i have confused in one topic and please help me,suppose that we have following function
$f(x)=x^3+x^2-2*x-3)$
we know that there is a tangent of this function which goes through point $(1,-3)$,we are required to find abscissa,where this tangent intersect ox axis.
what i have tried first it to find derivative,so we have
($f(x)'=3*x^2+2*x-2)$,now what does mean abscissa of intersection of tangent line with ox axis?if it means that,it includes every point where y=0,then answer is this
http://www.wolframalpha.com/input/?i=3*x%5E2%2B2*x-2%3D0
but if it is y value at $x=0$,then answer is $-2$,so i am confused about this and please help me,also what i should do if instead of ox axis,,we are required to find ordinate of intersection of tangent line with oy axis?
Best Answer
The derivative gives the slope of the tangent line at that point. In your case, if you evaluate $f'(1)$ you get $3$. The tangent line is then the line through $(1,-3)$ with slope $3$. The point-slope form of the line is then $y-(-3)=3(x-1)$. On the $x$ axis, $y=0$, so you substitute that in to find $x$.