Question:
If $y=x^{4}-10$ and if $x$ changes from 2 to $1.99$, the change in $y$ is
(a) $0.32$
(b) $0.032$
(c) $5.68$
(d) $5.968$
Solution:
Let $x=2$ and $x+\Delta x=1.99$
$\therefore \Delta x=1.99-2=-0.01$
$y=x^{4}-10$
Differentiating both sides with respect to $x$, we get
$\frac{d y}{d x}=4 x^{3}$
$\Rightarrow\left(\frac{d y}{d x}\right)_{x=2}=4 \times(2)^{3}=4 \times 8=32$
$\therefore \Delta y=\left(\frac{d y}{d x}\right) \Delta x$
$\Rightarrow \Delta y=32 \times(-0.01)=-0.32$
Thus, the change in $y$ is $0.32$.
Hence, the correct answer is option (a).