Mathematics Marathon (1 Viewer)

[tommykins]

回复: Re: 回复: Re: Mathematics Marathon

A³ + B³ = (A + B)(A² - AB + B²)

OMG. Silly me.

A^3+B^3 = (A+B)(A^2-AB+B^2) = (A+B)[(A+B)^2-2AB-AB] = (A+B)[(A+B)^2 -3AB]
A+B = 5/3, AB = -4/3

Thus A^3+B^3 = 85/9

 

[zzdfa]

The answer for the infinity series is 2.

The sum -> 2 as x (numerator) -> infinity and y (denominator) -> infinity

prove it =p

bonus points for general soln for the sum to infinity of i/x^i

 

[lyounamu]

prove it =p

bonus points for general soln for the sum to infinity of i/x^i

S = a/(1-r) but you cannot use that formula here because the r is not constant.

I will just say in the question that as x/y -> 0, the S -> 2.

 

[bored of sc]

An easy one:

Find all real numbers x which satisfy the equation

4x⁴ = 4x² + 3

 

[Js^-1]

4x⁴ = 4x² + 3
4x⁴ - 4x² - 3 = 0
Let x² = m
4m² - 4m - 3 = 0
4m² - 6m + 2m - 3 = 0
2m(2m - 3) + (2m - 3) = 0
(2m + 1)(2m - 3) = 0

So
2m + 1 = 0
m = -1/2
x² = -1/2
No real solutions.

2m - 3 = 0
m = 2/3
x² = 2/3
x = ± √(2/3)

.: x = ± √(2/3)

Think thats right...

If f (t) = mt² - 3e^t + 7
Find f ' (t) and F (t) where F (t) = ∫ f (t) dt.

 

[tommykins]

回复: Re: Mathematics Marathon

f'(t) = 2mt - 3e^t
F(t) = ∫ mt² - 3e^t + 7 = mt³/3 - 3e^t + 7t

 

[bored of sc]

State the domain and range of the function:

y = 2 root (25-x²)

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[lyounamu]

State the domain and range of the function:

y = 2 root (25-x²)

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-5<= x <= 5

0 < = y <= 10

Integrate -sin 2x

 

[Js^-1]

1/2 cos 2x + C

Find the Co-ordinates of the point of inflection for
f (p) = p³ - 3p² + 6p

 

[lyounamu]

1/2 cos 2x + C

Find the Co-ordinates of the point of inflection for
f (p) = p<SUP>3</SUP> - 3p<SUP>2</SUP> + 6p

f'(p) = 3p^2 - 6p + 6
f''(p) = 6p - 6

Point of inflection is at f''(p) = 0
i.e. when 6p-6 = 0
p = 1

When p<1, f''(p) < 0
When p >1, f''(p) > 0 Therefore, change in concavity.

Therefore, point of inflection is at p=1

I am running out of question..by the way...I don't have any textbooks here to refer to...sorry guys...

 

[tommykins]

There's not many questions that are difficult =\, only 3D trig and circle geo is what I struggle with but those require diagrams.

 

[Forbidden.]

There's not many questions that are difficult =\, only 3D trig and circle geo is what I struggle with but those require diagrams.

whoa theyre the same topics i struggled in too ... plus permuations and combinations back then

 

[tommykins]

whoa theyre the same topics i struggled in too ... plus permuations and combinations back then

goddammit i hate permutations and combinations. i always miss a detail here and there

 

[Ali92l]

A line passes through the origin touches the curve with equation y=4x^3 +1 . Find the equation of this line.



meh

 

[tommykins]

Err wouldn't there be like a finite amount of solutions to that?

 

[Js^-1]

Touches as in 'is tangential to'?

 

[xMrRand0m]

can't you use the 'family of lines' technique?...and wouldn't the line cut through the curve?

 

[lolokay]

A line passes through the origin touches the curve with equation y=4x^3 +1 . Find the equation of this line.

Spoiler

the gradient of the line is
y' = 12x²

hence it's equation is
y-0 = 12x²x - 0
y = 12x³
so, solving simultaneously
y = 4x³ + 1
8x³ = 1
x = 1/2 is where it touches the curve

so, solving for the equation of the line (passing through 1/2, y(1/2)=3/2 with gradient 3)
y - 3/2 = 3x - 3/2
y = 3x

 

[ratcher0071]

Spoiler

the gradient of the line is
y' = 12x²

hence it's equation is
y-0 = 12x²x - 0
y = 12x³
so, solving simultaneously
y = 4x³ + 1
8x³ = 1
x = 1/2 is where it touches the curve

so, solving for the equation of the line (passing through 1/2, y(1/2)=3/2 with gradient 3)
y - 3/2 = 3x - 3/2
y = 3x

that's pretty neat

 

[tommykins]

Still confused about 'touches' and intersects.