(e)h(x)5(11x)1/2h9(x)5 }12}(11x)21/2h0(x)52}14}(11x)23/2Since h(0)51,h9(1)5 }12}, and h0(1)14}, thecoefficients are b051,b15 }12}, and b2552}18}. The quadratic approximation is Q(x)511 }2x}2 }x82}.[21.35, 3.35] by [21.25, 3.25]As one zooms in, the two graphs quickly becomeindistinguishable. They appear to be identical.(f)The linearization of any differentiable function u(x) atx5ais L(x)5u(a)1u9(a)(x2a)5b01b1(x2a),where b0and b1are the coefficients of the constant andlinear terms of the quadratic approximation. Thus, thelinearization for f(x) at x50 is 11x; the linearizationfor g(x) at x51 is 12(x21) or 22x; and thelinearization for h(x) at x50 is 11 }2x}.53.Just multiply the corresponding derivative formulas by dx.(a)Since }ddx}(c)50,d(c)50.(b)Since }ddx}(cu)5c}ddux},d(cu)5c du.(c)Since }ddx}(u + v)5 }ddux} 1 }ddvx},d(u1v)5du1dv(d)Since }ddx}(u ?v)5u}ddvx} 1v}ddux},d(u ?
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