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That is qutie skilled Trebla. How did you manage to get this? What thought processes did you use? Also, not quite sure about ur first assumption. I do get how the area is between 2 rectangles. But why is then the area between 1/k+1 and 1/k Also, for the second part of Q. I do not get how...

bump?

hmmm... question is inequality? How did you sub into equn to make LHS = Also, perhaps more importantly, how did you get final greater then 1/(k+1) + how did you also get result of <=1, since k>=1

Use Induction to prove statement true for all positive integers n \frac{1}{n} \leq 1+ \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} - ln\ n \leq 1 Hence show that e * e^{\frac{1}{2}}*e ^{\frac{1}{3}}...*e^{\frac{1}{n}} does not converge to a limit

For this proof, can i do it like this? \int_{0}^{a} f(x)dx = \int_{0}^{a} f(a-x)dx \text{Let u=a-x }, du = -dx , x=a \ u=0, x=0 \ u=a\\ \int_{0}^{a}f(a-x) = \int_{a}^{0}f(u) -du(\text{Swapping Limits })\\ =-\int_{a}^{0}f(u) \\ =\int_{0}^{a}f(u) du =\int_{0}^{a}f(x) dx ( \text{since u is dummy...

Also... can some1 explain to me why the parametric equn of hyperbola has limited range: ie. -pi < theta < pi, and if P is on the left branch, -pi < theta < -pi/2, pi / 2< theta < pi

If they ask for the cartesian equn of an elipps, with x=3cost, y=2cost Would it be sufficent to say a=3, b=2 thus x^2/9 + y^2/4 = 1

But its finding a algerbric realtionship?

for ii) this is what ive got. Not sure if its rite |A| = rt3 (using a^2 + b^2 = modz - from x^2+2x+3=0) since |A| = |H| = |K| and H = -p+isqrt(q-p^2) sqrt(p^2+q-p^2) = sqrt3 q=3 from above 2p^2=q p=(sqrt6/2) thus subbing pq = 3(sqrt6)/2

How did you get from second last step to last step. Although thats the answer ive got too.

Is it applicable? Since it saves alot of time.

How are they defined? I had ideas that they were undefined and hence were critical pts. But also, i saw in Patel, that relative maxima, relative minima, and inflexion pts are criticals. Expand anyone?

Like times each pt together Say at y=x, x=1, y=1, and at y=f(x), x=1, y=5 then y=xf(x), at x=1 y =5 (1x5)

Just be interesting, if we could of found what hormones caused the mutation. Might relate back to humanity leading to hormone treatment of birth defects/ facial structural damages...

Untouchable is that how you would answer in exam?

use the calculataor. Except you dont say use a calculator, but put as x>>> inf, y>>> 0 etc... Also, i think that is division? Draw both graphs and calculate.

THe sum of the angles of a polygon of n sides is (2n-4) right angles, n>=3

\int \frac{1}{x \sqrt{x-1}} dx Using non-trig subsitutions.

How to graph: y = (x+1)^4/(x^4+1)

this is the moment you say; "please laugh..."