Exercise 8.1: Linear Approximation

EXERCISE 8.1

1. Let f (x) = 3√x. Find the linear approximation at x = 27 . Use the linear approximation to approximate 3√27.2 .

2. Use the linear approximation to find approximate values of

(i) (123)2/3

(ii) 4√15

(iii) 3√26

3. Find a linear approximation for the following functions at the indicated points.

(i) f (x) = x3 âˆ’ 5x + 12, x0 = 2

(ii) g ( x) = √[x2 + 9], x0 = −4

(iii) h(x) = x / x+1 , x0 = 1

4. The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. find the following in calculating the area of the circular plate:

(i) Absolute error (ii) Relative error (iii) Percentage error

5. A sphere is made of ice having radius 10 cm. Its radius decreases from 10 cm to 9.8 cm. Find approximations for the following:

(i) change in the volume (ii) change in the surface area

6. The time T , taken for a complete oscillation of a single pendulum with length l , is given by the equation T = 2π√(l/g) , where g is a constant. Find the approximate percentage error in the calculated value of T corresponding to an error of 2 percent in the value of l .

7. Show that the percentage error in the nth root of a number is approximately 1/n times the percentage error in the number

Answers:

1. (i) 3.0074

2. (i) 24.73  (ii) 1.9688  (iii) 2.963

3. (i) 7x âˆ’ 4 (ii) 9−4x / 4 (iii) x+1 /4

4. (i) 0.0225Ï€ cm2,  (ii) 0.006 cm2 (iii) 0.6%

5. (i) Volume decreases by 80Ï€ cm3 â€ƒ (ii) Surface area decreases by 16Ï€ cm2

6.  1%