Fourier series Tutorial - degree.vidhyadeep.orgdegree.vidhyadeep.org/civil/3110014...

VIDHYADEEP INSTITUTE OF ENGINEERING & TECH. ANITA KIM

Vidhyadeep Campus, Anita (Kim), Ta. Olpad, Dist. Surat Anita

Subject: Mathematics- I subject Code: 3110014 sem:1st (2018-2019)

Fourier series

Tutorial

1. Obtain the Fourier series of 𝑓(𝑥) = (𝜋−𝑥

2)

2in the interval 0 ≤ x ≤ 2𝜋. Hence, deduce

that 𝜋2

12=

1

12 −1

22 +1

32 − ⋯.

2. Find the Fourier series of the function 𝑓(𝑥) = 𝑥2; −𝜋 < 𝑥 < 𝜋.

3. Find the Fourier series of 𝑓(𝑥) = 𝑥 + in the interval −𝜋 ׀𝑥׀ < 𝑥 < 𝜋.

4. Find the Fourier series of periodic function 𝑓(𝑥) = −𝜋; −𝜋 < 𝑥 < 0

= 𝑥; 0 < 𝑥 < 𝜋

Hence, deduce that 𝜋2

8= ∑

1

(2𝑛−1)2∞𝑛=1 .

5. Find the Fourier series of 𝑓(𝑥) = 𝑥2; 0 < 𝑥 < 𝜋

= 0; 𝜋 < 𝑥 < 2𝜋.

6. Find the Fourier series of f(x) = eax in the interval – 𝜋 ≤ 𝑥 ≤ 𝜋. Here a is constant.

7. Obtain Fourier series for the function given by f(x) = 1 +2𝑥

𝜋; −𝜋 ≤ 𝑥 ≤ 0

= 1 −2𝑥

𝜋; 0 ≤ 𝑥 ≤ 𝜋. Hence

deduce that 1

12 +1

32 +1

52 + ⋯ = 𝜋2

8.

8. Find the Fourier series with period 2 to represent 𝑓(𝑥) = 𝑥2 + 𝑥 in the interval

-1<x<1.

9. Find the Fourier series with period 2 to represent 𝑓(𝑥) = 2𝑥 in the interval

-1<x<2.

10.Find the Fourier series of f(x) = 4 ;0<x<2

= -4 ;2<x<4

11. Find the Fourier series of f(x) =x2 , -2<x<2; f(x+4)=f(x).Hence, deduce the following.

(a) 1

12 −1

22 +1

32 −1

42 + ⋯ =𝜋2

12 (b)

1

12 +1

22 +1

32 +1

42 +1

52 + ⋯ =𝜋2

16

12. Find the Fourier series expansion of f(x)=x;−𝜋 ≤ 𝑥 ≤ 𝜋; 𝑓(𝑥 + 2𝜋) = 𝑓(𝑥).

13. Find half-range cosine series for f(x)=x;0<x<3.

14. Find (i) Fourier sine series (ii) Fourier cosine series and (iii) Fourier series of

f(x) =1;0<x<1

2

=0; 1

2< 𝑥 < 1.

15. Find the Fourier series of f(x)= 𝑥2

2,−𝜋 < 𝑥 < 𝜋.

16. Prove that ∫1−cos 𝜋𝑤

𝑤

∞

0sin 𝑥𝑤 𝑑𝑤 =

𝜋

2 𝑖𝑓 0 < 𝑥 < 𝜋 , 0 𝑖𝑓 𝑥 > 𝜋.

17. Find the Fourier series of f(x)= 𝜋 − 𝑥, 0 < 𝑥 < 𝜋.

18. Find the Fourier series of f(x)= │sin x│in – 𝜋 < 𝑥 < 𝜋.

19. Find the Fourier Cosine series of f(x)=ex, 0<x<L.

20. Find the Fourier series of the periodic function 𝜋 sin 𝜋𝑥 , 𝑝 = 2𝐿 = 1.

21. Obtain the Fourier series to represent the function 𝑓(𝑥) =1

4(𝜋 − 𝑥)2, 0 < 𝑥 < 2𝜋.

22. Find the Fourier series of 𝑓(𝑥) = 𝑥2 (0, 𝜋)

= 0 (𝜋, 2𝜋)


Vidhyadeep Campus, Anita (Kim), Ta. Olpad, Dist. Surat


Improper integrals

Tutorial

1. Evaluate∫𝑑𝑥

𝑥2+1

∞

0 .

2. Evaluate∫𝑑𝑣

(1+𝑣2)(1+𝑡𝑎𝑛−1𝑣)

∞

0.

3. Evaluate∫1

√3−𝑥𝑑𝑥

3

0.

4. Check the convergence of ∫1

𝑥2 𝑑𝑥5

0.

5. Check the convergence of ∫𝑑𝑥

1−𝑥

1

0. If convergent, then evaluate the same.

6. Check the convergence of ∫𝑑𝑥

√9−𝑥2

3

0.

7. Evaluate∫𝑑𝑥

(𝑥−1)23

3

0.

8. Find the value of Γ (−5

2).

9. Evaluate ∫ 𝑒−√𝑥∞

0𝑥

1

4𝑑𝑥.

10. Evaluate ∫ 𝑒−√𝑥∞

0𝑥

1

4𝑑𝑥.

11. Evaluate ∫ 𝑒−√𝑎𝑥∞

0𝑥𝑛𝑑𝑥.

12. B ( 4

3,

5

3)

13. Evaluate ∫𝑥2

√1−𝑥4

1

0𝑑𝑥 ∙ ∫

𝑑𝑥

√1−𝑥4

1

0

14. Prove that ∫𝑥8(1−𝑥6)

(1+𝑥)24

∞

0𝑑𝑥 = 0

15. Prove that ∫𝑒2𝑚𝑥+𝑒−2𝑚𝑥

(𝑒𝑥+𝑒−𝑥)2𝑛

∞

0𝑑𝑥 =

1

2𝐵(𝑛 + 𝑚, 𝑛 − 𝑚), 𝑛>m.

16. Find the volume of a right circular cone of base radius r and height h.

17. Use the method of slicing to finding the volume of solid with semicircular

base defined by y = 5√cos 𝑥 on the interval [−𝜋

2,

𝜋

2]. The cross sections of the solid

are squares perpendicular to the x-axis with base running from x-axis to the curve.

18. Find the length of the arc of the curve y = log sec x from x = 0 to x = 𝜋

3.

19. Find the length of the parabola x2 = 4y which lies inside the circle x2 = 4y

which lies inside the circle x2 + y2 = 6y.

20. Find the length of the curve x = eθ(sin𝜃

2+ 2 cos

𝜃

2) , 𝑦 = 𝑒𝜃 (cos

𝜃

2− 2 sin

𝜃

2)

measured from θ = 0 to θ = π.

21. Show that the length of one complete wave of the curve y = 𝑏𝑐𝑜𝑠 𝑥

𝑎 is equal to

the perimeter of the ellipse whose semi-axes are √𝑎2 + 𝑏2 and a.

22. Find the length of the cissoids r = 2a tan θ sin θ from θ = 0 to θ = 𝜋

4.

23. Find the length of the whole arc of the cardioids r = a ( 1 + cos θ) and show

that the upper half is bisected by the line θ = 𝜋

3.

24. Find the area of the surface of revolution of the solid generated by revolving

the ellipse 𝑥2

16+

𝑦2

4= 1 about the x- axis.

25. Find the surface area generated by revolving the loop of the curve

9ay2 = x(3a – x)2

26. Find the surface area of the solid generated by revolving the asteroid 𝑥2

3 +

𝑦2

3 = 𝑎2

3 about the x-axis.

27. Find the area of the surface of the solid generated by revolving upper half of

the cardioid r = a(1 – cos θ) about the initial line.

28. Find the surface area of the solid formed by the revolution of the loop about

the tangent at the pole of the curve r2 = a2 cos 2θ.




Indeterminate forms

Tutorial

1. Evaluate lim𝑥→0

𝑒𝑥+𝑒−𝑥−𝑥2−2

𝑠𝑖𝑛 2 𝑥−𝑥2 .

2. Evaluate lim𝑥→

1

2

cos2 𝜋𝑥

𝑒2𝑥−2𝑥𝑒.

3. Prove that lim𝑥→∞

(𝑎1

𝑥 − 1) 𝑥 = log 𝑎.


(𝑥

𝑥−1−

1

log 𝑥) .


(1

𝑥2 −1

sin2 𝑥) .

6. Prove that lim𝑥→0

(𝑎𝑥 + 𝑥)1

𝑥 = 𝑎𝑒.


(𝑎𝑥+𝑏𝑥+𝑐𝑥

3)

1

3𝑥= (𝑎𝑏𝑐)

1

9.


(1𝑥+2𝑥+3𝑥

3)

1

𝑥.


(cos 𝑥)cot 𝑥.

10. Evaluate: lim𝑥→0

(1

𝑥)

𝑡𝑎𝑛𝑥.


(1 − 𝑥2)1

log(1−𝑥) = 𝑒.


𝜋

2

(cos 𝑥)𝜋

2−𝑥

.


1

𝑥(1 − 𝑥𝑐𝑜𝑡 𝑥).

14. Evaluate lim𝑥→𝑎

log (𝑥−𝑎)

log(𝑒𝑥−𝑒𝑎).


𝜋

2

log sin 𝑥

(𝜋−2𝑥)2.

16. Evaluate (i) lim𝑥→0

(1

𝑥)

1−𝑐𝑜𝑠𝑥 (ii) lim

𝑥→0

tan 𝑥−𝑥

𝑥2 tan 𝑥.


𝑥𝑒𝑥−log(1+𝑥)

𝑥2 .


𝜋

2

2𝑥−𝜋

cos 𝑥.


(1+𝑥)1𝑥−𝑒+

1

2𝑒𝑥

𝑥2 .


(𝑒𝑥+𝑒2𝑥+𝑒3𝑥

3)

1

𝑥.



Anita


matrices

Tutorial

1. Find the inverse of the following matrices by Gauss-Jordan method.

(i) [1 2 32 5 31 0 8

] (ii) [1 0 1

−1 1 10 1 0

] (iii) [2 6 62 7 62 7 7

]

2. Solve the following linear systems of equations by Gauss elimination method,if

they possess a solution.

(i) x+y+2z = 9, 2x+4y-3z = 1, 3x+ 6y-5z = 0

(ii) x-y+z = 3, 2x-3y+5z = 10, x+y+4z = 4

(iii) x+2y+z = 8, 2x+3y+z = 13, x+y = 5

3. Solve the following linear systems of equations by Gauss -Jordan method.

(i) x+y+z=6 , x+2y+3z=14 , 2x+ 4y+ 7z=30

(ii) 3x+6y-3z=-2 , 6x+6y+3z=5,-2y+3z=1

(iii) x+2y+z-w=-2 , 2x+3y-z+2w=7, x+y+3z-2w=-6 , x+y+z+w=2.

4. Find the eigen values and corresponding eigen vectors for the following matrices.

(i) [1 2 20 2 1

−1 2 2] (ii) [

4 6 6−8 −10 −84 4 2

] (iii) [2 01 2

]

(iv) [6 −4 2

−2 5 −1−4 6 0

]

5. If A= [3 20 −7

], find the eigen values and eigen vectors for the following matrices:

(i) AT (ii) A-1 (iii) A3 (iv) A2-2A+I

6. Verify Cayley-Hamilton theorem for A=[−1 23 1

]. Hence find A5.

7. Using Cayley-Hamilton theorem, find A2,A-1, and A-2, from A= [2 4 −10 4 21 1 −2

].

8. Verify Cayley-Hamilton theorem for [−1 3 10 0 21 1 −1

].

9. If A=[1 2 00 −1 03 −7 4

], show that A4+A3+A2+A+I = 22A2+2A-19I.

10. Is the matrix A = [0 −2 21 2 03 2 0

] diagonalizable?

11. Determine the diagonal matrix from A = [1 0

−1 2]. Hence find A10 and A13.

12. Find an orthogonal matrix P that diagonalizes A = [1 44 1

].

13. Determine diagonal matrix orthogonally similar to the real symmetric matrix

A =[2 −1 −1

−1 2 −1−1 −1 2

].

14. Reduced the following matrices to row echelon form and hence find their ranks.

(i) [1 2 34 6 83 4 5

] (ii) [−2 1 31 4 50 1 2

]

15. Reduced the following matrices to reduced row echelon form and hence find their

ranks.

(i) [1 4 32 0 34 8 9

−11

−1] (ii) [

0 6 7−5 4 21 −2 0

]



Anita


Multiple integrals

Tutorial

1. Compute the integral ∬ 𝑥𝑦2𝐷

𝑑𝐴 where D is the rectangle definded by

0 ≤ 𝑥 ≤ 2 𝑎𝑛𝑑 0 ≤ 𝑦 ≤ 1.

2. Evaluate ∬ (4𝑥𝑦 − 𝑦3)𝐷

𝑑𝐴 where D is region bounded by y=√𝑥 𝑎𝑛𝑑 𝑦 = 𝑥3.

3. Evaluate the following integral: ∬ (1 − 6𝑥2𝑦)𝑅

𝑑𝐴 where R:0 ≤ 𝑥 ≤ 2, −1 ≤ 𝑦 ≤ 1

4. Evaluate the following integral: ∬ 𝑒𝑥

𝑦𝑅

𝑑𝐴 where 𝑅 = 1 ≤ 𝑦 ≤ 2, 𝑦 ≤ 𝑥 ≤ 𝑦3

5. Evaluate the following integral: Integrate f(u,v)=𝑣 − √𝑢 over the triangular region cut from the

first quadrant of the uv- plane by the u+v=1.

6. Evaluate the following integral: ∬ √𝑎2 − 𝑥2 − 𝑦2𝑑𝐴𝑅

where R is the positive quadrant of the

circle x2+y2=a2

7. Evaluate the following integral: ∬𝑟𝑑𝑟𝑑𝜃

√𝑎2+𝑟2𝑅 where R is a loop of r2 = a2 cos 2𝜃

8. Change the order of integration in ∫ ∫ 𝑑𝑥𝑑𝑦 𝑎+√𝑎2−𝑦2

𝑎−√𝑎2−𝑦2

𝑎

0 and hence evaluate the same.

9. Change the order of integration in ∫ ∫ 𝑒−𝑦2∞

𝑥

∞

0𝑑𝑦𝑑𝑥 and hence evaluate the same.

10. Sketch the region of integration, reverse the order of integration and evaluate the integral

∫ ∫𝑥𝑒2𝑦

4−𝑦

4−𝑥2

0

2

0𝑑𝑦𝑑𝑥

11. Evaluate ∬ 𝑥𝑦√𝑥2 + 𝑦2𝐷

𝑑𝑥𝑑𝑦 where D = {(𝑥, 𝑦) ∕ 1 ≤ 𝑥2 + 𝑦2 ≤ 4, 𝑥 ≥ 0, 𝑦 ≥ 0}

12. Evaluate ∬ 𝑥𝑦𝑅

𝑑𝑦𝑑𝑥 where R is the region between the circles x2+y2=1 and x2+y2=5

13. Evaluate ∬ 𝑒𝑥2+𝑦2𝑑𝑦𝑑𝑥

𝑅 where R is the semicircular region bounded by the X-axis and the

curve y= √1 − 𝑥2

14. Evaluate ∫ ∫ ∫ 𝑑𝑧𝑑𝑦𝑑𝑥𝑦−𝑥

0

1

𝑥

1

0

15. Evaluate ∫ ∫ ∫ (𝑥 + 𝑦 + 𝑧)2𝑥+2𝑦

0

𝑦

0

1

0𝑑𝑧𝑑𝑦𝑑𝑥

16. Evaluate ∫ ∫ ∫ (𝑟2 cos2 𝜃 + 𝑧2)2𝜋

0𝑟

√𝑧

0

1

0 𝑑𝜃𝑑𝑟𝑑𝑧

17. Find the jacobian of the transformation (i) x=rcos θ , y=rsinθ , z=z (ii) x=u2-v, y=u2+v

18. Evaluate ∬ (𝑥 + 𝑦)𝑅

𝑑𝐴 where R is the trapezoidal region with vertices given by

(0,0),(5,0),(5/2,5/2) and (5/2,-5/2) using the transformation x= 2u+3v, y= 2u-3v.

19. Given that x+y = u, y= uv, change the variables to u,v in the integral ∬ [𝑥𝑦(1 − 𝑥 − 𝑦)]1

2𝑑𝑥𝑑𝑦

taken over the area of the triangle with sides x=0,y=0, x+y = 1 and hence evaluate it.

20. Evaluate ∬ (𝑥2 − 𝑦2)2𝑑𝐴, over the area bounded by the lines │x│+ │y│=1 using the

transformations x+y = u, x-y = v.

21. Evaluate ∭ (𝑥2 + 𝑦2)2𝐸

𝑑𝑥𝑑𝑦𝑑𝑧 where E is the region bounded by the surface x2 + y2 ≤ 1 and

the planes z = 0 and z = 1.

22. ∭ 𝑦𝐸

𝑑𝑉 where E is the region that lies below the plane z = x+2 above the XY- plane and

between the cylinders x2 + y2 = 1 and x2 + y2 = 4

23. Evaluate ∭ 16𝑧𝐸

𝑑𝑉 where E is the upper half of the sphere x2 + y2 + z2 = 1

24. Use spherical coordinates to derive the formula for the volume of a sphere centered at the

origion and a.




Partial differentiations

Tutorial

1. Find lim(𝑥,𝑦)→(1,2)

5𝑥2𝑦

𝑥2+𝑦2.


𝑥2−𝑥𝑦

√𝑥−√𝑦.


𝑥−𝑦

𝑥+𝑦.


𝑥𝑦

𝑦2−𝑥2.


𝑥2𝑦

𝑥4+𝑦2.

6. Determine the set of points at which the given function is continuous:

f(x,y)=3𝑥2𝑦

𝑥2+𝑦2 , (𝑥, 𝑦) ≠ (0,0)

=0, (x,y) = (0,0)

7. Show that f (x,y) = 𝑥𝑦

√𝑥2+𝑦2, (𝑥, 𝑦) ≠ (0,0)

= 0, (x,y) = (0,0) is continuous at the origin.

8. Show that f (x,y) = (𝑥2 + 𝑦2) sin1

𝑥2+𝑦2 , (𝑥, 𝑦) ≠ (0,0)

= 0, (x,y) = (0,0) is continuous at the origin.

9. If u = log (tan x + tan y + tan z) then show that sin 2𝑥𝜕𝑢

𝜕𝑥+ sin 2𝑦

𝜕𝑢

𝜕𝑦+ sin 2𝑧

𝜕𝑢

𝜕𝑧= 2.

10. If u = 𝑥2+𝑦2

𝑥+𝑦 , show that (

𝜕𝑢

𝜕𝑥−

𝜕𝑢

𝜕𝑦)

2= 4 (1 −

𝜕𝑢

𝜕𝑥−

𝜕𝑢

𝜕𝑦).

11. If z = x+yx, prove that 𝜕2𝑧

𝜕𝑥𝜕𝑦=

𝜕2𝑧

𝜕𝑦𝜕𝑥.

12. If 𝜃 = 𝑡𝑛𝑒−𝑟𝑛

4𝑡 then find n so that 1

𝑟2

𝜕

𝜕𝑟(𝑟2 𝜕𝜃

𝜕𝑟) =

𝜕𝜃

𝜕𝑡.

13. If u = f (r) and r2 = x2+y2+z2, prove that 𝜕2𝑢

𝜕𝑥2 +𝜕2𝑢

𝜕𝑦2 +𝜕2𝑢

𝜕𝑧2 = 𝑓′′(𝑟) +2

𝑟𝑓′(𝑟).

14. If 𝑥2 = 𝑎√𝑢 + 𝑏√𝑣, 𝑦2 = 𝑎√𝑢 − 𝑏√𝑣 , where a and b are constants, find (𝜕𝑢

𝜕𝑥)y

(𝜕𝑥

𝜕𝑢)v

15. If u = ax+by, v = bx-ay, find the value of (𝜕𝑢

𝜕𝑥)y ∙ (

𝜕𝑥

𝜕𝑢)v ∙ (

𝜕𝑦

𝜕𝑣)x ∙ (

𝜕𝑣

𝜕𝑦)u .

16. If 𝑢 = sin (𝑥

𝑦) 𝑤ℎ𝑒𝑟𝑒 𝑥 = 𝑒𝑡 , 𝑦 = 𝑡2, 𝑓𝑖𝑛𝑑

𝑑𝑢

𝑑𝑡.

17. If 𝑢 = 𝑥2𝑦3, 𝑥 = log 𝑡 , 𝑦 = 𝑒𝑡 , 𝑓𝑖𝑛𝑑 𝑑𝑢

𝑑𝑡.

18. For 𝑧 = tan−1 𝑥

𝑦, 𝑥 = 𝑢 cos 𝑣 , 𝑦 = 𝑢 sin 𝑣, evaluate

𝜕𝑧

𝜕𝑢 𝑎𝑛𝑑

𝜕𝑧

𝜕𝑣 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (1.3,

𝜋

6)

19. If 𝑧 = 𝑓(𝑢, 𝑣)𝑎𝑛𝑑 𝑢 = 𝑥 cos 𝜃 − 𝑦 sin 𝜃 , 𝑣 = 𝑥 sin 𝜃 + 𝑦 cos 𝜃 , 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑥𝜕𝑧

𝜕𝑥+

𝑦𝜕𝑧

𝜕𝑦= 𝑢

𝜕𝑧

𝜕𝑢+ 𝑣

𝜕𝑧

𝜕𝑣 .

20. Find 𝜕𝑤

𝜕𝑟,

𝜕𝑤

𝜕𝑠 in terms of r and s if w = x+2y+z2, where x=

𝑟

𝑠, 𝑦 = 𝑟2 + log 𝑠 , 𝑧 = 2𝑟.

21. If 𝑢 = 𝑓 (𝑥

𝑦 ,

𝑦

𝑧 ,

𝑧

𝑥), prove that 𝑥

𝜕𝑢

𝜕𝑥+ 𝑦

𝜕𝑢

𝜕𝑦+ 𝑧

𝜕𝑢

𝜕𝑧= 0.

22. If (𝑐𝑜𝑠𝑥)𝑦 = (𝑠𝑖𝑛𝑦)𝑥,find 𝑑𝑦

𝑑𝑥.

23. If 𝑢 = 𝑥 log(𝑥𝑦)𝑤ℎ𝑒𝑟𝑒 𝑥3 + 𝑦3 + 3𝑥𝑦 = 1, 𝑓𝑖𝑛𝑑 𝑑𝑢

𝑑𝑥.

24. Find the equations of the tangent plane and normal line to the surface 2𝑥𝑧2 − 3𝑥𝑦 −

4𝑥 = 7 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (1, −1, 2).

25. Find the equations of the tangent plane and normal line to the surface cos 𝜋𝑥 − 𝑥2𝑦 +

𝑒𝑥𝑧 + 𝑦𝑧 = 4 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑃(0,1,2).

26. Show that the surfaces 𝑧 = 𝑥𝑦 − 2 𝑎𝑛𝑑 𝑥2 + 𝑦2 + 𝑧2 = 3 have the same tangent

plane at (1,1,-1).

27. Find all the stationary points of the function x3+3xy2-15x2-15y2+72x after examining

whether the function is maximum or minimum at those points.

28. Find the extreme values of x4+y4-2x2+4xy-2y2.

29.Find the minimum value of x3y2z subject to the condition x+y+z =1.

30. A rectangular box open at the top is to have a volume of 108 cubic metres. Find the

dimensions of the box if its total surface area is minimum.

31. Show that the rectangular solid of maximum volume that can be inscribed in a sphere

is a cube.

32. A rectangular box open at the top is to have a volume of 32 cubic units. Find the

dimensions of the box requiring least material for its construction.

33. A rectangular box without a lid is to be made from 12m2 of cardboard. Find the

maximum volume of such a box.

34. In a plane triangle ABC, find the extreme values of cos A cos B cos C.

35. Find the gradient of 𝑓(𝑥, 𝑦, 𝑧) = 2𝑧3 − 3(𝑥2 + 𝑦2)𝑧 + tan−1(𝑥𝑧) at (1,1,1).

36. If 𝑟 = 𝑥 î + 𝑦ˆ𝑗 + 𝑧ˆ𝑘, evaluate the following. (a)∇𝑟𝑛 (b) ∇ ׀𝑟𝑛 ׀

(c) ∇(3𝑟𝑛 − 4√𝑟 + 6𝑟−1

3) (d) ∇𝑟 (e) ∇(ln 𝑟) (f) ∇ (1

𝑟)

37. The temperature at any point in space is given by T = xy+yz+zx. Determine the

derivative of T in the direction of the vector 3î - 4ˆk at the point (1,1,1).

38. Find the scalar potential function f for A= y2î+2xyˆj-z2ˆk.

39. Find the angle between the surfaces 𝑥2 + 𝑦2 + 𝑧2 = 9, 𝑎𝑛𝑑 𝑧 = 𝑥2 + 𝑦2 − 3 at the

point (2,-1,2).

40. If the cone of revolution is z2 = 4(x2+y2),find a unit normal vector ˆn at the point

P(1,02).




Sequence & Series

Tutorial

1. Test the convergengence of the sequences { tanh n} , en and 1+(-1)n.

2. Show that the sequence {un} whose nth term is un = 1

1!+

2

2!+ ⋯ +

𝑛

𝑛! , n∈ N, is

monotonic increasing and bounded.Is it convergent?

3. Show that the sequence {𝑛

𝑛2+1} is monotonic decreasing and bounded. Is it convergent?

4. If x ∈ R with │x│< 1 then xn → 0 𝑎𝑠 𝑛 → ∞.

5. Check for convergence ∑𝑛+1

𝑛.∞

𝑛=1

6. Investigate the convergence of the series∑2𝑛+5

3𝑛∞𝑛=1 .

7. Show that {41/3n} converges to 1.

8. Test the convergence of the series ∑(2𝑛2−1)

1/3

(3𝑛3+2𝑛+5)1/4∞𝑛=1 .

9. Test the convergence of the series 1

1∙2∙3+

3

2∙3∙4+

5

3∙4∙5+ ⋯


2√1+

𝑥2

3√2+

𝑥4

4√3+

𝑥6

5√4+ ⋯.

11. Test the convergence of the series ∑𝑛𝑛 𝑥𝑛

(𝑛+1)𝑛∞𝑛=1 , x >0.

12. Test the convergence of the series ∑1

𝑛 log 𝑛 √log2 − 1

∞𝑛=1 .

13. Test the convergence of the series ∑2 tan−1 𝑛

1+𝑛2∞𝑛=1 .

14. Test the convergence of the series ∑1

2−

2

3+

3

4−

4

5+ ⋯∞

𝑛=1 .

15. Test the series for absolute or conditional convergence 1 −2

3+

3

32 +4

33 + ⋯.

16. Determine the interval of convergence for the series ∑2𝑛𝑥𝑛

𝑛!∞𝑛=0 And also, their behavior

at each end points.

17. For the series ∑(−1)𝑛 (𝑥+2)𝑛

𝑛∞𝑛=1 , find the radius and interval of convergence.For what

values of x does the series converge absolutely, conditionally?

18. Express 5+4(x-1)2-3(x-1)3+(x-1)4 in ascending power of x.

19. Find the expansion of tan ( 𝑥 +𝜋

4 ) in ascending powers of x up to terms in x4 and find

approximately the value of tan (43°).

20. If x = y(1+y2), prove that y = x-x3+3x5+…

21. Does the sequence whose nth-term is an = [(𝑛+1)

(𝑛−1)]

𝑛converges? If so,find lim

𝑛→∞𝑎n.

22.

The figure show the first seven of a sequence of squares. The outermost square has an area

4 m2. Each of the other squares is obtained by joining the midpoints of the sides of the

squres before it. Find the sum of areas of all squares in the infinite sequence.

23. Test the convergence of the series∑𝑛

𝑒−𝑛∞𝑛=1 .

24. Test the convergence of the series 1∙2

32∙42 +3∙4

52∙62 +5∙6

72∙82 + ⋯


12−3+

3

22−3+

3

32−3+

3

42−3+ ⋯



Subject: MATHEMATICS 1

Subject Code:3110014 Sem:1st(2019-2020)

Tutorial : 1 MATRIX (1) Reduced the following matrices to row echelon form and hence find their ranks.

(i) [1 2 34 6 83 4 5

] (ii) [−2 1 31 4 50 1 2

]

(2) Reduced the following matrices to reduced row echelon form and hence find their

ranks.

(i) [1 4 32 0 34 8 9

−11

−1] (ii) [

0 6 7−5 4 21 −2 0

]

(3) Solve the following linear systems of equations by Gauss elimination method,

If they possess a solution.

(i) x+y+2z = 9, 2x+4y-3z = 1, 3x+ 6y-5z = 0

(ii) x-y+z = 3, 2x-3y+5z = 10, x+y+4z = 4

(iii) x+2y+z = 8, 2x+3y+z = 13, x+y = 5

(4) Solve the following linear systems of equations by Gauss -Jordan method.

(i) x+y+z=6 , x+2y+3z=14 , 2x+ 4y+ 7z=30

(ii) 3x+6y-3z=-2 , 6x+6y+3z=5, -2y+3z=1

(iii) x+2y+z-w=-2 , 2x+3y-z+2w=7, x+y+3z-2w=-6 , x+y+z+w=2.

(5) Find the inverse of the following matrices by Gauss-Jordan method.

(i) [1 2 32 5 31 0 8

] (ii) [1 0 1

−1 1 10 1 0

] (iii) [2 6 62 7 62 7 7

]

(6) Find the eigen values and corresponding eigen vectors for the following matrices.

(𝑖) [1 2 20 2 1

−1 2 2] (ii) [

4 6 6−8 −10 −84 4 2

] (iii) [2 01 2

] (𝑖𝑣) [6 −4 2

−2 5 −1−4 6 0

]

(7) If A= [3 20 −7

], find the eigen values and eigen vectors for the following

matrices: (i) AT (ii) A-1 (iii) A3 (iv) A2-2A+I

(8) Verify Cayley-Hamilton theorem for A=[−1 23 1

]. Hence find A5.

(9) Using Cayley-Hamilton theorem, find A2,A-1, and A-2, from A= [2 4 −10 4 21 1 −2

].

(10) Verify Cayley-Hamilton theorem for [−1 3 10 0 21 1 −1

].

(11) If A=[1 2 00 −1 03 −7 4

], show that A4+A3+A2+A+I = 22A2+2A-19I.

(12) Is the matrix A = [0 −2 21 2 03 2 0

] diagonalizable?

(13) Determine the diagonal matrix from A = [1 0

−1 2]. Hence find A10 and A13.

(14) Find an orthogonal matrix P that diagonalizes A = [1 44 1

].

(15) Determine diagonal matrix orthogonally similar to the real symmetric matrix

A =[2 −1 −1

−1 2 −1−1 −1 2

].

Fourier series Tutorial - degree.vidhyadeep.orgdegree.vidhyadeep.org/civil/3110014...

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