading a text after it was written and after those political and economic conditions changed allows insight. For example, one might suggest that Fitzgerald's personal mental health issues, that we are now more aware of, were perhaps symptomatic of the larger moral emptiness that underlay the hedonism and materialism by which the lives of the Nouveau Rich seemed to be controlled in the 1920s in cities like New York. As such, a student might reach the conclusion that, based on an analysis of various contexts, much American literature was characterised by the importance of hope in adversity but while John Steinbeck was responding to adverse economic conditions, Fitzgerald was depicting a less visible, moral struggle against a larger elitist negligence and egotism, which essentially responds to the criterion that answers must integrate analysis between two or more texts.
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Saoirse G.

Answered by Saoirse, English tutor with MyTutor

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