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Applications of Derivatives — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsApplications of Derivatives3 Marks
Question
Find the approximate value of
$f(x)=x^3+5 x^2-2 x+3 \text { at } x=1.98 \text {. }$
✓

Answer

Let $f(x)=x^3+5 x^2-2 x+3$
Differentiate w.r.t. $x$.
$
f^{\prime}(x)=3 x^2+10 x-2
$
Let $a=2, h=-0.02$
For $x=a=2$, from (I) we get
$
\begin{aligned}
f(a) & =f(2)=(2)^3+5(2)^2-2(2)+3 \\
\therefore \quad f(a) & =27 \quad \ldots
\end{aligned}
$
For $x=a=2$, from (II) we get
$
\begin{aligned}
f^{\prime}(a) & =f^{\prime}(2)=3(2)^2+10(2)-2 \\
\therefore \quad f^{\prime}(a) & =30
\end{aligned}
$
We have, $f(a+h) \doteqdot f(a)+h f^{\prime}(a)$
$
\begin{gathered}
f[(2)+(-0.02)] \doteqdot f(2)+(-0.02) \cdot f^{\prime}(2) \\
f(1.98) \doteqdot 27-(0.02) \cdot(30) \ldots[\text { From }
\end{gathered}
$
(III) and (IV)]
$
\begin{aligned}
& \doteqdot 27-0.6 \\
f(1.98) & \doteqdot 26.4
\end{aligned}
$

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