22866 questions

What is the geometrical meaning of the derivative of a function?

Since the definition of derivative is the limit of the incremental rapport as h approaches 0 of ( f(x + h) - f(x) ) / h ), the geometrical meaning of a derivative is the slope of the tangent line in a general point (x, y) of the function (we assume the the function is sharp through all the points in its domain).
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Michelangelo P.

Answered by Michelangelo, Further Mathematics A Level tutor with MyTutor


What's the difference between fixed and variable costs?

Fixed are costs that do not change with the level of output. For example, heating bills would be a fixed cost. Whether we make 100 computers or 300 computers, our fixed costs do not change. Variable costs are costs that do change with output, for example packaging. If we make 100 computers and our packaging costs are £5 per computer, the our packaging costs would be £500. This would then change if we were to make 300 computers as our cost is not £900.
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Will N.

Answered by Will, Business Studies GCSE tutor with MyTutor


For what values of k does the line y=kx-1 have two distinct points of intersection with the circle (x-2)^2+(y-3)^2=2?

sub y=kx-1 into circle equation, get (k^2+1)x^2-(8k+4)x+18=0for 2 distinct solutions need b^2-4ac>0, ie -8k^2+64k-56>0iff k^2-8k+7<0complete the square: intersections of equation in k satisfy(k-4)^2=9 so inequality satisfied when 1<k<7. strict inequality for distinct intersection
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Christopher B.

Answered by Christopher, who has applied to tutor Maths GCSE with MyTutor

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Prove by induction that the nth triangle number is given by n(n+1)/2

base case: (1 x 2)/2 = 1 as required inductive step: assuming statement holds for n=k, the (k+1)th triangle number is given by k(k+1)/2 + (k+1) by definition=(k^2+3k+2)/2=(k+1)(k+2)/2=(k+1)((k+1)+1)/2result follows by induction
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Christopher B.

Answered by Christopher, who has applied to tutor Maths A Level with MyTutor


Write 8^2(4^2 / 2^7) in the form 2^x

We know that 82 = (23)2 = 26 and 42 = (22)2 = 24 So we can write the original expression as 26 (24/27) Then using indices laws we get 26(2-3) = 23
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Charlotte C.

Answered by Charlotte, Maths GCSE tutor with MyTutor


Solve the equation 2log (base 3)(x) - log (base 3)(x+4) = 2

First express as a single logarithm as follows. The number in front of the logarithm remembering log rules can be rewritten as the power of the number in the brackets
So rewriting the RHSlog3(x2) - log3(x+4)log3(x2/(x+4))
remember inverse log3 is to the power of 3
2 = log3(x2/(x+4))32=(x2/(x+4))
expanding and solving
x=12 as cannot do a negative logarithm of a number
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Theranjit S.

Answered by Theranjit, Maths A Level tutor with MyTutor


Write a balanced equation for the oxidation of Iron from the 2+ oxidation state to the 3+ oxidation state using the manganate ion.

Firstly, we need to write a balanced equation for the oxidation of Fe2+ to Fe3+ . We know that when something is oxidised, it is losing electrons, making the half equation: Fe2+ ---> Fe3+ + e- (we can check this by ensuring the charges on each side of the equation are equal). If iron is being oxidised, the manganate ion must be being reduced and so we write a half equation for the reduction of Mn(VII) to Mn(II): MnO4- --->  Mn2+. We must now balance the equation, first for number of atoms of each element by adding water and hydrogen ions: MnO4- + 8H+ --->  Mn2+ + 4H2O. We then balance for charge by adding electrons: MnO4- + 8H+ + 5e- --->  Mn2+ + 4H2O. Finally, we must combine the two half equations by multiplying everything in the equation for oxidation of iron by 5 so that, once combined, the number of electrons on each side of the equation cancel each other out. This leave us with the final equation: 5Fe2+ + MnO4- + 8H+ ---> 5Fe3+ + Mn2+ + 4H2O.
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Paige M.

Answered by Paige, Chemistry A Level tutor with MyTutor


“Is “Ethical” Consumerism A Solution To Poverty, Or A Dangerous Distraction?”

Argumentation is fundamental for a TSA essay - the purpose of the essay is NOT to give a balanced view of the argument demonstrating a sensibility to both sides, the purpose IS to create an argument that can convince the reader of your viewpoint and stance.
To that effect, the first step is to pick a side of the argument to fall down on - often it is better to pick a side less frequently argued and more conventionally controversial to demonstrate your unique ability to reason and argue. Also if it’s an argument that you don’t actually believe in, it will force you to follow the logic more rationally in order to make it as consistent as possible.
Example plan of argumentation:Introduction: frame the argument in a clear and precise way that gives you the freedom to expound - E.g. “Whether or not “ethical” consumerism, namely, the disregarding of consumption in accordance with the maximally efficient price (as determined by free market forces) in favour of consumption at artificially higher prices (as determined by socially-minded economic interventionists) rests on the definition of poverty undertaken in the argument. Poverty must be considered to refer to absolute levels of income and wealth not relative terms of inequality. To this end, “ethical” consumerism must be evaluated against whether or not it can alleviate the resource deprivation and economic isolation associated with absolute poverty, not whether it acts as a plausible method of progressive economic redistribution.”Argument 1 - the price mechanism is the most efficient way of allocating resources, any impediment on the price mechanism will create an inefficient and sub-optimal usage of resources. Argument 2 - “ethical” consumerism does nothing to address the causes of poverty; there is no focus towards education, healthcare, long-term economic development, or security that will alleviate the causes of poverty. Argument 3 - focusing the effort and financial capital wasted in a programme of “ethical” consumerism harms the consumer and, rather hypocritically, prices out of consumption those consumers with the least financial resources to begin with. As more goods were to see their prices rise with “ethical” consumerism as the driving force of price allocation, those economic agents we seek to assist out of poverty will only be further trapped by the higher pricing of “ethical” consumerism on an array of products. Optional Dismissed Counter-Argument 4 - whilst is can be argued that “ethical” consumerism can provide the means for incubating an industry that, in its conception, would have been eradicated from the comparative advantage other established competitors might have, and that those without the economic resources to consume “ethically” would not be forced to, this argument misses the point that any sub-optional misapplication of resources harms both consumers and producers on the margin of poverty first. Any inefficient allocation of resources has to be felt somewhere, and it is not with those with the means to consume “ethically”. Conclusion: take a firm stand and conclude concisely along the lines of your argumentation. It may be worthwhile saving a subtle new insight for the conclusion to provide the reader with a longer food for thought at the end - “E.g. “”Ethical” consumerism should, therefore, be considered in the same light as international emergency aid. An exogenous, financial injection designed to alleviate short-term economic suffering. If, however, “ethical” consumerism is to be considered as an authentic long-term solution to poverty, it’s price-mechanism distortion and efficiency incentivising corrosion will only deepen the causes of entrenched structural poverty, and fail to alleviate their symptoms. “Ethical” consumerism is nothing more than a well intentioned economic misdirect aimed at the symptom of poverty - regrettably, such a simple solution will do next to nothing to alleviate it’s cause.”
If you keep in the forefront of your mind that the TSA essay is a place to demonstrate your ability to argue through reason, with logic as the guiding force, you’ll write a sterling essay in ample time.
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Daniel W.

Answered by Daniel, TSA Oxford University tutor with MyTutor

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