caculus questions...?

Question

1. A vertical circular cylinder has radius r ft. and height h ft. If the height and radius both increase at the constant rate of 2 ft/sec, then the rate in square feet per second at which the LATERAL SURFACE AREA increase is???

2. If f(x) = 10^x and
10^1.04 aprrox. equals 10.96
wat is the approx. value of f'(1)?

hmm. i actually got the solutions.. can ya guys check if its right??

1. 4pi (r+h)
2. 24

Answer 1

Both of the solutions you gave are correct. Good work!

Answer 2

2 is right, but 1 isn't, and I have to look at it some more to determine what the right answer is.

Anyway, the reason 2 isn't right is that the initial area is 2πrh and the area at any given time is 2π(r+Δr)(h+Δh) = 2π(rh + rΔh + hΔr + ΔrΔh) = 2πrh + 2π(rΔh + hΔr + ΔrΔh), so ΔA = 2π(rΔh + hΔr + ΔrΔh). Still trying to come up with a derivative.

Even after the first second, the change isn't 4π(r+h), because substituting 2 for Δr and 2 for Δh yields 2π(2r+2h+4) = 4π(r+h+2). It's also not linear, because substituting 4 for Δr and 4 for Δh yields 2π(4r+4h+16) = 8π(r+h+4), which is more than double the value at 1 sec.