If the solve the problem

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).

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