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 :

If x³y² = (x - y)⁵, find dy/dx at (1, 2).

Solution

Problem 6 :

x = 2t + 5 and y = t² - 5, then dy/dx = ?

Solution

Problem 7 :

The cost function for the production of x units of a commodity is given by

C(x) = 2x³ - 15x² + 36x + 15

The cost will be minimum when x is equal to 

a)  3     b)  2    c)  1    d)  4

Solution

Problem 8 :

For the functions y = x³ - 3x, the value of d²y/dx² at dy/dx is zero, is

a)  ± 1      b)  ±3      c)  ± 6      d)  none

Solution

Problem 1 :

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 2 :

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 3 :

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 4 :

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 5 :

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 6 :

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 7 :

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

Problem 8 :

A particle is moving along the curve y² = x³ + 2x. When it passes the point (1, √3), we have dy/dt = 1. Find dx/dt.

Solution

Problem 9 :

Find

a) linear approximation of f(x) = √x at x = 16

b) use to approximate √15.9

Solution

Problem 10 :

The radius of a sphere is increased from 10 cm to 10.1 cm. Estimate the change in volume.

Solution

Problem 1 :

A particles moves along a line according to the law

s(t) = 2t³ – 9t² +12t – 4, where t > 0.

(i) At what time the particle changes direction?

(ii) Find the total distance travelled by the particle in the first 4 seconds.

(iii) Find the particle’s acceleration each time the velocity is zero.

Solution

Problem 2 :

If the volume of a cube of side length x is v = x³. Find the rate of change of the volume with respect to x when x = 5 units.

Solution

Problem 3 :

If the mass m(x) (in kilogram) of a thin rod of length x (in meters) is given by, m(x) = √(3x)   then what is the rate of change of mass with respect to the length when it is x=3 and x = 27 meters.

Solution

Problem 4 :

A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water?

Solution

Problem 5 :

An airplane is flying towards a radar station at a constant height of 6 km above the ground. If the distance between the airplane and the radar station is decreasing at the rate of 400 km per hour when s = 10 km, what is the horizontal speed of the plane ?

Solution

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