TN Maths, 12th Chapter 8 Important Questions

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Class : XII

Subject : Maths

Chapter : 8

Chapter : Differentials And Partial Derivatives

IMPORTANT QUESTIONS

      

2 & 3 Marks Questions

1. 1.Let f,g=(a,b)→R be differentiable functions. Show that d(fg)=fdg+gdf

2. Let g(x)=x² +sin x calculate the differential dg.

3. If the radius of a sphere with radius 10cm, has to decrease by 0.1 cm approximately how much will its volume decrease.

4. Find the differential dy for each of the following function

i) y= (1-2x)³/(3-4x)

ii) y=(3+sin(2x)⅔)

iii) y=e^(x2-5x+7) cos (x²-1)

5. Find df for f(x) =x²+3x and evaluate if for

i) x=r and dx=0.1

ii) x=3 and dx=0.02

6. Assuming log10 e=0.44343 find approximate value of log10 1003.

7. Find ∆f and df for function f for the indicated values of x, ∆x and compare

i) f(x) = x³+ 2x²; x=R ∆r=dx=0.5

ii)f(x)= x²+2x+3; x= -0.5, ∆x=dx=0.1

8. Let f(x)= √x Find the linear approximation of x=27 Use the linear approximateto 3√27.2

9. Use the linear approximation to find approximate values of 

i) (123)⅔

ii) ⁴√15

iii) ³√26

10. Find a liner approximation for the following function of the indicated points 

i) f(x) =x³-5x+12, x0=2

ii) g(x)= √ (x²+9) , x0=4

iii) h(x)= (x/x+1) , x0=1

11. Find the liner approximation for f(x) = √(1+x), x ≥ -1 at x0=3 .Use the liner approximation to estimate f(30.2) 

12. Use linear approximation to find an approximate value of √ 9.2 without using a 

 Calculator. 

5 Mark Questions

1. The radius of a circular plate is measured as 12.65 cm instead of the length 12.5cm .Find the following in calculating the area of the circular palate.

i) Absolute error

ii) Relative error

iii) Percentage error

2. A sphere is made of ice having radius 10cm. Its radius decreases from 10cm to 9.8 cm find approximation for the surface area.

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

4. Show that the parentage error in the nth root of a number is approximately 1/n time the percentage error in the number.

5. In a newly developed city it is estimated that the voting population (in thousands) will increase according to V(t) =30+12t²-t³,o ≤ L ≤ 8 where t is the time in years. Find the approximate change in voters for the time change from 4 to 4 1/6 year.

6. 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 change number of words learned when x changes from

i) 1 to 1.1 hour 

ii) 4 to 4.1 hour 

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