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MATHEMATICS( MA10001 ), ASSIGNMENT-I, SESSION-2006-2007

Note: Students are advised to submit the Assignments to their respective

Tutorial Teachers in Tutorial Class Perodically.

1. A function f is thrice differentiable on [a, b] and f(a)=f(b)=0 and f

(a) =

f

(b) = 0. Prove that f

(c) = 0 for some c (a, b).

2. Verify Rolles theorem for the function f(x) = a(x b)(x c)(x d) on theinterval [b, d] where b < c < d.

3. Prove that for the function f, f be zero at some point in (a, b), but all the

conditions of the Rolles theorem do not hold together:

f(x) =

2x 1 if 1 < x 21 if 0 x 11 2x if1 x < 0

4. If ann+1

+ an1n

+ . . . + a0=0, (n = -1) then prove that the equations anxn +an1xn1 + . . . + a0x=0 has a real root lying between 0 and 1.

5. If p(x) is a polynomial and k R, prove that between any two real roots ofp(x)=0, there is a root of p

(x) + kp(x) = 0.

6. The functions u, v, u, v are all continuous on R and uv uv = 0 in R. Prove

that between any two consecutive real roots of u=0 lies one real root of v=0

and between any two consecutive real roots of v=0 lies one real root of u=0.

7. If f(x) and g(x) are continuous functions in [a, b] and they are differentiable

in (a, b), then prove that,

f(a)g(b) g(a)f(b) = (b a)

f(a)g

(c) g(a)f(c)

where a < c < b.

8. Use M. V.T to prove, 0 < 1x

log ex1x

< 1, for x > 0.

9. Show that,

v u1 + v2

< tan1 v tan1 u < v u1 + u2

and deduce that,

/4 + 3/25 < tan1 4/3 < /4 + 1/6

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10. A function f is differentiable on [a, b] and f(a)=f(b)=0 and f(c)< 0(> 0) for

some c

(a, b). Prove that there is at least one point

(a, b) for which

f

() > 0(< 0).

11. Show that

sin sin cos cos = cot , where 0 < < < < /2

12. If f is differentiable on [0, 1], show by Cauchy mean value theorem that f(1)f(0) = f

(x)

2xhas at least one solution in (0, 1).

13. If f

(x) 0 on [a, b], prove that fx1+x22

12

f(x1) + f(x2)

, for any two

points x1, x2 [a, b].

MATHEMATICS( MA10001 ), ASSIGNMENT-II

1. If f

is continuous at a and f

(a) = 0 prove that limh0 = 12 , where isgiven by f(a + h) = f(a) + hf

(a + h), 0 < < 1.

2. If f(x) = sin x prove that limh

0 =1

3, where is given by

f(h) = f(0) + hf

(h), 0 < < 1.

3. Applying Taylors theorem with remainder prove that

1 + x2 x3

8 0

4. Applying Maclaurins theorem with remainder expand

(i)log(1 + x) (ii)(1 + x)m

5. Find the value of

(i) limx0sin2 xx2x2 sin2 x

(ii) limx0(tanxx

)1/x2

6. Find the value of a such that

limx0a sinxsin2x

tan3 x

is finite. Find the limit.

7. Find a and b such that

limx0x(1+acos2x)+b sin2x

x3= 1

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MATHEMATICS( MA10001 ), ASSIGNMENT-III

1. Examine the curve given by y = x42x3 + 1 for concavity and convexity. Alsodetermine its point of inflection.

2. Show that the curve y = 3x540x3+3x20 is concave upwards for 2 < x < 0and 2 < x < , but convex upwards for < x < 2 and 0 < x < 2. Alsoshow that x=-2, 0, 2 are its points of inflection.

3. Find the range of values of x for which the curve y = x4 6x3 + 12x2 + 5x + 7is concave upwards or downwards.

4. Find the points of inflection of the curve f(x) = e4x2

and determine the

intervals of convexity and concavity of the curve.

5. Find the asymptotes of the curves

(i) y3 x3 = 6x2.

(ii) y2(x2 a2) = x2(x2 4a2).

(iii) y2(x 6) = x3 27.

(iv) x = t2

1+t3, y = t

2+21+t

.

6. Find the points of inflection of the curve f(x) = (8x2 x3)13 and determinethe intervals of convexity and concavity of the curve.

7. Determine the asymptotes of the curve f(x) = x2+2x+2

.

8. Find the radius of curvature of the curves

(i) x = 6(t sint), y = 6(1 cost).

(ii) x2

a2+ y

2

b2= 1.

(iii) x = a cos3 , y = a sin3 .

(iv) r = 1+ecos

at = .

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MATHEMATICS( MA10001 ), ASSIGNMENT-IV

1. Show that

f(x, y) =

xyx2+y2

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

is not differentiable at (0, 0).

2. Let

f(x, y) =

x4+y4

xy , x = y0 ,x=y

Show that fx and fy exist at (0, 0) but f is not continuous at (0, 0).

3.

f(x, y) =

x sin 1

y+ x

2y2x2+y2

if y = 00 if y=0

Show that limy0 limx0 f(x, y) exists but neither lim(x,y)(0,0) f(x, y) nor

limx0 limy0 f(x, y) exists.

4. Let

f(x, y) =

xyx

2y2x2+y2

, if x2 + y2 = 00 , if x2 + y2=0

Prove that 2f

xy= 2f

yx, (x, y) = (0, 0).

5. Prove that the function f(x, y) =|xy| is not differentiable at the point (0,

0) but fx

and fy

exist at (0, 0).

6. Let

f(x, y) =

x sin 1y + y sin 1x , if xy = 00 , if xy=0

Show that the limit exists at the origin.

7. If

(x, y) =

(x2 + y2) log(x2 + y2) , if x2 + y2 = 00 , if x2 + y2=0

Show that xy(0, 0)=yx(0, 0), although neither xy nor yx is continuous at

(0, 0).

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8. If

h(x, y) =

x2 sin 1x

+ y2 sin 1y

, when xy = 0x2 sin 1x , when x = 0, y=0y2 sin 1

y, when x=0, y = 0

0 , when x=0, y=0

Show that neither hx nor hy is continuous at (0, 0) but h(x, y) is differentiable

at (0, 0).

9. If

g(x, y) = x3+y3

xy , if x = y

0 , if x=y

Show that g is not continuous at (0, 0) but both gx(0, 0) and gy(0, 0) exist.

MATHEMATICS( MA10001 ), ASSIGNMENT-V

1. If u=f(x, y), x=r cos , y=r sin show that

ux

2+u

y

2=u

r

2+

1

r2

u

2

2. If z=f(x, y), where x=u+v and y=uv, prove that

2z

u2 2

2z

uv+

2z

v2= (x2 4y)

2z

y2 2 z

y

3. If z is a function of two variables x and y and x=c cosh u cos v, y=c sinh u sin v,

prove that

2z

u2+

2z

v2=

1

2c2(cosh 2u cos2v)

2z

x2+

2z

y24. Show that the transformation x=r cos , y=r sin reduces the equation

xy2u

x2

2u

y2

(x2 y2)

2u

xy= 0 to r

2u

r u

= 0

5. Transform the equation

x2z

x+ y2

z

y= z2

by introducing new independent variables

u = x, v =1

y 1

xand w =

1

z+

1

x.

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6. A function f(x, y) having continuous second order partial derivatives when

expressed in terms of the new variables u and v defined by x= 12

(u + v) and

y2 = uv becomes g(u, v); Prove that

2g

uv=

1

4

2fx2

+ 2x

y

2f

xy+

2f

y2+

1

y

f

y

7. If u=tan1 x3+y3

xy , show that

x22u

x2+ 2xy

2u

xy+ y2

2u

y2= (1 4sin2 u) sin2u

8. If H(x, y) be homogeneous function of x and y of degree n having continuous

first order partial derivatives and

u(x, y) =

x2 + y2)n/2

show that

x(H

u

x) +

y(H

u

y) = 0

MATHEMATICS( MA10001 ), ASSIGNMENT-VI

1. Show that for 0 < < 1,

3 sin x sin y = xy 16

x3 + 3xy2

cos x sin y +

y3 + 3x2y

sin x cos y

2. Show that for 0 < < 1,

eax sin by = by+abxy+1

6a3x33ab2xy2 sin(by)+3a

2bx2yb3y3 cos(by)eax

3. Find the minimum value of x2 + y2 + (x + y + 1)2.

4. Find all maxima and minima of the function x3 + y3 63(x + y) + 12xy.

5. Find the shortest distance between the skew lines

x + 2

3=

y 24 =

z 21

= s (say)

x + 2

1 =y

1

2 =z

1

3 = t (say)

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6. Prove that the volume of the greatest rectangular parallelopiped that can be

inscribed in the ellipsoid

x2

a2+ y

2

b2+ z

2

c2= 1

is 8abc33

.

MATHEMATICS( MA10001 ), ASSIGNMENT-VII

(i) x2 dydx

+ xy =

1 x2y2

(ii) y3 2xy2dx + 2xy

2 x3dy = 0(iii)

xy sin xy + cos xy

ydx +

xy sin xy cos xy

xdy = 0

(iv)

y + y3

3+ x

2

2

dx + 1

4

x + xy2

dy = 0

(v)

2xy4ey + 2xy3 + y

dx +

x2y4ey x2y2 3x

dy = 0

(vi)

y2 + 2x2y

dx +

2x3 xy

dy = 0

Solve

(i) sin xdydx

+ 3y = cos x

(ii)

1 + y2

dx =

tan1 y x

dy

(iii) x

1 x2

dy +

2x2y y ax3

dx = 0

(iv) dzdx

+ zx

log z = zx2

(log z)2

(v)dy

dx y tan x = y2

sec x

(vi) x3 dydx x2y + y4 cos x = 0

(vii)

x3y2 + xy

dx = dy

MATHEMATICS( MA10001 ), ASSIGNMENT-VIII

(1) Using the method of variation of parameters, solve

(i) y2 + 4y = 4 tan 2x

(ii) y2 y = 21+ex

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(iii) y2 3y1 + 2y = ex1+ex

(iv) y2 2y1 = ex

cos x

(v) x2y2 + 3xy1 + y =1

(1x)2

(2) Solve the Eulers equations

(i)

D2 + 1x

D

y = 12logxx2

(ii)

x4D3 + 2x3D2 x2D + x

y = 1

(iii)

x2D2 3xD + 5y = x2 sin log x(iv)

x4D4 + 6x3D3 + 9x2D2 + 3xD + 1

y = (1 + log x)2

(v)

x2D2 3xD + 1

y = logx sinlogx+1x

(3) Solve the system of differential equations

(i) dxdt 3x 4y = 0, dy

dt+ x + y = 0

(ii) dydx

+ y = z + ex, dzdx

+ z = y + ex

(iii) dxdt

+ dydt 2y = 2 cos t 7 sin t, dx

dt dy

dt+ 2x = 4 cos t 3sin t

(iv) dxdt

+ 2 dydt 2x + 2y = 3et, 3dx

dt+ dy

dt+ 2x + y = 4e2t

(v) dxdt

+ 4x + y = te3t, dydt

+ y 2x = cos2 t

MATHEMATICS( MA10001 ), ASSIGNMENT-IX

1. Show that the function f(z) = |z| is nowhere differentiable but continuouseverywhere in C.

2. Show that the function f(z)=u+iv given by

f(z) =

x3(1+i)y3(1i)

x2+y2, z = 0

0 , z=0

is continuous and C-R equations are satisfied at (0, 0) but not differentiable.

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3. Show that the function f given by

f(z) = x2y5(x+iy)

x4+y10 , z = 00 , z=0

is not analytic at (0, 0) though it satisfies C-R equations at (0,0).

4. Find the points where the functions are differentiable and hence deduce that

they are nowhere analytic

(i)f=u+iv, u=x2 + y2, v=xy.

(ii)f=u+iv, u=x2y2, v=2x2y2.

5. State all the complex functions those are differentiable every where.

6. Let C. Give an example of a function which is differentiable only at thepoint and nowhere else.

MATHEMATICS( MA10001 ), ASSIGNMENT-X

1. Prove that u = x33xy2 is harmonic and find its harmonic conjugate and thecorresponding analytic function f(z).

2. If a function is analytic, show that it is independent of z.

3. Find the analytic functions whose real part are

(i) u = ex(x cos y y sin y)

(ii) u = ex{(x2 y2) cos y + 2x sin y}

4. Find a analytic function for which v(x, y) = log(x2 + y2) is the imaginary part.

5. Evaluating dz

z + 2

over the circle C: |z|=1, show that20

1 + 2 cos

5 + 4 cos = 0

6. f is analytic within and on C:|z | = r, then show that

f() =

20

1

2f( + rei)d

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7. Show that

(i)|z|=3/2

ez

(z 1)(z + 3)2dz =ei

8, (ii)

|z|=1

sin zz

dz = 0

8. Find the values of

(i) C

z3 4z 1z + 2i

dz, C : |z| = 3

(ii) C

z + 4

z2 + 2z + 5dz, C : |z + 1 i| = 2.

Coordinator Mathematics-I, 2006-2007:

Prof. M.P. Biswal

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