Question #179791

Let P and Q be points on the curve 𝑦=𝑥2−2𝑥 while x=2 and x=2+h respectively. Express the gradient of pq in terms of h and hence find the gradient of the curve 𝑦=𝑥2−2𝑥 at x= 2


1
Expert's answer
2021-04-15T06:31:25-0400
(x+h)22(x+h)(x22x)x+hx\dfrac{(x+h)^2-2(x+h)-(x^2-2x)}{x+h-x}

=x2+2xh+h22x2hx2+2xh=\dfrac{x^2+2xh+h^2-2x-2h-x^2+2x}{h}

=2x2+h=2x-2+h


y(x)=limh0(x+h)22(x+h)(x22x)x+hxy'(x)=\lim\limits_{h\to 0}\dfrac{(x+h)^2-2(x+h)-(x^2-2x)}{x+h-x}

=limh0(2x2+h)=2x2+0=2x2=\lim\limits_{h\to 0}(2x-2+h)=2x-2+0=2x-2

grad y=y=2x2\text{grad}\ y=y'=2x-2

x=2x=2


grad yx=2=y(2)=2(2)2=2\text{grad}\ y|_{x=2}=y'(2)=2(2)-2=2

grad yx=2=2\text{grad}\ y|_{x=2}=2





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