Answer
$a.$ The slope is $m_{tan}=2.$
$b.$ The equation is $y=2x+1.$
Work Step by Step
$a.$ Using the formula from definition (2) with $a=0$ and $f(a)=1$ (coordinates of the point $P(0,1)$) we have
$$m_{tan}=\lim_{h\to0}\frac{f(0+h)-f(0)}{h}=\lim_{h\to0}\frac{2(0+h)+1-(2\cdot 0+1)}{h}=\lim_{h\to0}\frac{2h}{h}=\lim_{h\to0}2=2.$$
$b.$ Using the formula $y-f(a)=m_{tan}(x-a)$ with the same values for $a$ and $f(a)$ as in part $a$ and the calculated value $m_{tan} = 2$ we get
$$y-1=2(x-0)\Rightarrow y-1=2x$$ which gives
$$y=2x+1.$$
