WORKSHEET ON LINEAR APPROXIMATION

Problem 1 :

Find differential dy for each of the following functions :

(i)  y  =  (1-2x)³/(3-4x)

(ii)  y  =  (3 + sin2x)^2/3

(iii)  y  =  e^{x^2-5x+7} cos (x²-1)

Solution

Problem 2 :

Find df for f(x)  =  x² + 3x and evaluate it for

(i) x = 2 and dx = 0.1

(ii) x = 3 and dx = 0.02

Solution

Problem 3 :

Find ∆f and df for the function f for the indicated value of x and ∆x and compare

(i)  f(x)  =  x³ - 2x² ; x  =  2, ∆x  =  dx  =  0.5

(ii)  f(x)  =  x²+2x+3; x  =  -0.5, ∆x  =  dx  =  0.1

Solution

Problem 4 :

Assuming log₁₀ e  =  0.4343, find the approximate value of log ₁₀ 1003         Solution

Problem 5 :

The trunk of a tree has diameter 30 cm. During the following year, the circumference grew 6 cm.

(i) Approximately, how much did the tree’s diameter grow?

(ii) What is the percentage increase in area of the tree’s cross-section?

Solution

Problem 6 :

An egg of a particular bird is very nearly spherical. If the radius to the inside of the shell is 5 mm and radius to the outside of the shell is 5.3 mm, find the volume of the shell approximately.

Solution

Problem 7 :

Assume that the cross section of the artery of human is circular. A drug is given to a patient to dilate his arteries. If the radius of an artery is increased from 2 mm to 2.1 mm, how much is cross-sectional area increased approximately?

Solution

Problem 8 :

In a newly developed city, it is estimated that the voting population (in thousands) will increase according to 

V (t) = 30 + 12t² − t³

0 ≤ t ≤ 8 where t is the time in years. Find the approximate change in voters for the change from 4 to 4  1/6 years.

Solution

Problem 9 :

The relation between the number of words y a person learns in x hours is given by 

y  =  52√x, 0 ≤ x ≤ 9 

What is the approximate number of words learned when x changes from

(i) 1 to 1.1 hour? (ii) 4 to 4.1 hour?

Solution

Problem 10 :

A circular plate expands uniformly under the influence of heat. If it’s radius increases from 10.5 cm to 10.75 cm, then find an approximate change in the area and the approximate percentage change in the area.

Solution

Problem 11 :

A coat of paint of thickness 0 2 cm is applied to the faces of a cube whose edge is 10 cm. Use the differentials to find approximately how many cubic centimeters of paint is used to paint this cube. Also calculate the exact amount of paint used to paint this cube.        Solution

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