20 Calculus Questions

Q1 : If f(x) = 2xx + 5/x2x. Find f (4)Sol : We can also write it asf(x) = 2x3/2 + 5/x5/2f(x) = 2x3/2 + 5x-5/2Taking derivative w.r.t x, we getf (x) = d/dx(2x3/2 + 5x-5/2)f (x) = 3x1/2 25/2 * x-7/2f (x) = 3x1/2 25/2x7/2Nowf (x) = 3(4)1/2 25/2(4)7/2f (x) = 3(4)1/2 25/2((4)1/2)7f (x) = 3(22)1/2 25/2((22)1/2)7f (x) = 3(2) 25/2(2)7f (x) = 6 25/256f (x) = (1536 25) / 256f (x) = 1511 / 256HenceQ2 : f(x) = 5x6lnx. Find f (e3)Sol : Taking derivative w.r.t x, we getf (x) = d/dx(5x6lnx)f (x) = 5x6 * 1/x + lnx * 30x5f (x) = 5x5 + 30x5lnxNowf (e3) = 5(e3)5 + 30(e3)5ln(e3)f (e3) = 5(e15) + 30(e15)3ln(e)f (e3) = 5e15 + 90e15(As ln(e) = 1)f (e3) = 95e15HenceQ3 : Find the derivative of f(x) = e-4xsinxSol : Taking derivative w.r.t x, we getf (x) = d/dx(e-4xsinx)f (x) = e-4xcosx - 4e-4xsinxHenceQ4 : f(x) = x3tan-1(2x) , find f (x)Sol : f (x) = d/dx(x3tan-1(2x))f (x) = 3x2tan-1(2x) + x3 * 1 / ((2x)2+ 1) * 2f (x) = 3x2tan-1(2x) + 2x3 / (4x2+ 1)HenceQ5 : Evaluate 8 (7)(7)Sol : - NowWhere A is unit vector and magnitude of unit vector is 1, so 8 (7)(7)Now put u = 7x, then du = 7dx or dx = 1/7 du, we get1 8 ()() 71 8 ()()7Now put v = sin ...

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