Recreational Problems (SM level)

[enwiabe]

Let the record show that I was wrong! I misdivided something. I have brought shame to the spec community *commits suicide*

[Ahmad]

7. If f is continuous and f(x) + f(-x) = x^2. What is int (-1 to 1) f(x) dx?

8. Let f(x) = (3x^4 + 16)/(2x^4 + 11). Find the whole number nearest to f(2^32).

9. Find the smallest possible constant A such that ln x <= Ax^2 for all x > 0.

[cara.mel]

for #4, why does tan(x)*tan(90-x) = 1?

That and #1 are the only ones I can work out how to do so far (without simply asking the calculator)

[enwiabe]

9. A = e^(1/2)

[Ahmad]

tan(x) * tan[90 - x] = sin(x) / cos(x) * sin[90 - x] / cos[90 - x]

Using complementary angles:

= sin(x) / cos(x) * cos(x) / sin(x) = 1

What sort of problems do you guys want? And what sort of difficulty? If I get feedback I can attract a wider spectrum of people.

[enwiabe]

8. Should be 1.

[Ahmad]

Try to post solutions, that way we can discuss methods, spot mistakes etc. I got something different for that one.

[Toothpaste]

7. If f is continuous and f(x) + f[-x] = x^2. What is int (-1 to 1) f(x) dx?

f(x) = x^2 / 2  , making f(x) + f[-x] = x^2 true.

int (-1 to 1) f(x) dx

=   [x^3 / 6](-1 to 1)
= 1/6 -  -1/6
= 2/6
= 1/3

I got this by just looking at it, so um, I'm probably wrong.

[AppleXY]

Used symmetry and int. properties for 7. Stil trying. :p

[Ahmad]

7. If f is continuous and f(x) + f[-x] = x^2. What is int (-1 to 1) f(x) dx?

f(x) = x^2 / 2  , making f(x) + f[-x] = x^2 true.

int (-1 to 1) f(x) dx

=   [x^3 / 6](-1 to 1)
= 1/6 -  -1/6
= 2/6
= 1/3

I got this by just looking at it, so um, I'm probably wrong.

You proved it for one such f(x), which yields the correct answer. This is a correct method, but can you prove it for f(x) in general? 

[Ahmad]

10.  Evaluate int (-pi/2 to pi/2) sin(x) sin[2x] dx.

11. Evaluate int (-pi/4 to pi/4) sin^2(x) / cos^2(x) dx.

12. Let f(x) = sin^4(x) + cos^4(x). Find the 100th derivative of f. [hide=hint](sin^2(x) + cos^2(x))^2 = 1 [/hide]

[Khangfu]

7. 2/3 - (-1,1)f(-x)dx ?

[Ahmad]

You should be able to evaluate it exactly. Here's a hint, split it up into (-1 to 0) and (0 to 1). On (-1 to 0) make the substitution u = -x.

=)

[asa.hoshi]

10. 4/3
11. -pi/2 + 2

[asa.hoshi]

5. x=1, x= [-3+squareroot(5)]/2, x= [-3-squareroot(5)]/2