1. ## differentiation rules #2

I need help on these problems. Thanks in advance.
1. Use differentials to estimate the amount of paint needed to apply a coat of paint .05 cm thick to a hemispherical dome with diameter 50 m.
2. Use a linear approximation or differentials to estimate the given number. L(x) = f(a)+f '(a)(x-a) I have tried to use the formula and pluged in the numbers but I still couldn't figure out the correct answer.
a) e^-.015
b) tan 44 degrees

2. 1. Use differentials to estimate the amount of paint needed to apply a coat of paint .05 cm thick to a hemispherical dome with diameter 50 m.

$\displaystyle dV = 2\pi r^2 \, dr$

$\displaystyle dV = 2\pi (25 \, m)^2 \, (.0005 \, m)$

dV will be in cubic meters

2. Use a linear approximation or differentials to estimate the given number. L(x) = f(a)+f '(a)(x-a) I have tried to use the formula and pluged in the numbers but I still couldn't figure out the correct answer.

a) $\displaystyle e^{-.015}$

use the line tangent to $\displaystyle y = e^x$ at $\displaystyle (0,1)$

$\displaystyle m = f'(0) = e^0$

$\displaystyle y - e^0 = e^0(x - 0)$

$\displaystyle y - 1 = x$

$\displaystyle y = x + 1$

$\displaystyle e^{-.015} \approx -.015 + 1 = .985$

b) tan 44 degrees

note that derivatives for trig functions are only valid for angles in radians.

$\displaystyle f(x) = \tan{x}$

$\displaystyle f'(x) = \sec^2{x}$

use line tangent to the point $\displaystyle \left(\frac{\pi}{4},1\right)$

$\displaystyle m = \sec^2\left(\frac{\pi}{4}\right) = 2$

$\displaystyle y - 1 = 2\left(x - \frac{\pi}{4}\right)$

$\displaystyle y = 2\left(x - \frac{\pi}{4}\right) + 1$

$\displaystyle \tan\left(\frac{44\pi}{180}\right) \approx 2\left(-\frac{\pi}{180}\right) + 1 = 1 - \frac{\pi}{90}$