TUTORIAL-01
TUTORIAL-01
[SUCCESSIVE DIFFERENTIATION]
1) If [pic], then P.T. [pic]
2) Find nth derivative of y =[pic]
3) Find yn if y= [pic]
4) If [pic]then show that yn = (-1)n-2 (n-2)![ [pic]-[pic]]
5) Find nth order derivative of [pic]
6) y= tan -1 [[pic]] P.T yn= (-1)n-1(n-1)! Sin( nθ ) sin(n) θ, where θ = tan -1 ([pic])
7) U = eax cos(bx+c) then P.T. Un= (a2+b2)n/2 eax cos[bx + c + n tan-1(b/a)]
8) Find yn if y = e5x cosx cos3x
9) If y= xn logx then P.T yn+1 = [pic]
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 x2 yn+2 + (2n+1)x yn+1+ (n2+1)yn =0
12) If y =[pic] P.T (1-x2) yn -[2(n-1)x+1]yn-1- (n-1)(n-2)yn-2=0
13) If y= eacos[pic]x then 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 =[pic] then prove that yn =[pic][ log x- (1+[pic]+[pic]+---+[pic])]
18) U = (x2-1)n then prove that [pic][(x2-1)[pic]Un] = n(n+1) Un where Un=[pic]U
19) y = tan-1[[pic]] 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 [pic] then prove that [pic] and also find the value of [pic]
2) If u = log (x2+ y2 + z2) then prove that x [pic] = y [pic] = z [pic]
3) If xx yy zz = c, show that at x = y = z, [pic]= -[xlog(ex)][pic] and also P.T
[pic]- 2xy [pic]+ [pic] = [pic] at x = y= z
4) x = rcosθ , y = rsinθ then prove that [pic]+ [pic] = [pic][([pic])[pic]+([pic])[pic]]
5) If u= (8x2+ y2)[logx-logy] then find x[pic]+y [pic]
6) If u= sin[pic][[pic]] then show that [pic]= -[pic][pic]
7) Verify the Euler’s theorem for u = x2 tan -1 ([pic]) - y2 tan -1 ([pic]) ,xy [pic]0
and also show that [pic]= [pic]
8) If F(x,y) = [pic]+[pic]+[pic] then show that x[pic]+y[pic]+2F(x,y) =0
9) If u= tan -1 [[pic]] Then prove that x2[pic]+ 2xy[pic]+ y2[pic] = 2cos3u sinu
10) If u=[pic] then prove that
x2[pic]+ 2xy[pic]+ y2[pic]= [pic]tan u[[pic][pic]]
11) If x = e[pic]cos(rsinθ) and y = e[pic]sin(rsinθ) then P.T.
[pic]= [pic] and [pic]= -[pic] Hence deduce that [pic]+[pic]+[pic][pic]= 0
12) If z = f(u,v) where u = x2-2xy-y2 and v=y then show that
(x+y) [pic]+ (x-y) [pic]= 0
is equivalent to [pic]=0
13) If logeθ = r-x where r2= x2+ y2 then show that [pic]= [pic]
14) If F=F(x,y) and If x+y = 2eθcos(Ф) , x-y= 2ieθsin(Ф) then prove that
[pic]
15) State and Prove Euler’s Theorem on Homogeneous Functions of Three
Independent Variables.
16) If θ= t[pic]e[pic] then find the value of ‘n’ for which [pic]
17) If u = f(r) and x = rcoθ , y = rsinθ , then P.T.[pic], where
[pic][pic]
18) If [pic], and lx+my+nz=0 then prove that
[pic]
TUTORIAL NO -03
[Mean Value Theorem, Errors and Approximations]
1) Verify Rolle’s Theorem for function ex (sinx-cosx) in [ [pic],[pic]]
2) State & Verify Rolle’s Theorem for f(x) = [pic] , [0,Π ]
3) P.T. between any two real roots of the equations ex sinx = 1, there is at least one root of the equation ex cos x + 1= 0
4) Verify Rolle’s Theorem , F(x) = log [[pic]] in (2,3)
5) Using Lagrange’s Mean Value Theorem prove that if a < 1, b< 1 and a< b, then
[pic] < sin-1b – sin-1a < [pic]
Hence deduce that (i) [pic] < sin-1([pic]) < [pic]
(ii) [pic] < sin-1([pic]) < [pic]
6) If f(x+h)= f(x) + [pic] f ’(x)+ [pic]f ”(θh+a) Find θ as x [pic]a, f(x) deing (x-a)[pic]
7) Use appropriate Mean Value Theorem to Prove that
[pic]= cot c ; a ................
................
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