Differential equationsFind the slope of the tangent line to the parabola
3 months ago
Find the slope of the tangent line to the parabola y=4x-x^(2) at the point (1,3), i) using Definition 1, ii) using Equation 2
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jeffdunhamticketseugene Staff answered 3 months ago
Definition 2: m=lim_(x-> a)(f(x)-f(a))/(x-a) i) m=lim_(x-> 1)(f(x)-f(1))/(x-1) =lim_(x-> 1)((4x-x^(2))-(4(1)-1^(2)))/(x-1) =lim_(x-> 1)(-x^(2)+4x-3)/(x-1) =lim_(x-> 1)((x-1)(3-x))/(x-1) =lim_(x-> 1)(3-x)=3-1 =2 Equation 3: m=lim_(h-> 0)(f(a+h)-f(a))/(h) ii) m=lim_(h-> 0)(f(1+h)-f(1))/(h) =lim_(h-> 0)([4(1+h)-(1+h)^(2)]-[4(1)-1^(2)])/(h) =lim_(h-> 0)(2h-h^(2))/(h) =lim_(h-> 0)(2-h)=2-0 =2 Result: m=2
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