Question:
By application of derivatives, approximate value of
\[
\sqrt[5]{242}
\]
is
By application of derivatives, approximate value of
\[
\sqrt[5]{242}
\]
is
Show Hint
For approximation by derivatives, always choose the nearest number whose exact value is easy to compute.
Updated On: Jun 15, 2026
- \(2.9085\)
- \(2.9975\)
- \(2.9527\)
- \(2.8529\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
Approximation formula:
\[
f(a+h)\approx f(a)+hf'(a)
\]
Take function
\[
y=x^{1/5}
\]
Choose nearby perfect power.
Step 1: Choose nearest value.
\[ 243=3^5 \] Thus \[ 242=243-1 \] Take \[ f(x)=x^{1/5} \]
Step 2: Find derivative.
\[ f'(x)=\frac15x^{-4/5} \] At \[ x=243 \] \[ f'(243)=\frac1{5(3^4)} = \frac1{405} \]
Step 3: Apply approximation.
\[ f(242) \approx f(243)-f'(243) \] \[ = 3-\frac1{405} \] \[ = 3-0.002469 \] \[ = 2.99753 \] Hence \[ \boxed{2.9975} \]
Step 1: Choose nearest value.
\[ 243=3^5 \] Thus \[ 242=243-1 \] Take \[ f(x)=x^{1/5} \]
Step 2: Find derivative.
\[ f'(x)=\frac15x^{-4/5} \] At \[ x=243 \] \[ f'(243)=\frac1{5(3^4)} = \frac1{405} \]
Step 3: Apply approximation.
\[ f(242) \approx f(243)-f'(243) \] \[ = 3-\frac1{405} \] \[ = 3-0.002469 \] \[ = 2.99753 \] Hence \[ \boxed{2.9975} \]
Was this answer helpful?
0
0
Top TS EAMCET Mathematics Questions
- A bag contains $5$ red balls, $3$ black balls and $4$ white balls are drawn at random. The probability that they are not of same colour is
- If $cosec\, \theta - \cot \theta = 2017$, then quadrant in which $\theta $ lies is
- If $a$ is a unit vector, then $| a \times \hat{ i }|^{2}+| a \times \hat{ j }|^{2}+| a \times \hat{ k }|^{2}=$
- $A$ straight line makes an intercept on the $Y$-axis twice as long as that on $X$-axis and is at a unit distance from the origin. Then the line is represented by the equations
- Let $S$ and $S'$ be the foci of an ellipse and B be one end of its minor axis. If $SBS'$ is an isosceles right angled triangle then the eccentricity of the ellipse is
View More Questions
Top TS EAMCET Conic sections Questions
- The equation of the locus of a point which is at a distance of 5 units from a fixed point (1,4) and also from a fixed line 2x+3y-1=0 is
- If the normals drawn at the points $P\left(\frac{3}{4}, \frac{3}{2}\right)$ and $Q(3,3)$ on the parabola $y^2 = 3x$ intersect again on $y^2=3x$ at R, then R =
- If $\theta$ is the acute angle between the tangents drawn from the point (1,5) to the parabola $y^2 = 9x$ then
- Let P be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and let the perpendicular drawn through P to the major axis meet its auxiliary circle at Q. If the normals drawn at P and Q to the ellipse and the auxiliary circle respectively meet in R, then the equation of the locus of R is
- The midpoint of the chord of the ellipse $x^2+\frac{y^2}{4}=1$ formed on the line $y=x+1$ is
View More Questions
Top TS EAMCET Questions
- A bag contains $5$ red balls, $3$ black balls and $4$ white balls are drawn at random. The probability that they are not of same colour is
- If $cosec\, \theta - \cot \theta = 2017$, then quadrant in which $\theta $ lies is
- If $a$ is a unit vector, then $| a \times \hat{ i }|^{2}+| a \times \hat{ j }|^{2}+| a \times \hat{ k }|^{2}=$
- $A$ straight line makes an intercept on the $Y$-axis twice as long as that on $X$-axis and is at a unit distance from the origin. Then the line is represented by the equations
- Let $S$ and $S'$ be the foci of an ellipse and B be one end of its minor axis. If $SBS'$ is an isosceles right angled triangle then the eccentricity of the ellipse is
View More Questions