I get B B A B A...

24. Lay your ruler through the points on the curve where x = -2 and x = 4. Which tangent might be parallel?

31. d/dr(r^3) must = 12. Why?

41. I agree with you.

43. Separate variables, giving 1/F * dF/dt = -2. Now integrate each side with respect to t. Call the constant of integration ln(A). Then take the exp of each side and simplify. See what occurs when t = 0.

44. Is your fraction flipped? Agreeing with you again, then.

Edit:

Just in case a picture helps with 43...

Spoiler:

... where

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to t, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

And for 31... Related rates usually involves the same shapes, but travelling down instead of up...

Spoiler:

__________________________________________

Don't integrate - balloontegrate!

http://www.ballooncalculus.org/examples/gallery.html

http://www.ballooncalculus.org/asy/doc.html