Question 58211
First, find f(x+h):
{{{f(x+h)=3(x+h)^2-5(x+h)+2}}}
{{{f(x+h)=3(x^2+2xh+h^2)-5x-5h+2}}}
{{{f(x+h)=3x^2+6hx+3h^2-5x-5h+2}}}
{{{f(x+h)=highlight(3x^2+(6h-5)x+(3h^2-5h+2))}}}
Now write the problem:
{{{(f(x+h)-f(x))/h}}}={{{(highlight(3x^2+(6h-5)x+(3h^2-5h+2))-(3x^2-5x+2))/h}}}
={{{(cross(3x^2)+(6h-5)x+(3h^2-5h+2)-cross(3x^2)+5x-2)/h}}}
={{{((6h-5+5)x+(3h^2-5h+2-2))/h}}}
={{{(6hx+3h^2-5h)/h}}}
{{{highlight((f(x+h)-f(x))/h=6x-5+3h)}}}
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Note that when you take the limit as h nears 0, you have the formula for the slope of the curve at each point (x) along the curve.