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Unit VI- DifferentiationA. Partial DerivativesB. Mean Value TheoremsC. Successive DifferentiationD. Approximation and ErrorsE. Maxima and MinimaA. Partial Derivatives Let z=f(x,y) be a function in two variables x and y. ?z?x = derivative of z wrt x keeping y constant. ?z?y = derivative of z wrt y keeping x constant.Remarks: 1. Higher order partial derivatives: 1. ?2z?x2 =??x(?z?x) 2. ?2z?y2 =??y(?z?y) 3. ?2z?xy =??x(?z?y) 4. ?2z?yx =??y(?z?x) 2. Euler’s Theorem Let z be a homogeneous function of two variables x and y of degree n then I. x ?z?x+y ?z?y =nz II. x2?2z?x2 +2xy ?2z?xy +y2?2z?y2 =n(n-1)zExercise 6.1Q.1 Find ?z?x and ?z?y of the following: a. z= x3+4xy2+y3 b. z= xy+yx c. z= xsiny +ysinx d. z= log(x2+y2) e. z= x2y3+x/y f. z= exycos(xy) g. z= tan-1(y/x) h. z= 10(3x+4y)Q.2 Find higher order partial derivatives of the following: a. z= x5+4xy+y5 b. z=cos(xy) c. z= x+y d. z= tan-1(xy)Q.3 Verify Euler’s Theorem for the following: a. z= x4+xy3+y4 b. z=3x2-5xy+4y2 c. z=xy Q.4 If z= xy+yx then verify that ?2z?x?y = ?2z?y?x.Q.5 If z=tan-1(y/x), prove that ?2z?x2 +?2z?y2=0.B. Mean Value Theorems:1. Rolle’s Theorem If f(x) is continuous on [a,b], differentiable on (a,b) and f(a)=f(b) then there exists c∈(a,b) such that f ’(c)=0.2. Lagrange’s Mean value theorem If f(x) is continuous on [a,b] and differentiable on (a,b) then there exists c∈(a,b) such that f ‘(c)=(f(b)-f(a))(b-a).3. Cauchy Mean Value theorem If f(x) and g(x) are continuous on [a,b] and differentiable on (a,b) then there exist c∈(a,b) such that fb-f(a)gb-g(a)=f'(c)g'(c) Exercise 6.2Q.1 Verify Rolle’s theorem for the following:(a) f(x)=x2-3x+2 for [1,2](b) f(x)=(x-3)(x-7) for [3,7](c) f(x)=e-xsinx for [0,π](d) f(x)=x2-5x+7 on [0,5]Q.2 Verify Lagrange’s theorem for the following:(a) f(x)= (x-1)(x-2)(x-4) on [0,4](b) f(x)=2x2-7x+10 on [2,5](c) f(x)= logx on [1,e](d) f(x)= lx2+mx+n on [a,b]Q.3 Verify CMVT.(a) f(x)=3x+2, g(x)=x2+1 on [1,4](b) f(x)=sinx, g(x)=cosx on [0,π/2](c) f(x)=ex, g(x)=e-x on [0,1](d) f(x)=x , g(x)= 1x on [4,9]C. Successive Differentiation Let y=f(x). yn= nth order derivative of y ie f(x)= dnydxn = f(n)(x).Sr. N.y=f(x)yn1.c=constant02.(ax+b)mnpm an(ax+b)m-n if n≤motherwise zero3.1ax+b(-1)nn!an(ax+b)n+14.eaxaneax5.sin(ax+b)ansin(ax+b+nπ2)6.cos(ax+b)ancos(ax+b+nπ2)7.log(ax+b)-1n-1n-1!an(ax+b)nExercise 6.3Find nth order derivative of the following:1. y= x(x-1)(x-3)2. y=2x-1x-2(x-3)3. y= 4x2+14. y=x2x+15. y=sin3x6. y=cos2x7. y= cos2x.cos5x8. y= sinx.sin2x.sin3x9. y=sin3x.sin5xD. Approximations and ErrorsLet z=f(x,y) be a differentiable function.Let δx, δy, δz be amount of error (or increment) in x, y, z respectively.Then, δz=?z?x δx+?z?y δyPercentage error: If δx is an error in x then δxx×100 is known as percentage error in x.Exercise 6.4Q.1 If f(x,y)=x2+y2+xy, find f(2.01,2.001) approximately.Q.2 If f(x,y)=exy, compute f(1.1,2.01) approximately, given e2=7.389.Q.3 Find percentage error in area of an ellipse of 1% error is made in measuring major and minor axes of the ellipse.Q.4 The period of simple pendulum is T=2πlg. Find the percentage error in T due to possible errors of 1% in l and 2.5% in g.Q.5 Find the possible percentage error in computing the parallel resistance R from two resistances R1 and R2 if in both the resistances error of 2% is made.Q.6 The sides of a triangle can be measured as 12cm and 15cm and included angle is 60o. If the possible errors in the sides are 1% and in the angle is 2% then find the percentage error in determining the area.Q.7 Find the approximate value of [(0.98)2+(2.01)2].E. Maxima/MinimaSteps to find maxima/minima of f(x,y)1. Solve ?f?x =0 and ?f?y =0 simultaneously. Let one of the solution be (a,b)2. Let ?2f?x2=r , ?2z?y2 =s, ?2z?x?y =t at (a,b).3. (a,b) gives maxima if rs-t2>0 and r<0.4. (a,b) gives minima if rs-t2>0 and r>0.Exercise 6.5Q.1 Find the extreme values of f(x,y)=x2y2(1-x-y).Q.2 Find the extreme values of xy(1-x-y).Q.3 Discuss the maxima and minima of x3+y3-9xy.Q.4 Divide 640 into three parts such that the sum of their products taken at a time is maximum.Q.5 Find maxima and minima of x3+3xy2 – 15x2-15y2+72x. ................
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