The area in m 2 of a blot of ink is growing such that after t second, its area is given by, A = 2 t 2 + t 3 + 5 . Determine the rate of increase of area at t = 10 second.

The general formula of basics of differentiation, d x n d x = n x n - 1 . Given the equation of area in terms of time is, A = 2 t 2 + t 3 + 6 , the rate of increase of area, d A d t = d 2 t 2 + t 3 + 6 d t = 4 t + 1 3 , t = 10 s , then the rate of change of area, d A d t = 4 × 10 + 1 3 = 40.3 m 2 s - 1

Obtain the differential coefficient of the following (i) x – 1 2 x + 5 (ii) 1 2 x + 1 (iii) 3 x + 4 4 x + 5 (iv) x 2 x 3 + 1

(i) d d x ( x - 1 ) ( 2 x + 5 ) Using differentiating by parts method: = ( x - 1 ) d d x ( 2 x + 5 ) + ( 2 x + 5 ) d d x ( x - 1 ) = ( x - 1 ) ( 2 ) + ( 2 x + 5 ) = 2 x - 2 + 2 x + 5 = 4 x + 3 (ii) d d x 1 ( 2 x + 1 ) = d d x ( 2 x + 1 ) - 1 = - 2 ( 2 x + 1 ) - 2 = - 2 ( 2 x + 1 ) 2 (iii) d d x 3 x + 4 4 x + 5 = ( 4 x + 5 ) d d x ( 3 x + 4 ) - ( 3 x + 4 ) d d x ( 4 x + 5 ) ( 4 x + 5 ) 2 = 12 x + 15 - 12 x - 16 ( 4 x + 5 ) 2 = - 1 4 x + 5 2 (iv) d d x x 2 x 3 + 1 = ( x 3 + 1 ) d d x ( x 2 ) - x 2 d d x ( x 3 + 1 ) ( x 3 + 1 ) 2 = 2 x 4 + 2 x - 3 x 4 ( x 3 + 1 ) 2 = 2 x - x 4 x 3 + 1 2

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Beta Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>BEGINNER'S BOX - 9

Embibe Experts Solutions for Chapter: Mathematical Tools in Physics, Exercise 9: BEGINNER'S BOX - 9

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Beta Question Bank for Medical: Physics>Chapter 2 - Mathematical Tools in Physics>BEGINNER'S BOX - 9>Q 1

The area in $m^{2}$ of a blot of ink is growing such that after $t$ second, its area is given by,
$A = 2 t^{2} + \frac{t}{3} + 5 .$
Determine the rate of increase of area at $t = 10$ second.