F.E. Sem-I Dept. of Applied Mathematics
TUTORIAL-01
[SUCCESSIVE DIFFERENTIATION]
1)If , then P.T.
2)Find nth derivative of y =
3)Find yn if y=
4)If then show that yn = (-1)n-2 (n-2)![ -]
5)Find nth order derivative of
6)y= tan -1 [] P.T yn= (-1)n-1(n-1)! Sin( nθ ) sin(n)θ, where θ = tan -1 ()
7)U = eaxcos(bx+c) then P.T. Un= (a2+b2)n/2eaxcos[bx+c + n tan-1(b/a)]
8) Find yn if y = e5x cosx cos3x
9) If y= xnlogx then P.T yn+1 =
10) If U1/k + U-1/k = 2x then show that (x2-1)Un+2+ (2n+1)xUn+1+ (n2-k2) Un=0
11) If y = a cos (logx) + b sin (logx) then show that x2yn+2 + (2n+1)x yn+1+ (n2+1)yn =0
12)If y = P.T (1-x2) yn-[2(n-1)x+1]yn-1- (n-1)(n-2)yn-2=0
13) If y= eacosxthen show that (1-x2)yn+2 + (2n+1)x yn+1- (n2+a2)yn =0
14)If x = sinθ , y=sin(2θ) prove that (1-x2)yn+2 - (2n+1)x yn+1- (n2-4)yn =0
15)If x = sinh(y) prove that (1+x2)yn+2 + (2n+1)x yn+1+ n2yn =0
16) Leibnitz’s Theorem and use it to find yn for y = e2x(x3+x+1)
17) If y = then prove that yn =[ log x- (1+++---+)]
18)U = (x2-1)nthen prove that [(x2-1)Un] = n(n+1) Un where Un=U
19) y = tan-1[] then prove that (x2+a2)yn+2 + (2n+1)x yn+1+ n(n+1)yn =0
20)If u= sin [ log(1+2x+x2)] then prove that
(x+1)2Un+2+ (2n+1) (x+1)Un+1+ (n2+4) Un=0
TUTORIAL-02
[PARTIAL DIFFERENTIATION]
1) If then prove that andalso find the value of
2) If u = log (x2+y2 +z2) then prove that x = y = z
3) If xx yy zz = c, show that at x = y = z, = -[xlog(ex)] and also P.T
- 2xy + = at x = y= z
4) x = rcosθ , y = rsinθ then prove that + = [()+()]
5) If u= (8x2+y2)[logx-logy] then find x+y
6) If u= sin[] then show that = -
7) Verify the Euler’s theorem for u = x2tan -1 () - y2tan -1 () ,xy 0
and also show that =
8) If F(x,y) = ++ then show that x+y+2F(x,y) =0
9) If u= tan -1 [] Then prove that x2+ 2xy+y2 = 2cos3u sinu
10) If u= then prove that
x2+ 2xy+y2= tan u[]
11) If x = ecos(rsinθ) and y = esin(rsinθ) then P.T.
= and = - Hence deduce that ++= 0
12)If z = f(u,v) where u = x2-2xy-y2and v=y then show that
(x+y)+ (x-y) = 0
is equivalent to =0
13) If logeθ = r-x where r2= x2+y2 then show that =
14) If F=F(x,y) and If x+y = 2eθcos(Ф) , x-y= 2ieθsin(Ф) then prove that
15) State and Prove Euler’s Theorem on Homogeneous Functions of Three
Independent Variables.
16) If θ= te then find the value of ‘n’ for which
17) If u = f(r) and x = rcoθ , y = rsinθ , then P.T., where
18) If , and lx+my+nz=0 then prove that
TUTORIAL NO -03
[Mean Value Theorem, Errors and Approximations]
1)Verify Rolle’s Theorem for function ex (sinx-cosx) in [ ,]
2) State & Verify Rolle’s Theorem for f(x) = , [0,Π ]
3) P.T. between any two real roots of the equations exsinx = 1, there is at least one root of the equation excos x + 1= 0
4)Verify Rolle’s Theorem , F(x) = log [] in (2,3)
5)Using Lagrange’s Mean Value Theorem prove that if a < 1, b< 1 and a< b, then
< sin-1b – sin-1a <
Hence deduce that (i) < sin-1() <
(ii) < sin-1() <
6) If f(x+h)= f(x) + f ’(x)+ f ”(θh+a) Find θ as x a, f(x) deing (x-a)
7) Use appropriate Mean Value Theorem to Prove that
= cot c ; a <c <b; state theorem you use.
8)If 1 <a <b S.T there exist ‘c’ satisfying a <c < b such that log [ ] =
9)Using appropriate Mean Value Theorem Prove that for a < c < b
= , Hence deduce that ec = sinx = (ex-1) cos c
10] Using differential, calculate approximate value of f(0.999) where
f(x) = 2x4+7x3-8x2+3x+1
11] consider area of a circular ring as an increment of area of a circle, find approximately the area of a ring whose inner and outer radii are 3 and 3.02 inches respectively
12] If the radius of the sphere is measured as 5 inches with a possible error of 0.02 inches, find approximately the greatest possible error and percentage error in the computed value of the volume.
13] In calculating the volume of right circular cone errors of 2% and 1% are
made in height and radius of base respectively, find the percentage error in
volume.
14] Find approximately
15] Find approximately by using theory of approximation.
16] If , find the approximate value of f when x=1.99, y=3.01,
z=0.98
TUTORIAL NO -04
[Applications of Partial Derivatives & Maxima and Minima]
1] Find minimum and maximum value of the function
f(x,y) = (x+y).e6X+2X
2] Show that the minimum value of u =xy+a(+) is 3a
3] Find the dimensions of the rectangular box with open top of maximum
capacity whose surface area is 432 cm2
4] Find maximum value of xm yn zp where x + y +z = a
5] Find the minimum and maximum distances from the origin to the
curve 3x2+4xy+6y2=140
6] Divide 24 into three parts such that the continued product of the first,
square of the second and cube of the third may be maximum.
7] Determine the point on paraboloid z = x2+y2 which is closest to the
point ( 3, -6, 4)
8] Find the minimum distance of any point from the origin on the plane
x +2y +3z = 14.
9] Find the minimum and maximum distance of the point (3,4,12) from the
Sphere x2+y2+z2 = 1
10] Examine the minimum and maximum on the surface
11] Examine the minimum and maximum on the surface
12] A rectangular box , open at top is to have a volume of 108 m3. Find the
dimensions of the box so that the surface area is minimum.
TUTORIAL NO -05
[Indeterminate Forms]
A] Evaluate the following limits:
1) 2) 3) 4)
5) Find values of a,b,c such that
6) Find the constants a,b,c if
7) Find value of ‘p’ if is finite.
8) Evaluate 9) Evaluate
10) Find a,b if 11) Evaluate
12) Prove that 13) Evaluate
14) Evaluate 15) Evaluate
16) Evaluate 17) Evaluate
18) Evaluate
TUTORIAL NO -06
[Expansions of functions]
1)Test the series for convergence 1+
2)Test the series
3)Examine the convergence of series
4)Test the convergence of series whose nth term is
5)Test the convergence of for x >0 and x1
6)Test the convergence of series
7)Test the convergence of
8)Prove that ecosbx = 1+ ax +
9) Prove that
10)Show that log(secx)=
11)P.T sin[e-1]=
12)P.T. log[1+e]=
13)P.T. sinx = x+
14)S.T. cos[tanh(logx)]=
15)Expand in powers of x and Hence show that
16)Expand log[] upto term in x
17)If y = then prove that x=
18) Show that xcosecx =
19) Express 2x +3x-8x+7 in terms of (x-2) by using Taylor’s Theorem.
20) By Using Taylor’s Theorem arrangethe function in Powers of x
[7+(x+2) +3(x+2)+(x+2)-(x+2)]
21)Calculate the value of to four decimal places using Taylor’s
Theorem
22) Use TSE to calculate approximately (63.7)
23) Use TSE to calculate approximately (2.98)
TUTORIAL NO -07
[Complex Numbers-I]
1)Express the complex numbers in the form x+iy
i) ii) iii)
2) Find modulus and amplitude of
i) ii) iii)
3) Prove that the statements ‘ Re(z)>0’ and ‘/z-1/ < /z+1/’ are equivalent
4) Express in polar form i)1-cosα+isinα ii)
5) If α & β are roots of x2-2x+2=0 then show that
6) If z=cosθ+isinθ , show that
7) If z1, z2 are two complex numbers then show that
8)If x and y are real, solve the equation
9)If (1+cosθ+isinθ)(1+cos2θ=isin2θ)=u+iv prove that
i) ii)
10) If sinθ+sinФ=0 & cosθ+cosФ=0 then prove
cos2θ+cos2Ф= 2cos(Π+θ+Ф) & sin2θ+sin2Ф = 2 sin(Π+θ+Ф)
11) Use De`moivre`s Theorem to solve equation x-x+1=0
12) solve
13) Express cosθ in term of multiple of θ
14) Express sinθ in term of multiple of θ
15) Express in powers of sinθ only
16) Show that
17) Show that
18) Simplify
TUTORIAL NO -08
[Complex Numbers-II]
1] Find tanhx if 5sinhx-coshx=5
2] S.T = cosh6θ+sinh6θ
3] If y=log(tanx) P.T i) sinhny =
ii) 2coshnycosec2x = cosh(n+1)y+cosh(n-1)y
4] If u= log tan[] P.T i) tanh = tan ii) coshu = secθ
5] S.T
6] If tan(α+iβ)=x+iy show that i) x2+y2+2xcot(2α)=1
ii) x2+y2-2ycoth(2β)+1=0
7] P.T i) ii) sinh(tanx) = log tan[]
8] P.T sin(cosecθ)=
9] If log[ cos(x-iy)]=α+iβ , P.T α= and find β.
10] Separate into real & imaginary part of ii
11] S.T
12] P.T
13] S.T i) sin(log i)=1 ii) cos(log i)=0
14] If Then S.T (α+β)= e and tan()=
15] P.T
16] P.T , is it defined for all values of z
TUTORIAL NO -09
[Vector Calculus]
1) Find the unit vector joining the points (2,0,-4) to (0,2,-3)
2) Find the distance between the points (-4,1,5) and (1,2,3)
3) Find the angle between the vectors i+4j+8k and 2i-j+2k
4) Find the projection of the vector 2i+3j+4k on 3i-2j+6k
5) Find the Value of p such that pi+2j-3k is perpendicular to the vector i-pj-2k
6) Find the area of the parallelogram whose adjacent sides are 3i+4j-k and 4i-j+2k
7) Find the sides and angles of the triangle whose vertices are (1,-2,2),(2,1,-1),
(3,-1,2).
8) If then show that x = x = x
9) Show that (x )x + (x )x = (x )x= 0
10) (+) {(+) x(x)} = 2 []
11) Show that [ (-) (-)(-)]=0
12) Show that [()x(+)].= []
13) i x (x i) +j x (x j)+k x (x k)=2 Prove
14) (x ).( x)+ (x ).(x)+( x ).(x) =0
15) Show that [].[ x ]=
16) Show that ( x ) x ( x )=0
17) Show that ( x).(x)=
18) If =2i+j+k , =i-2j+k , =i+k then show that
[ x , x , x ]=[]
TUTORIAL NO -10
[Vector Differentiation]
1) If = then show that x=k
2) Find the unit tangent vector to the curve x = t, y = t,z = tat t=1.
3) Prove that where =xi+yj+zk
4) A particle moves along the curve r=ecost i+esint j+ek find the tangent at t=0
and its projection on 4i-j+8k
5) Find the normal vector to the surface at (1,-2,-1)
6) Find the angle between the surfaces x2+y2+z2=9 & z = x2+y2-3 at (2,-1,2)
7)Find normal to at (1,1,1) and its projection on the vector =i+4j+8k
8) Find the directional derivatives of the surface xy3z2- 4 at (2,-2,1) in the
direction of 2i+3j+6k
9) In what direction from (3,1,-2) is the directional derivatives of
Maximum and what is its magnitude
10) Find the divergence of at (2,-1,1)
11) If find div. at (1,1,1)
12) Show that the normal vector to Ф = xy+yz+zx is both irrotational and solenoidal
13) Find the value of a such that is solenoidal
14) If is solenoidal,then find the
corresponding scalar potential
15) Show that , is irrotational for all n but is solenoidal only for n = -3
16) Find gradient vector to the surface Ф = log(yz+zx+xy)
17) Find the angle between the normals to the surface xy=z2 at (4,1,2) and (3,3,-3)
18) Prove that i) ii)
References :
1] Applied Mathematics –I by G. V. Kumbhojkar , C. Jamnadas
2] Applied Mathematics –I by Dr. Sai Subramaniam , Tech-Max
3] A text book of Applied Mathematics Vol-1&2 by P. N. Wartikar
& J. N. Wartikar , Vidyarthi Griha , Pune
4] Advanced Engineering Mathematics by Jain & Iyengar , Narosa
5] Advanced Engineering Mathematics by Zill & Kullen , Narosa
6] Higher Engineering Mathematics by H. K. Dass , S. Chand
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