# Math Help - Five problems, derivatives and rates

1. ## Five problems, derivatives and rates

I'm reviewing so I can test out of calculus. I was given a study guide with answers included. These five I don't get. In numerical order, the answers are B, B, B, A, and B respectively. But why?

24. ...What?
31. ...What?
41. I keep getting 1/13.
43. ...What?
44. I keep getting 16pi/25.

Thanks

2. Originally Posted by Sir Cumphrence
In numerical order, the answers are B, B, B, A, and B respectively.
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