18652 questions

Write down the equations of the three asymptotes and the coordinates of the points where the curve y = (3x+2)(x-3)/(x-2)(x+1) crosses the axes.

Step 1: find the asymptotesThere are two kinds of asymptotes we look for: vertical and horizontal.The vertical ones happen when y goes to negative or positive infinity, which is when the denominator of the fraction is 0. Here, the denominator of the fraction is 0 when x = 2, and x = -1.The horizontal ones happen when x goes to negative or positive infinity. We can find out the value of y by looking at what the fraction tends to when x goes to infinity or negative infinity. We can make arguments like this: 3x+2 is approximately equal to 3x when x is large, and x-3, x-2, x+1 are approximately equal to x for the same reason. So we can put that into the fraction, and say that y is approximately equal to 3x * x / x * x = 3 when x is negative/positive infinity. So the horizontal asymptote is at y = 3.Overall answer to step 1: x = 2, x = -1, y = 3Step 2: Coordinates of the points where the curve touches the axisThe curve touches the axis when x=0 or y=0.When x=0, y = (2)(-3)/(-2)(+1) = -6 / -2 = 3. So the coordinate is (0,3).When y=0, 0 = (3x+2)(x-3)/(x-2)(x+1). Multiplying both sides by (x-2)(x+1), we seethat 0 = (3x+2)(x-3). We can expand the brackets to find that this is just a quadratic equation: 0 = 3x^2 - 7x - 6. Solving this, we get x = 3 and x = -2/3. So the coordinates are (3, 0), (-2/3, 0). Overall answer to step 2: (0,3), (3,0), (-2/3, 0)
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Meg T.

Answered by Meg, Further Mathematics tutor with MyTutor


How can I find the stationary point of y = e^2x cos x?

At a stationary point of y, dy/dx = 0.Step 1: Let's find dy/dx first by differentiating. To differentiate the product of two functions, we can use the product rule:d(fg)/dx = f * dg/dx + df/dx * g. So dy/dx = d(e^2x cos x)/dx = (e^2x) * (-sin x) + (2e^2x) * cos x = 2e^2x cos x - e^2x sin x.Step 2: Now we've found dy/dx, we can set it to 0. So we can set 2e^2x cos x - e^2x sin x = 0. Therefore 2e^2x cos x = e^2x sinx. We can cancel e^2x from each side because it is never equal to zero, therefore 2cos x = sin x. Dividing by cos x gives 2 = tan x. We can use arctan now to find x: arctan 2 = arctan(tan x) = x. Now finally we know x, so we can find y by plugging into our original equation: y = e ^ (2*arctan2) * cos (arctan2) = 4.09
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Meg T.

Answered by Meg, Maths tutor with MyTutor



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2t 2.

Answered by 2t, Philosophy tutor with MyTutor


How do I use "ser" and "estar"?

These two verbs translate to the english verb "to be" and, yes, it can be tricky to understand when we should use one or the other. As a general rule, a) "Ser" is used when we talk about a permanent feature that characterises something. -1. Dates and hours: "Son las ocho y media" (It's half past eight) / "Hoy es Jueves" (Today is Thursday). -2. Place of origin: "Soy de Alemania" (I'm from Germany). -3. Nationality: "Soy alemán/alemana" (I'm German). -4. Occupation: "Soy profesor" (I'm a teahcer). -5. Religious or political affiliation: "Soy musulmán" (I'm a muslim) / "Soy comunista" (I'm a communist). -6. The material something is made of: "Es de oro" (It's made of gold). -7. Possesion: "Es tuyo" (It's yours). -8. Relations: "Soy su padre" (I'm his/her father). -9. Location of an event: "El concierto es en Málaga" (The concert is in Malaga). -10. Essential qualities: "Soy humano" (I am human). b) "Estar" is used for temporary states. -1. Geographic location: "El Everest está en Nepal" (The Everest is in Nepal). -2. Emotional state: "Estoy un poco triste" (I'ma bit sad). -3. Progressive tenses (-ing): "Estoy cocinando" (I'm cooking).
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Joanes B.

Answered by Joanes, who has applied to tutor Spanish with MyTutor


Integrade the function f(x) = 1/(x^2 - 5x +6) with respect to x.

First, we need to factorise the denominator and therefore we use this formula: x1,2 =[ -b + sqrt(b2 -4 a c) ]/ (2a). You obtain two roots, x = 2 and x=3 and the integration is simplified to : 1/[(x-2)(x-3)], but still cannot be solved.By using the method of partial factions the integration becomes solvable as it can be split in two separate functions: -1/(x-2) + 1/(x-3). The solution is: -ln(x-2) + ln(x-3) + cI skipped few steps as I cannot show all the working using this text box. In a tutorial that when I will be using the whiteboard it will become clear as I will show all the steps and methodology.
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Efstathios E.

Answered by Efstathios, Maths tutor with MyTutor


a ball is dropped from rest off a cliff of height 50m, determine the final velocity of the ball assuming no air resistance.

Consider the appropriate equation of motion to solve this (use SUVAT method). The equation v^2=u^2+2*a*s is appropriate to solve. Substituting in the knowns gives v=+sqrt(2*9.81*50)=31.32 which, quoted to the correct number of significant figures is 30m/s
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Cameron J.

Answered by Cameron, Physics tutor with MyTutor


Describe the architectural decoration of the Parthenon.

The Parthenon is a temple dedicated to Athena and is built in both Doric and Ionic architectural styles. Built by Phidias under Pericles, the building was funded by the left over money from the Delian League and stands on the Acropolis in Athens. The pediments of the Parthenon show the birth of Athena on the East side and the West depicts the battle for patronage of the city between Poseidon and Athena. Beneath these and surrounding the top of the whole structure in a band are the metopes, with triglyphs in between each one. The South metopes depict the Centauromachy, the West the Amazonomachy, the East the Gigantomachy and finally, the North the battle between the Greeks and the Persians. The imagery on these Metopes depicts either the triumph of civilisation over the Barbarian, most obvious in the grotesque and violent depictions of all of the 'barbarians' versus the serene youth and calm of the Greeks.On the upper side of the space between the inner and outer colonnades is the Ionic frieze of the Parthenon that depicts some kind of procession, which is not entirely clear. It could be the first Panathenaic procession, since the temple is dedicated to Athena, or it could just be a generic depiction of this. One other possibility is the depiction of the myth of Erechtheus, the autochthonous Athenian figure. The inclusion of so many gods on this temple shows how Athens is under the protection of not only Athena, but the Pantheon of gods too.
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Riyana K.

Answered by Riyana, Classical Civilisation tutor with MyTutor


What are the potential disadvantages of Trades Unions?

1. Creates UnemploymentIf labour markets are competitive, and trade unions are successful in pushing for higher wages, it can cause disequilibrium unemployment. Union members can benefit from higher wages, but outside the union, there will be higher unemployment.
2. Lost ProductivityIf unions go on strike and work unproductively (work to rule) it can lead to lost sales and output. Therefore their company may go out of business and be unable to employ workers at all. In many industries, trade unions have created a situation of a confrontational approach.
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Anish P.

Answered by Anish, Economics tutor with MyTutor

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