
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)(xa) 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

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)(xa) 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 $\displaystyle \left(\frac{\pi}{4},1\right)$
$\displaystyle m = \sec^2\left(\frac{\pi}{4}\right) = 2$
$\displaystyle y  1 = 2\left(x  \frac{\pi}{4}\right)$
$\displaystyle y = 2\left(x  \frac{\pi}{4}\right) + 1$
$\displaystyle \tan\left(\frac{44\pi}{180}\right) \approx 2\left(\frac{\pi}{180}\right) + 1 = 1  \frac{\pi}{90}$