differentiation rules #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.

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

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

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) $e^{-.015}$

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

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

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

$y - 1 = x$

$y = x + 1$

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

b) tan 44 degrees

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

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

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

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

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

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

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

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