Download OU (Osmania University) B.Sc Computer Science 1st Sem Differential Calculus Important Question Bank For 2021 Exam

BSc Odd Semester Question Bank

Paper-1: Differential Calculus

UNIT-I

2

1. If y = log, show that 2 =2log-3

3

+

2. If y =+ then show that 2y

2

1y3 = 3y2 .

2

3. If 2+2+2=1, show that 2 = 2-

(+)3

2

4

1

4. Find the nth derivative of (i) y = (+2)(2+3) (ii) y = (-1)(-2) (iii) y = 2-2

5. If y = Sin ax + Cos ax, prove that yn=[1-(-1) 2]

6. State and prove Leibnitz theorem.

7. If 1/+-1/=2, prove that (2-1)+2+(2+1)+1+(2-2)=0

8. If y = (-1), prove that (1-2)+2=(2+1)+1+(2-2) and

also find yn(0).

9. State and prove Maclaurin's theorem.

10. Using Maclaurin's theorem expand (i) Sin x (ii) Tan x (iii)log(1+sinx)

(iv) exsec x

11. State and prove Taylor's theorem.

12. Expand log(sin x) in powers of (x-2) by Taylors theorem.

13. State and prove (i) Rolle's (ii) Lagrange's (iii) Cauchy Mean Value

theorems.

14. Find `c' by Lagrange's theorem for f(x)=2-4, a=2 and b=3.

15. Verify Cauchy MVT for f(x)=ex and g(x)=e-x between x [,]

UNIT-II

1. Determine the limits (i) lim0-1-

2 (ii) lim0log (1-2)

logcos (iii)lim0-

log (-)

2. Determine the limit of log (-)

3. Determine (i) lim0[12- 12] (ii) ) lim0[12-2]

4. Determine (i) lim0[cos]1/2 (ii) lim0[cos]cot (iii) lim[2-/] /2

5. Define curvature, radius of curvature, total curvature, length of an arc as a

function.

6. Find the radius of curvature in different forms.

7. For a cycloid =(+),=(1-), prove that =4acos(12)

8. Show that the curvature at the point (3a/2,3a/2) on the curve

3+3=3 -823.

2

9. In the ellipse 2+22=1, show that the radius of curvature at an end of the

major axis is equal to the semi-latus rectum of the ellipse.

10. Show that the radius of curvature at any point of the astroid =3,

=3 is equal to twice the length of the perpendicular from the origin to

the tangent.

11. Define circle of curvature, chord of curvature, centre of curvature, evolute,

involute.

12. Prove that the evolute of the hyperbola 2=2 (+)2/3-(-)23 =22/3

13. Show that the circle of curvature at the point (2,2) of the parabola

y2=4ax as its equation as 2+2-62-4+43=324

UNIT-III

1. Define Partial Differential equation, homogeneous function

2

2. If = 2-1()-2-1(), 0, prove that =2-2

2+2

3

3. If = , show that =(1+3+222)

4. If =(3+3+3-3), show that

2

(a) [+ + ]2= -9

(++)2 (ii) 2+22+22= -3

(++)2

5. State and prove Euler's theorem for homogeneous functions.

6. Corollary of Euler's theorem.

7. If =-1[3+3

- ] , then show that

(i) += 2 (ii) 222+222+2 2=(1-42)2

8. Verify Euler's theorem for =2+2+2.

9. If H = (y ?z, z ?x, x ? y) prove that ++=0

10. If 1-2+ 1-2=, 22=

(1-2)3/2

11. Obtain Taylor's formula for the function ex+y at (0,0) for n=3.

12. Expand the function f(x,y)=x2+xy-y2 by Taylor's theorem in powers of (x-1)

and (y+2).

UNIT-IV

1. Define maximum and minimum value, stationary value, extreme value.

2. Find the extreme values of 5x6+18x5+15x4-10.

3. Show that the maximum and minimum values of (x+1)(x+4)(x-1)(x-4) are -9

and -1/9 respectively.

4. Show that the maximum value of x1/x is e1/e.

5. Using Lagrange's condition discuss the maximum and minimum values of u

where u = (x3y2)(1-x-y).

6. show that the minimum value of u = xy+ (a3/x)+(a3/y) is 3a2.

7. Lagrange's method of undetermined multipliers.

8. Find the max and min of x2+y2+z2 subject to ax2+by2+cz2=1.

9. Find the minimum value of x2+y2+z2, given that ax+by+cz=p.

10. In a plane triangle find the maximum value of u = Cos A CosB CosC

11. Find the asymptotes parallel to the coordinate axes for the curves

(x2+y2)x- ay2 and x2y2-a2.

12. Find the asymptotes of x3+2xy-xy2-2y3+xy-y2-1=0.

13. Find the asymptotes of x3-x2y-xy2+y3+2x2-4y2+2xy+x+y+1=0.

14. Find the envelope of a straight line x cos + ysin =l sin cos, for being

parameter.

15. Find the envelope of the family of curves -=2-2

This post was last modified on 23 January 2021