Differential Calculus (B.Sc notes)
275 videos • 835 views • by Doctor of Mathematics
Differential Calculus (B.Sc notes)
1
Trace the curve y^2(1-x^2)=x^2(1+x^2)
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2
If u=(x^2+y^2+z^2)/x;v=(x^2+y^2+z^2)/y;w=(x^2+y^2+z^2)/z;find the jacobian of x,y,z with respect to
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3
Expand sinx in power of (x-π/2) [Taylor's series]
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4
Expend 2x^3+7x^2+x-1 in power of (x-2) by Taylor's theorem
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5
Use Taylor's theorem to prove that tan-1(x+h)=tan-1(x)+(hsinθ).sinθ/1-(hsinθ)^2.sin2θ/2+(hsinθ)^3.Si
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6
Find the volume of solid generated by revolution of one arc of the cycloid x=a(θ-sinθ);y=a(1-cosθ)
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7
Find the surface generated by revolution of arc of a catenary y=c.cosh(x/c) about the x-axis.
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8
A particle moves along the curve x=a.cost; y=a.sint; z=b.t. Find the velocity and acceleration at t=
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9
Let f(x,y)=e^x.siny,x=t^3+1 and y=t^4+t.Then df/dt at t=0 is....(rounded off to two decimal places)
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10
IIT JAM :Which of the following is FALSE (MATHEMATICS)
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11
Show that the following function is continuous but not differentiable at x=0. f(x)=x.sin(1/x);if x=/
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12
B.Sc. 1st year very important question of differential calculus
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13
If y=a.cos(logx)+b.sin(logx), show that x²y2+xy1+y=0 and x2yn+2+(2n+1)xyn+1+(n²+1)yn=0
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14
Expand tanx by Taylor's theorem upto three terms in power of (x-pi/4)
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15
Find the expansion of sin2x.siny about (0,0) upto and including the terms of fourth degree in (x,y).
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16
Differentiate y=x^coshx+x^sinhx w.r.t x
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17
Prove that function f(x) is continuos over R.Where f(x)=|x|+x/3 x greater than or equal to 3 f(x)
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18
Evaluate lim x tends to 0[1/x^2-1/sin^2(x)]
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19
Evaluate :Limit x tends to 1/2 ,1/x[1/x] where [x] is greatest integer function.
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20
Find a,b and c so that:limt n tends to 0 [axe^x-bsinx+cxe^-x]/x^3=2
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21
Evaluate Lt (2-x/a)^tan(π/2a) as x tends to a
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22
(sinx)^tanx as x tends to infinity
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23
(inverted video) Lt (tanx/x)^1/x as x tends to infinity
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24
Lt [(1+x)^1/x-e+1/2ex]/x^2=11e/24 Prove that
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25
Lt log[tan^2(2x)]tan^2x as x tends to infinity
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26
Find the values of a and b so that Lt [x(1+acosx)-bsinx]/x^3=1
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27
If Lt [(1+axsinx)-bcosx]/x^4 be finite ,find the values of a and b and the limit.
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28
Evaluate (1+a/x)^x as x tends to infinity
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29
Lt x tends to 0 [xe^x-log(1+x)]/x^2 Evaluate
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30
Lt [x.e^x-log(1+x)]/x^2 as x tends to 0.
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31
Discuss the differentiability of the function f(x)=x+2x^2.sin(1/x);x≠0 otherwise 0;at point x=0 and
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32
Expand tan-1x (tan inverse x ) in the power of (x-π/4) by Taylor's theorem.
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33
Find the asymptotes of curve x^2.y^2-x^2.y-x.y^2+x+y+1=0
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34
Prove that the asymptotes of the curve (x^2-y^2).y-2ay^2+5x-7=0 form a triangle of area a^2.
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35
condition that xcosα+ysinα=p should touch x^m.y^n=a^(m+n) is p^(m+n).m^m.n^n=(m+n)^(m+n).a^m+n.sin^α
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36
Find the condition that the line x.cosα+y.sinα=p may touch the curve x^m/a^m+y^m/b^m=1
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37
If the normal at any point to the curve x^2/3+y^2/3=a^2/3 makes angle phi with x-axis ,show that its
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38
If the tangent at (x1,y1) to the curve x^3+y^3=a^3 meets the curve again in (x2,y2),show that x2/x1+
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39
State and prove Euler's theorem.If u=sin-1(x+y)/(√x+√y) then show that xdelu/delx+ydelu/dely=1/2sinu
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40
If y=e^m.cos-1x,then show that (1-x^2)yn+2-(2n+1)x.yn+1-(n^2+m^2)yn=0
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41
A tent on a square base of side x,has its sides vertical of height y and the top is a regular pyrami
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42
Lt x tends to 0 logx^2/logcot^2x
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43
Find the asymptotes of curve x^3-2x^2y+xy^2+x^2-xy+2=0
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44
Find the asymptotes of the curve (x^2-y^2)(x+2y+1) +x+y+1=0.
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45
❌Lt x tends to 0 sinx.logx[I committed a mistake so differentiation of cosec x =-cosecx.cotx]
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46
Lt x tends to 0. [1/2x^2-1/2xtanx]
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47
Expand tan-1x in terms of (x-1) upto non-zero three terms||Top Mathematics channel on YouTube||Best
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48
Find the radius of curvature of curve x^3+y^3=3axy
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49
Trace thw curve y^2=x.(x-1)^2 and find the area of the loop.
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50
If y=sin(sinx); prove that d^2y/dx^2+tanx.dy/dx+y.cos^2x=0
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51
Analyse the equation 2x^2-3x+5y+4=0 and sketch the curve
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52
Find (x^2.e^x.cosx)n.
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53
Find the equation of the tangent at the point (x1,y1) of ellipse x^2/a^2+y^2/b^2=1.
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54
limit x tends to a log(x-a)/log(e^x-e^a
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55
The sum of three positive number is unity.What is the maximum value of their product.
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56
Show that f(x)=cos(1/x) when x≠0,f(0)=0 is discontinuous at origin.
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57
Show that the following function is discontinuous at the origin: f(x)=1/1+e^1/x
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58
Examine the following function for continuity at x=2, f(x)=(x^2-4)/(x-2),f(2)=0
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59
With usual notations,for any curve prove d^2r/ds^2=sin^2(ϕ)/r-sin(ϕ)/ρ ;d^2r/ds^2=sin^2ϕ/r-sinϕ/ρ
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60
e^ax.sinbx=bx+abx^2+(3a^2b-b^3)x^3/3!+....
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61
IF u=(x^2+y^2+z^2)^-1 then prove that del^2u/delx^2+del^2u/del^2y+del^2u/delz^2=
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62
Using (,) epsilen-delta definition show that lim x tends to x.sin(1/x)=0
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63
Find all the asymptotes of the curve given by:x^2y-xy^2+xy+y^2+x-y=0
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64
If y=x^2.sinx,prove that D^n(y)=(x^2-n^2+n)sin(x+n.π/2)-2nxcos(x+n.π/2).
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65
Trace the curve (Cissoid) r=a.sin^2θ/cosθ
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66
B.Sc. Mathematics : Show that the whole length of the cardiod r=a (1+cos@) is 8a.
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67
If y=sin-1x/√1-x^2, prove that (1-x^2)yn+1-(2n+1)xyn-n^2yn=0
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68
Apply Maclaurin theorem to obtain the term upto x^4 in the expansion of log(1+sin^2)
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69
Find the radius of curvature at any point 't' on the curve x=a.cost, y=b.sint.
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70
If x=cosh(1/m.logy) , prove that (x^2-1) yn+2+(2n+1) xyn+1+(n^2-m^2) yn=0
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71
Explanation of Shab-e-miraj and Isra on basis on special theory of relativity
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72
Show that the functions u=x/y-z, v=y/z-x, w=z/x-y are not independent of one another. find relation
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73
Find the length of the arc of the curve x=t^2.cost, y=t^2.sint from origin to t I. e. 0 to t.
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74
Show that the functions u=x^3+x^2y+x^2z-z^2(x+y+z),v=z+x,w=x^2-z^2+xy-zy are not independent.find re
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75
Find maximum and minimum of x^3y^2(1-x-y)
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76
Find pedal equation of the curve x^2/3+y^2/3=a^2/3
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77
dr/dθ+r/=aθ^n
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78
Find the intervals in which curve y=(x^2+4x+5) e^-x is concave upward or downward.find inflexion poi
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79
Using Taylor's Theorem show that tan(π/4+x) =1+2x+2x²+8/3x^3+....
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80
Using Taylor's theorem show that tan(π/4+x) =1+2x+2x^2+8/3.x^3+....
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81
Find the radius of curvature of curve the cardiod r=a(1-cosθ)
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82
Expand log[cos(x+π/4)]
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83
Trace the curve a^2.y^=x^3.(2a-x)
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84
Lt x tends to infinity x^n/e^x ; n being a positive integer.
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85
If u=(x^1/4+y^1/4)/(x^1/5+y^1/5) ;then the value of x∂u/∂x+y∂u/∂y
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86
If u=sin-1(√x-√y/√x+√y) , prove that ∂u/∂x=-y/x.u∂/∂y
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87
If u=cos-1(x+y/√x+√y) , prove that xdelu/delx+ydelu/dely+1/2.cotu=0
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88
If u=tan-1(x²+y²/x+y), prove that x∂u/∂x+y∂u/∂y=1/2.sin2u
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89
Verify Euler's theorem for the homogenous function:u=x^n.log(y/x)
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90
If u=ax²+2hxy+by², prove that x∂u/∂x+y∂u/∂y=2u
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91
If u=sin-1(x+y/√x+√y) , prove that xdelu/delx+ydelu/dely=1/2.tanu
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92
Verify Euler's theorem for the function u=1/√(x^2+y^2)
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93
If u=log[(x^3+y^3) /(x+y)] , prove that x.delu/x+y.delu/dely=2.
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94
If u=x^2y^2/(x^2+y^2), prove that
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95
u=x²y²/(x²+y²), prove that x²∂²u/∂x²+2xy∂²u/∂x∂y+y²∂²u/∂y²=2u.
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96
If V=logsin {π(2x^2+y^+xz)^1/2/2(x^2+xy+2yz+z^2)^1/3}, find x∂V/∂x+y∂V/∂y+z∂V/∂z
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97
If u=tan-1y/x, prove that x∂u/∂x+y∂u/∂y=0
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98
Show that in the exponential curve y=be^x/a, the subtangent is of constant length and subnormal vari
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99
Show that the pedal equation of the lemniscate r^2=a^2.cos2θ is r^3=a^2p.
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100
Determine the position and nature of the double points of the curve x^3-y^2-7x^2+4y+15x-13=0
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101
Subtangent and subnormal
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102
Find the subtangent, subnormal, tangent and it's intercept on the cycloid x=a(t+sint),y=a(1-cost).
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103
If (1+x)y=ln(1+x) then show that(1+x)^2yn+2+(2n+3) (1+x) yn+1+(n+1)^2yn=0
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104
Find the radius of curvature of x=acos^3t, y=asin^3t
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105
Find radius of curvature for the Cartesian curve y=f(x)
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106
Proof of Euler's theorem and prove x^2del^2u/delx^2+2xydel^2u/delxdelu+y^2del^2u/dely^2=n(n-1)u
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107
Find the radius of curvature for the curve√x+√y=√a at the point where y=x cuts it.
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108
Prove that every differentiable function is continuous.
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109
Check the following function for x=05, f(x)=e^1/x-1/e^1/x+1
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110
Test the continuity and differentiabelity of the function f(x) at x=2 where f(x)=1+x for x less than
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111
Evaluate limit (3x+|x|)/(7x-5|x|)
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112
Show that limit sinax/sinbx=a/b
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113
Find first order partial derivative of u w. r. t. x if u=y^x.
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114
Apply the method of Laplace transform to solve differential equation: d^2x/dt^2-2dx/dt+x=e^t with x=
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115
Find the maxima and minima of the function x^3+y^3-12x-3y+20.
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116
Second derivative of {3/(s^2+9)} w. r. t. s
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117
Using Laplace transform to solve D. E. d^2y/dt^2-3dy/dt+2y=4t+e^3t
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118
Pedal equation of r=a(1-cosθ)
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119
Lagrange's conditions for maxima and minima (Unique Solution)
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120
State Lagrange's mean value theorem
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121
If y=ae^mx+be^-mx then show that y2=m^2y
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122
Find radius of curvature at the point on the paranola y^2=4ax.
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123
Find pedal equation of the curve r=a/θ.
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124
limited x tends to 0 [ xe^x-log(1+x)]/x^2
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125
limit (a^xsinbx-b^xsinax) /(tannx-tanax)
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126
Limit (x,y) tends to (0,0) xy/x+y
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127
If y=sin-1x/√1-x^2 , show that (1-x^2)yn+2-(2n+3)xyn+1-(n+1)^2yn=0
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128
If y=e^tan-1x , prove that (1+x^2) yn+2+{2(n+1)x-1}yn+1+n(n+1)yn=0
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129
Derivative of sinh-1x
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130
If y=cos(logx) prove that x^2yn+2+(2n+1)xyn+1+(n^2+1)yn=0
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131
∫x^7/√(a^2-x^2) =16/35a^7 limit x=0 to x=a
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132
Given that 2x+y-3z-2u=0, x+2y+z+u=0.Find the following partial derivatives: (delx/dely)z,
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133
If u=f(z), z being a homogenous function of degree n in x, y, prove that
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134
Find the radius of curvature at any point of the curve r=a(1+cosθ)
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135
Find the pedal equation of the curve (x^2+y^2)c^2=x^2y^2.||Unique Solution||Top B. Sc. Mathematics
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136
If u=log(x^3+y^3+z^3-3xyz ), show that del^2u/del2+del^2u/dely^2+del^2u/delz^2=-3/(x+y+z)^2
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137
Prove that rdθ/ds=sinΦ
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138
Find the radius of the curvature of the cycloid s=4asinΨ.
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139
Find the envelope of the straight line y=m x+cm^3, m being the parameter.
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140
Find the asymptote of the curve y=2/x-3.
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141
Find the length of perpendicular from the pole on the tangent to the curve r^n=a^ncosnθ.
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142
Evaluate lim[π/4x-π/2x(e^πx+1)], x tends to 0.
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143
Evaluate limit secπx/tan3πx
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144
If u=sin-1x/y+tan-1y/x, show that xdel u/del x+ydel u/del y=0
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145
as If u=(1-2xy+y^2)^-1/2, prove that xdelu/d elx-y delu/dely=y^2u^3
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146
If z=x^2tan-1(y/x) -y^2tan-1(x/y), prove that del^2z/delydelx=x^2-y^2/x^2+y^2
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147
Find all the asymptotes of the curve (x^2-y^2)^2+(x^3+y^3)+xy+7=0.
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148
Ify=(1/x)^x , show that y2(1)=0
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149
If V=(x²+y²+z²)^m/2, then find the value of m(m≠0) which will make (∂²V/∂x²+del^2V/dely^2+
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150
If u=(1-2xy+y^2)^-1/2 , prove that del/delx[(1-x^2)delu/delx]+del/dely[y^2delu/dely]=0
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151
Two equal forces are inclined at an angle 2θ.Their resultant is 3 times as great as when they are
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152
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153
Prove that if f(x, y)=1/√y.e^-(x-a)^2/4y, fxy(x, y)=fyx(x,y)
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154
Ifz=(x+y)+(x+y)Φ(y/x), then prove that x(d^2z/dx^2-d^2z/dxdy)=y(d^2y/dy^2-d^2z/dxdy) all partial
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155
If z(z^2+3x)+3y=0, prove that d^2z/dx^2+d^2z/dy^2=2z(x-1)/(z^2+x)^3 all partial doffferentiation
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156
If x+y=1,prove that d^n(x^ny^n)/dx^n=n![y^n-(nC1)^2y^n-1x+(nC2)^2y^n-2x^2+.....+]
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157
If z^3-3yz-3x=0,. show that zdz/dx=dx/dy,z[d^2z/dxdy+(dx/dx)^2]=d^2z/dy^2 all partial differentiatio
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158
If u(x,y,z)=1/(x^2+y^2+z^2), find the value of d^2u/dx^2+d^2u/dy^2+d^2u/dz^2 all partial differentia
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159
∫x^4e-xdx from 0 to infinity
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160
Evaluate ∫e-4x.x^3/2dx from 0 to infinity
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161
Find the asymptotes of the curve:y^3+x^2y-y+1=0
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162
Expand log(1+x) by Maclaurin's theorem.
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163
Find the coordinates of the centre of the conic 41x^2+24xy+9y^2-130ax-60ay+116a^2=0
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164
e^ax.sinbx.coscx का n वा्ं अवकल गुणांक ज्ञात करो।
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165
x/(1+3x+2x^2) का वां अवकल गुणांक ज्ञात करोFind n th derivative of x/(1+3x+2x^2)
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166
If u=x^2tan-1y/x-y^2tan-1x/y show that del^2u/delx.dely=(x^2-y^2)/((x^2+y^2)
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167
समघातीय फलन की परिभाषा उदाहरण सहित लिखो। Homogeneous Function in two varieties
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168
यदि x=rcosθ,y=rsinθ हो तो दिखाइए del(r,θ)/del(x,y)
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169
निम्न के लिए Euler's Theorem को सत्यापित करो-. u=(x^1/4+y^1/4)/(x^1/5+y^1/5)
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170
Solve:(D^2-1)y=cos^2x
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171
सिद्ध करो कि del(u,v)/del(x,y).del(x,y)/del(u,v)=1
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172
Expand e^asin-1x by Maclaurin 's theorem general term.Hence show e^θ=1+sinθ+1/2!sin^2θ+2/3!sin^3θ+.
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173
If y=tan-1x,find the value of (yn)0 and show that when n=2p,4p+1,4p+3 then its value is 0,n-1!,-n-1!
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174
Determine the position and character of double points on the curve a^4y^2=x^4(2x^2-3a^2)
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175
Find the radius of curvature at any point of the cycloid x=a(θ+sinθ),y=a(1-cosθ).
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176
Find the angle between the curves r^2=a^2cos2θ and r^2=b^2sin2θ.
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177
Find the angle between the curves r=asin2θ and r=acos2θ .
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178
Show that the radius of curvature at any point on curve √x/a+ √y/b=1 is 2(ax+by)^3/2/ab
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179
Obtain pedal equation for the curve r^2=a^2cos2θ.
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180
Show that for the curve by^2=(x+a)^3,the square of the subtangent varies as subnormal.
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181
Show that curves ax^2+by^2=1and a'x^2+b'y^2=1cut orthogonally if 1/a-1/b=1/a'-1/b'.
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182
Find the intrinsic equation of the spiral r=aθ,the arc being measured from the pole.
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183
Trace the curve y^3=a^2x-x^3. (curve tracing)
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184
Formula of radius of curvature of the curve in intrinsic form
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185
Formula of radius of curvature for Cartesian equations
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186
Formula of radius of curvature for Parametric Equations
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187
Let a function f be defined as f(x)=sin2x/x, when x≠0;=1when x=0.Examine whether f is continuous at
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188
find the asymptote of x^3+y^3-a^2x=0
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189
Find the asymptotes of the following curve x^3+2x^2y-xy^2-2y^3+xy-y^2=1.
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190
Find the value radius of curvature of the curve y=ax²+bx+c at x=1/2a(√(a²-1)-b).
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191
Obtain length of tangent and normal of same curve and show subnotnal/subtangent=(L.of.norm/L.tang)^2
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192
Find the maximum value of where u=sinxsinysin(x+y)
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193
If u=log(x^4+y^4)/x+y to prove xdu/dx+ydu/dy=3
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194
Find the envelope of the family of straight lines x/a+y/b=1,where parameters connected a^n+b^n=k^n
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195
Find the envelope of the straight lines xcosα+ysinα=lsinαcosα.
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196
Prove that for the ellipse x^2/a^2+y^2/b^2=1,ρ=a^2b^2/p^3,p being the perpendicular from the centre
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197
Find the radius of curvature for the curve r=2acosθ.
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198
Find ∂z/∂x,∂z/∂y of the following two variables function z=3x^2/√y-2y/x^2
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199
Find ∂z/∂x,∂z/∂y of the two variables function z=3e^-x2cos(3xy^2).
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200
Determine the second order partial derivatives ∂ 2 z/ ∂x 2,∂ 2 z/ ∂y2,∂ 2 z / ∂x ∂y of z=ln(x^2+3y).
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201
A company manufactures two types of athletic shoes: jogging shoes and cross-trainets.The total reven
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