Differentials practice problems
dy = f′(x) dx, and using it to estimate a small change. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
The radius of a sphere changes from \( \displaystyle 3 \) to \( \displaystyle \frac{301}{100} \). Use differentials to estimate the change in its volume, \( \displaystyle dV \).
Find the differential \( \displaystyle dy \) of \( \displaystyle y = \frac{7}{x} \), and evaluate it at \( \displaystyle x = -3 \), \( \displaystyle dx = - \frac{1}{10} \).
The radius of a circle changes from \( \displaystyle 11 \) to \( \displaystyle \frac{1101}{100} \). Use differentials to estimate the change in its area, \( \displaystyle dA \).
The radius of a circle changes from \( \displaystyle 10 \) to \( \displaystyle \frac{1001}{100} \). Use differentials to estimate the change in its area, \( \displaystyle dA \).
Find the differential \( \displaystyle dy \) of \( \displaystyle y = - 2 x^{2} + 2 x + 3 \), and evaluate it at \( \displaystyle x = 3 \), \( \displaystyle dx = \frac{1}{10} \).
Find the differential \( \displaystyle dy \) of \( \displaystyle y = \frac{3}{x} \), and evaluate it at \( \displaystyle x = 4 \), \( \displaystyle dx = \frac{1}{20} \).
The radius of a sphere changes from \( \displaystyle 12 \) to \( \displaystyle \frac{121}{10} \). Use differentials to estimate the change in its volume, \( \displaystyle dV \).
Find the differential \( \displaystyle dy \) of \( \displaystyle y = - 2 x^{3} - x - 1 \), and evaluate it at \( \displaystyle x = -2 \), \( \displaystyle dx = - \frac{1}{50} \).
The side of a cube changes from \( \displaystyle 7 \) to \( \displaystyle \frac{349}{50} \). Use differentials to estimate the change in its volume, \( \displaystyle dV \).
The radius of a circle changes from \( \displaystyle 9 \) to \( \displaystyle \frac{449}{50} \). Use differentials to estimate the change in its area, \( \displaystyle dA \).