Download OU B.Sc Computer Science 1st Sem Differential Calculus Important Questions

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BSc Odd Semester Question Bank

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Paper-1: Differential Calculus

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UNIT-I

1. If y = (e^x + e^-x)/2, show that y_n = 2log23
2. If y = x²e^x, then show that 2y₁y₃ = 3y₂².
3. If ax² + 2hxy + by² = 1, show that d²y/dx² = (h²-ab)/(hx+by)³
4. Find the n^th derivative of (i) y = 1/(x+2)(2x+3) (ii) y = 1/(x-1)(x-2) (iii) y = 1/(a²-x²)

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5. If y = Sin ax + Cos ax, prove that y_n=aⁿv[1 — (—1)ⁿSin 2ax]
6. State and prove Leibnitz theorem.
7. If y^1/m + y^-1/m = 2x, prove that (x² — 1)y_n+2 + (2n + 1)xy_n+1 + (n² —m²)y_n =0
8. If y = Sin(msin^-1x), prove that (1 — x²)y_n+2 = (2n + 1)xy_n+1 + (n² —m²)y_n and also find y_n(0).
9. State and prove Maclaurin’s theorem.

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10. Using Maclaurin’s theorem expand (i) Sin x (ii) Tan x (iii)log(1+sinx) (iv) e^xsec x
11. State and prove Taylor’s theorem.
12. Expand log(sin x) in powers of (x-p/2) by Taylor's 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)=vx² — 4, a=2 and b=3.

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15. Verify Cauchy MVT for f(x)=e^x and g(x)=e^-x between x ? [a, b]

UNIT-II

1. Determine the limits (i) lim_x?0(1-cos x)/x² (ii) lim_x?0 (?₀^x sin(t²)dt)/x³ (iii)lim_x?0 x^x
2. Determine the limit of (e^x-e^{sin x})/(x-sin x)

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3. Determine (i) lim_x?0[1/x² — cot²x] (ii) lim_x?0[1/sin²x — 1/x²]
4. Determine (i) lim_x?0[cos x]^1/x2 (ii) lim_x?8[cosx]^x (iii) lim_x?8[2 — x/a]^Tan(px/2a)
5. Define curvature, radius of curvature, total curvature, length of an arc as a function.
6. Find the radius of curvature in different forms.

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7. For a cycloid x = a(t + sint),y = a(1 — cost), prove that ? = 4acos (t/2)
8. Show that the curvature at the point (3a/2,3a/2) on the curve x³ +y³ =3axy is -8v2/a
9. In the ellipse x²/a²+ y²/b² =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 x = acos³?, y = asin³? 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.

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12. Prove that the evolute of the hyperbola 2xy = a² is (x + y)^2/3 — (x — y)^2/3 = 2a^2/3
13. Show that the circle of curvature at the point (am², 2am) of the parabola y²=4ax as its equation as x² + y² - 6am²x — 4ax + 4am³y = 3a²m⁴

UNIT-III

1. Define Partial Differential equation, homogeneous function
2. If u= x²Tan^-1(x/y) - y²Tan^-1(y/x),where xy ? 0, prove that ?²u/?x?y = (x²-y²)/(x²+y²)

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3. If u = e^xyz, show that ?³u/?x?y?z = (1 + 3xyz + x²y²z²)e^xyz
4. If u = Log(x³ + y³ + z³ — 3xyz), show that
   (a) [?/?x + ?/?y + ?/?z] u= 3/(x+y+z) (b) [?²/?x² + ?²/?y² + ?²/?z²] u = -9/(x+y+z)²
5. State and prove Euler’s theorem for homogeneous functions.
6. Corollary of Euler’s theorem.

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7. If u=Tan^-1 [ (x³+y³)/(x-y) ] where x ? y, then show that
   (I) x?u/?x+y?u/?y=2sin2u (ii) x²?²u/?x² +2xy?²u/?x?y+y²?²u/?y²=(1+sin²u)sin2u
8. Verify Euler’s theorem for z = ax² + 2hxy + by².
9. If H=(y -z, z-x, x-y) prove that ?H/?x+?H/?y+?H/?z=0
10. If xv(1 —y²) + yv(1—x²)=a,show that dy/dx= -v(1-y²)/v(1-x²)

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11. Obtain Taylor’s formula for the function e^x+y at (0,0) for n=3.
12. Expand the function f(x,y)=x⁴+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 5x⁴+18x³+15x²-10.

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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 x^1/x is e^1/e.
5. Using Lagrange’s condition discuss the maximum and minimum values of u where u = (xy²)(1-x-y).
6. show that the minimum value of u ='xy+ (a³/x)+(a³/y) is 3a³.
7. Lagrange’s method of undetermined multipliers.

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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.

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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 cosa + ysina =| sina cosa, for a being parameter.
15. Find the envelope of the family of curves axseca — bycoseca = a² — b².

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