The point P(5,34)\displaystyle {P}{\left({5},{34}\right)} lies on the curve y=x2+x+4\displaystyle {y}={x}^{{2}}+{x}+{4}. If Q\displaystyle {Q} is the point (x,x2+x+4)\displaystyle {\left({x},{x}^{{2}}+{x}+{4}\right)}, find the slope of the secant line PQ\displaystyle {P}{Q} for the following values of x\displaystyle {x}.

If x=5.1\displaystyle {x}={5.1}, the slope of PQ\displaystyle {P}{Q} is:  

and if x=5.01\displaystyle {x}={5.01}, the slope of PQ\displaystyle {P}{Q} is:  

and if x=4.9\displaystyle {x}={4.9}, the slope of PQ\displaystyle {P}{Q} is:  

and if x=4.99\displaystyle {x}={4.99}, the slope of PQ\displaystyle {P}{Q} is:  

Based on the above results, guess the slope of the tangent line to the curve at P(5,34)\displaystyle {P}{\left({5},{34}\right)}.
 

License