Hi

I got a calculus exam soon and during revision I noticed that some ofthe solutions were unclear to me. Could you please explain what is happening on areas marked red such as what rules are used.

1. Find the derivative of the following functions

d/dy Artanh (arctan (y^2))

= (Artanh)' (arctan (y^2)) * d/dy * arctan(y^2)

= [1/(1 - (arctan(y^2))^2] * [(2y)/(1 + y^4)]

I do not understand how did the 3rd line was made, I know that:

arctan dfferentiated is 1/(1+x^2). I presume the numerator of the 2nd part is 1 + y^2 and denom' is ((1 + y^2)^2)

2. Prove that the function x -> 1/x is an asymptotics to x -> x/[(1 + x^4)^(1/2)] for

x -> infin {x/[(1 + x^4)^(1/2)] } / (1/x) = x^2 / [(1 + x^4)^(1/2)]

and

= Lim x -> infin x^2 / [(1 + x^4)^(1/2)]

= Lim x -> infin x^2 / [(1 + [1/x^4])^(1/2)] = 1

At this stage I presumed you divide all the x's by the highest power of x which seems to be x^2 but the denominator does not show that it seems.

3. Give a short argument based on symmetry considerations that

Integ (lims of 2(pi) by -(pi)) : (x + (sin x)(cos^4x))dx =

Integ (lims of (pi) by (pi)) : (x + (sin x)(cos^4x))dx

__Integ (lims of 2(pi) by -(pi)) : (x + (sin x)(cos^4x))____dx__ +Integ (lims of (pi) by (pi)) : (x + (sin x)(cos^4x))dx

First part is equal to zero since this is odd. How do you confirm if the function is odd or even?

4. When is the Sandwich (Squish) Rule is used? Is it just for limits only?

5. Use comparison test to determine the convergence or divergence of

{Sum of}_infin(n=1) [1/( n + (n)^(1/2))]

Quite a few subjects im doing OK on just these seems unclear to me any help would be great.

Thanks